Open main menu
Key Stage Wiki β

Difference between revisions of "Testpage"
Line 1: Line 1:
−
{{Use dmy dates|date=February 2015}}
+
{{Infobox physical quantity
−
{{chembox
+
|bgcolour={default}
−
| Verifiedfields = changed
+
|name = Kinetic energy
−
| Watchedfields = changed
+
|image=[[File:Wooden roller coaster txgi.jpg|220px]]
−
| verifiedrevid = 477003420
+
|caption=The cars of a [[roller coaster]] reach their maximum kinetic energy when at the bottom of the path. When they start rising, the kinetic energy begins to be converted to gravitational [[potential energy]]. The sum of kinetic and potential energy in the system remains constant, ignoring losses to [[friction]].
−
| Name = Calcium carbonate
+
|unit = [[joule]] (J)
−
| ImageFileL1 = calcium carbonate.png
+
|symbols = KE, ''E''<sub>k</sub>, or T
−
| ImageFileR1 = Calcium-carbonate-xtal-3D-SF.png
+
|derivations = ''E''<sub>k</sub> = ½''[[mass|m]][[velocity|v]]''<sup>2</sup> <br>
−
| ImageFile2 = Calcium carbonate.jpg
+
''E''<sub>k</sub> = ''E''<sub>t</sub>+''E''<sub>r</sub>
−
| IUPACName = Calcium carbonate
 
−
| OtherNames = [[calcite]]; [[aragonite]]; [[chalk]]; [[Lime (material)]]; [[Limestone]]; [[marble]]; [[oyster]]; [[pearl]];
 
−
|Section1={{Chembox Identifiers
 
−
| UNII_Ref = {{fdacite|correct|FDA}}
 
−
| UNII = H0G9379FGK
 
−
| ChEMBL_Ref = {{ebicite|changed|EBI}}
 
−
| ChEMBL = 1200539
 
−
| KEGG_Ref = {{keggcite|correct|kegg}}
 
−
| KEGG = D00932
 
−
| InChI = 1/CH2O3.Ca/c2-1(3)4;/h(H2,2,3,4);/q;+2/p-2
 
−
| ChEBI_Ref = {{ebicite|correct|EBI}}
 
−
| ChEBI = 3311
 
−
| SMILES = [Ca+2].[O-]C([O-])=O
 
−
| InChIKey = VTYYLEPIZMXCLO-NUQVWONBAS
 
−
| SMILES1 = C(=O)([O-])[O-].[Ca+2]
 
−
| StdInChI_Ref = {{stdinchicite|correct|chemspider}}
 
−
| StdInChI = 1S/CH2O3.Ca/c2-1(3)4;/h(H2,2,3,4);/q;+2/p-2
 
−
| StdInChIKey_Ref = {{stdinchicite|correct|chemspider}}
 
−
| StdInChIKey = VTYYLEPIZMXCLO-UHFFFAOYSA-L
 
−
| CASNo = 471-34-1
 
−
| CASNo_Ref = {{cascite|correct|CAS}}
 
−
| ChemSpiderID_Ref = {{chemspidercite|correct|chemspider}}
 
−
| ChemSpiderID = 9708
 
−
| EINECS = 207-439-9
 
−
| PubChem = 10112
 
−
| RTECS = FF9335000
 
 
}}
 
}}
−
|Section2={{Chembox Properties
+
{{Classical mechanics}}
−
| Formula = CaCO<sub>3</sub>
+
In [[physics]], the '''kinetic energy''' of an object is the [[energy]] that it possesses due to its [[motion (physics)|motion]].<ref>{{cite book
−
| MolarMass = 100.0869 g/mol
+
|title=Textbook of Engineering Physics (Part I)
−
| Appearance = Fine white powder; chalky taste
+
|first1=Mahesh C.
−
| Odor = odorless
+
|last1=Jain
−
| Density = 2.711 g/cm<sup>3</sup> ([[calcite]])<br />2.83 g/cm<sup>3</sup> ([[aragonite]])
+
|year=2009
−
| Solubility = 0.013 g/L (25 °C)<ref>{{cite book|title=SI Chemical Data Book (4th ed.) |publisher=John Wiley & Sons Australia, Ltd. |author1=Aylward, Gordon |author2=Findlay, Tristan |isbn=978-0-470-81638-7|year=2008}}</ref><ref>{{cite book|title=Calcium Carbonate: From the Cretaceous Period Into the 21st Century|publisher=Springer Science & Business Media|year=2001|url=https://books.google.com/books?id=pbkKGa19k5QC&pg=RA1-PR2|author=Rohleder, J. |author2=Kroker, E. |isbn=3-7643-6425-4}}</ref>
+
|isbn=978-81-203-3862-3
−
| SolubilityProduct = 3.3{{e|-9}}<ref>{{cite book|last =Benjamin|first=Mark M.|year=2002|title=Water Chemistry |publisher=McGraw-Hill|isbn =0-07-238390-9|url=https://books.google.com/?id=67anQgAACAAJ}}</ref>
+
|page=9
−
| Solvent = dilute acids
+
|url=https://books.google.com/books?id=wKeDYbTuiPAC}}, [https://books.google.com/books?id=wKeDYbTuiPAC&pg=PA9#v=snippet&q=kinetic&f=false Chapter 1, p. 9]
−
| SolubleOther = soluble
+
</ref>
−
| MeltingPt = 1,339 °C (2,442 °F; 1,612 K) (calcite) <br> 825 °C (1517 °F; 1,098 K) (aragonite) <ref>{{cite web|url=https://www.cdc.gov/niosh/docs/81-123/pdfs/0090.pdf|title=Occupational safety and health guideline for calcium carbonate|publisher=US Dept. of Health and Human Services|accessdate=31 March 2011}}</ref>
+
It is defined as the [[work (physics)|work]] needed to accelerate a body of a given mass from rest to its stated [[velocity]]. Having gained this energy during its [[acceleration]], the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.
−
| BoilingPt = decomposes
+
 
−
| RefractIndex = 1.59
+
In [[classical mechanics]], the kinetic energy of a non-rotating object of [[mass]] ''m'' traveling at a [[speed]] ''v'' is {{smallmath|f=\frac{1}{2}mv^2}}. In [[Special relativity|relativistic mechanics]], this is a good approximation only when ''v'' is much less than the [[speed of light]].
−
| pKa = 9.0
+
 
−
| pKb =
+
The standard unit of kinetic energy is the [[joule]].
−
| MagSus = -38.2·10<sup>−6</sup> cm<sup>3</sup>/mol
+
 
−
}}
+
==History and etymology==
−
|Section3={{Chembox Structure
+
The adjective ''kinetic'' has its roots in the [[Ancient Greek|Greek]] word κίνησις ''[[-kinesis|kinesis]]'', meaning "motion". The dichotomy between kinetic energy and [[potential energy]] can be traced back to [[Aristotle]]'s concepts of [[actuality and potentiality]].<ref>{{cite book |title=Logic in Reality |edition=illustrated |first1=Joseph |last1=Brenner |publisher=Springer Science & Business Media |year=2008 |isbn=978-1-4020-8375-4 |page=93 |url=https://books.google.com/books?id=Jnj5E6C9UwsC}} [https://books.google.com/books?id=Jnj5E6C9UwsC&pg=PA93 Extract of page 93]</ref>
−
| CrystalStruct = Trigonal
+
 
−
| SpaceGroup = <span style="text-decoration: overline">3</span>2/m
+
The principle in [[classical mechanics]] that ''E ∝ mv<sup>2</sup>'' was first developed by [[Gottfried Leibniz]] and [[Johann Bernoulli]], who described kinetic energy as the ''living force'', ''[[vis viva]]''. [[Willem 's Gravesande]] of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, [[Willem 's Gravesande]] determined that their penetration depth was proportional to the square of their impact speed. [[Émilie du Châtelet]] recognized the implications of the experiment and published an explanation.<ref>{{Cite book|author=Judith P. Zinsser |title=Emilie du Chatelet: Daring Genius of the Enlightenment|publisher=Penguin|year= 2007|isbn=0-14-311268-6}}</ref>
−
}}
+
 
−
|Section5={{Chembox Thermochemistry
+
The terms ''kinetic energy'' and ''work'' in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to [[Gaspard-Gustave Coriolis]], who in 1829 published the paper titled ''Du Calcul de l'Effet des Machines'' outlining the mathematics of kinetic energy. [[William Thomson, 1st Baron Kelvin|William Thomson]], later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51.<ref>{{cite book| author=Crosbie Smith, M. Norton Wise|title=Energy and Empire: A Biographical Study of Lord Kelvin|publisher=Cambridge University Press|pages=866| isbn=0-521-26173-2}}</ref><ref>{{cite book|author=John Theodore Merz|title=A History of European Thought in the Nineteenth Century|publisher=Blackwood|year=1912|page= 139|isbn=0-8446-2579-5}}</ref>
−
| DeltaHf = −1207&nbsp;kJ·mol<sup>−1</sup><ref name=b1>{{cite book| author = Zumdahl, Steven S.|title =Chemical Principles 6th Ed.| publisher = Houghton Mifflin Company| year = 2009| isbn = 0-618-94690-X|page=A21}}</ref>
+
 
−
| Entropy = 93&nbsp;J·mol<sup>−1</sup>·K<sup>−1</sup><ref name=b1 />
+
==Overview==
−
}}
+
[[Energy]] occurs in many forms, including [[chemical energy]], [[thermal energy]], [[electromagnetic radiation]], [[gravitational energy]], [[electric energy]], [[elastic energy]], [[nuclear binding energy|nuclear energy]], and [[rest energy]]. These can be categorized in two main classes: [[potential energy]] and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy.<ref>{{Cite web|url=https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-kinetic-energy|title=Khan Academy|website=Khan Academy|access-date=2016-10-09}}</ref>
−
|Section6={{Chembox Pharmacology
+
 
−
| ATCCode_prefix = A02
+
Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a [[cyclist]] uses [[food energy|chemical energy provided by food]] to accelerate a [[bicycle]] to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome [[drag (physics)|air resistance]] and [[friction]]. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist.
−
| ATCCode_suffix = AC01
+
 
−
| ATC_Supplemental = {{ATC|A12|AA04}}
+
The kinetic energy in the moving cyclist and the bicycle can be converted to other forms.  For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top.  The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill.  Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively, the cyclist could connect a [[Bottle dynamo|dynamo]] to one of the wheels and generate some electrical energy on the descent.  The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy.  Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as [[heat]].
−
}}
+
 
−
|Section7={{Chembox Hazards
+
Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's [[frame of reference]]. Thus, the kinetic energy of an object is not [[Galilean invariance|invariant]].
−
| ExternalSDS = [http://www.inchem.org/documents/icsc/icsc/eics1193.htm ICSC 1193]
+
 
−
| MainHazards =
+
[[Spacecraft]] use chemical energy to launch and gain considerable kinetic energy to reach [[orbital speed|orbital velocity]].  In an entirely circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However, it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is [[elliptic orbit|elliptical]] or [[hyperbolic trajectory|hyperbolic]], then throughout the orbit kinetic and [[potential energy]] are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Without loss or gain, however, the sum of the kinetic and potential energy remains constant.
−
| NFPA-H = 0
+
 
−
| NFPA-F = 0
+
Kinetic energy can be passed from one object to another. In the game of [[billiards]], the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically, and the ball it hit accelerates its speed as the kinetic energy is passed on to it. [[Collisions]] in billiards are effectively [[elastic collision]]s, in which kinetic energy is preserved. In [[inelastic collision]]s, kinetic energy is dissipated in various forms of energy, such as heat, sound, binding energy (breaking bound structures).
−
| NFPA-R = 0
+
 
−
| NFPA-S =
+
[[Flywheel]]s have been developed as a method of [[flywheel energy storage|energy storage]].  This illustrates that kinetic energy is also stored in rotational motion.
−
| RPhrases =
+
 
−
| SPhrases =
+
Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by [[Newtonian mechanics|Newtonian (classical) mechanics]] is suitable. However, if the speed of the object is comparable to the speed of light, [[special relativity|relativistic effects]] become significant and the relativistic formula is used. If the object is on the atomic or [[sub-atomic scale]], [[quantum mechanical]] effects are significant, and a quantum mechanical model must be employed.
−
| LD50 = 6450 mg/kg (oral, rat)
 
−
| PEL = TWA 15 mg/m<sup>3</sup> (total) TWA 5 mg/m<sup>3</sup> (resp)<ref>{{PGCH|0090}}</ref>
 
−
}}
 
−
|Section8={{Chembox Related
 
−
| OtherAnions = [[Calcium bicarbonate]]
 
−
| OtherCations = [[Magnesium carbonate]]<br />[[Strontium carbonate]]<br />[[Barium carbonate]]
 
−
| OtherCompounds = [[Calcium sulfate]]
 
−
}}
 
−
}}
 
−
[[File:Calcite.png|thumb|right|Crystal structure of calcite]]
 
  
−
'''Calcium carbonate''' is a [[chemical compound]] with the [[Chemical formula|formula]] [[Calcium|Ca]][[Carbon|C]][[Oxygen|O]]<sub>3</sub>. It is a common substance found in [[Rock (geology)|rocks]] as the [[mineral]]s [[calcite]] and [[aragonite]] (most notably as [[limestone]], which is a type of sedimentary rock build mainly of calcite) and is the main component of [[pearl]]s and the [[Seashell|shells of marine organisms]], [[snail]]s, and eggs. Calcium carbonate is the active ingredient in [[agricultural lime]] and is created when calcium ions in [[hard water]] react with [[carbonate ion]]s to create [[limescale]]. It is medicinally used as a [[calcium]] supplement or as an [[antacid]], but excessive consumption can be hazardous.
+
==Newtonian kinetic energy==
  
−
==Chemistry==
+
===Kinetic energy of rigid bodies===
−
Calcium carbonate shares the typical properties of other carbonates. Notably,
+
In [[classical mechanics]], the kinetic energy of a ''point object'' (an object so small that its mass can be assumed to exist at one point), or a non-rotating [[rigid body]] depends on the [[mass]] of the body as well as its [[speed]]. The kinetic energy is equal to 1/2 the [[Multiplication|product]] of the mass and the square of the speed. In formula form:
−
* it reacts with [[acid]]s, releasing [[carbon dioxide]]:
 
−
:CaCO<sub>3</sub>(s) + 2H<sup>+</sup>(aq) → Ca<sup>2+</sup>(aq) + CO<sub>2</sub>(g) + H<sub>2</sub>O (l)
 
−
* it releases carbon dioxide upon heating, called a [[thermal decomposition]] reaction, or [[calcination]] (to above 840&nbsp;°C in the case of CaCO<sub>3</sub>), to form [[calcium oxide]], commonly called [[quicklime]], with reaction [[enthalpy]] 178 kJ/mole:
 
−
:CaCO<sub>3</sub> (s) → CaO (s) + CO<sub>2</sub> (g)
 
  
−
Calcium carbonate will react with water that is saturated with carbon dioxide to form the soluble [[calcium bicarbonate]].
+
:<math>E_\text{k} =\tfrac{1}{2} mv^2 </math>
−
:CaCO<sub>3</sub> + CO<sub>2</sub> + H<sub>2</sub>O → Ca(HCO<sub>3</sub>)<sub>2</sub>
+
where <math>m</math> is the mass and <math>v</math> is the speed (or the velocity) of the body. In [[SI]] units, mass is measured in [[kilogram]]s, speed in [[metres per second]], and the resulting kinetic energy is in [[joule]]s.
  
−
This reaction is important in the [[erosion]] of [[carbonate rock]], forming [[cavern]]s, and leads to [[hard water]] in many regions.
+
For example, one would calculate the kinetic energy of an 80&nbsp;kg mass (about 180&nbsp;lbs) traveling at 18&nbsp;metres per second (about 40&nbsp;mph, or 65&nbsp;km/h) as
 +
:<math>E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12960 \,\text{J} = 12.96 \,\text{kJ}</math>
  
−
An unusual form of calcium carbonate is the hexahydrate, [[ikaite]], CaCO<sub>3</sub>·6H<sub>2</sub>O. Ikaite is stable only below 6&nbsp;°C.
+
When you throw a ball, you do [[work (physics)|work]] on it to give it speed as it leaves your hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: '''net force × displacement = kinetic energy''', i.e.,
  
−
==Preparation==
+
:<math>F s =\tfrac{1}{2} mv^2</math>
−
The vast majority of calcium carbonate used in industry is extracted by mining or quarrying. Pure calcium carbonate (e.g. for food or pharmaceutical use), can be produced from a pure quarried source (usually [[marble]]).
 
  
−
Alternatively, calcium carbonate is prepared from [[calcium oxide]]. Water is added to give [[calcium hydroxide]] then [[carbon dioxide]] is passed through this solution to precipitate the desired calcium carbonate, referred to in the industry as precipitated calcium carbonate (PCC):<ref name="PCC">{{cite web|title = Precipitated Calcium Carbonate |accessdate = 11 January 2014|url = http://www.lime.org/uses_of_lime/other_uses/precip_cc.asp}}</ref>
+
Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed.
  
−
: CaO + H<sub>2</sub>O → Ca(OH)<sub>2</sub>
+
The kinetic energy of an object is related to its [[momentum]] by the equation:
−
:<chem>Ca(OH)2 + CO2 -> CaCO3(v) + H2O</chem>
+
:<math>E_\text{k} = \frac{p^2}{2m}</math>
  
−
==Structure==
+
where:
−
The thermodynamically stable form of CaCO<sub>3</sub> under normal conditions is hexagonal β-CaCO<sub>3</sub>, (the mineral [[calcite]]).<ref name ="Ropp">{{cite book|last=R C Ropp Elsevier|title=Encyclopedia of the alkaline earth compounds|publisher=Elsevier|isbn=9780444595508|pages=359–370}}</ref> Other forms can be prepared, the denser,(2.83 g/cc) orthorhombic λ-CaCO<sub>3</sub> ( the mineral [[aragonite]]) and μ-CaCO<sub>3</sub>, occurring as the mineral [[vaterite]].<ref name ="Ropp"/> The aragonite form can be prepared by precipitation at temperatures above 85&nbsp;°C, the vaterite form can be prepared by precipitation at 60&nbsp;°C.<ref name ="Ropp"/> Calcite contains calcium atoms coordinated by 6 oxygen atoms, in aragonite they are coordinated by 9 oxygen atoms.<ref name ="Ropp"/> The vaterite structure is not fully understood.<ref name="DemichelisRaiteri2013">{{cite journal|last1=Demichelis|first1=Raffaella|last2=Raiteri|first2=Paolo|last3=Gale|first3=Julian D.|last4=Dovesi|first4=Roberto|title=The Multiple Structures of Vaterite|journal=Crystal Growth & Design|volume=13|issue=6|year=2013|pages=2247–2251|issn=1528-7483|doi=10.1021/cg4002972}}</ref> Magnesium carbonate MgCO<sub>3</sub> has the calcite structure, whereas strontium and barium carbonate (SrCO<sub>3</sub> and BaCO<sub>3</sub>) adopt the aragonite structure, reflecting their larger ionic radii.<ref name ="Ropp"/>
+
:<math>p\;</math> is momentum
 +
:<math>m\;</math> is mass of the body
  
−
==Occurrence==
+
For the ''translational kinetic energy,'' that is the kinetic energy associated with [[rectilinear motion]], of a [[rigid body]] with constant [[mass]] <math>m\;</math>, whose [[center of mass]] is moving in a straight line with speed <math>v\;</math>, as seen above is equal to
  
−
[[File:Calcium carbonate chunks.JPG|thumb|Calcium carbonate chunks from clamshell]]
+
:<math> E_\text{t} =\tfrac{1}{2} mv^2 </math>
  
−
===Geological sources===
+
where:
−
[[Calcite]], [[aragonite]] and [[vaterite]] are pure calcium carbonate minerals. Industrially important source rocks which are predominantly calcium carbonate include [[limestone]], [[chalk]], [[marble]] and [[travertine]].
+
:<math>m\;</math> is the mass of the body
 +
:<math>v\;</math> is the speed of the [[center of mass]] of the body.
  
−
[[File:Silfurberg.jpg|thumb|Calcite is the most stable polymorph of calcium carbonate. It is transparent to opaque. A transparent variety called [[Iceland spar]] (shown here) is used for optical purposes.{{clarify|date=January 2018}}]]
+
The kinetic energy of any entity depends on the reference frame in which it is measured. However the total energy of an isolated system, i.e. one in which energy can neither enter nor leave, does not change over time in the reference frame in which it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the [[Oberth effect]]. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames would however disagree on the value of this conserved energy.
  
−
===Biological sources===
+
The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the [[center of momentum]] frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the [[invariant mass]] of the system as a whole.
−
Eggshells, snail shells and most seashells are predominantly calcium carbonate and can be used as industrial sources of that chemical.<ref>{{cite web |title=How are seashells created? |author=Horne, Francis |date=23 October 2006 |work=Scientific American |accessdate=25 April 2012 |url=http://www.scientificamerican.com/article.cfm?id=how-are-seashells-created}}</ref> Oyster shells have enjoyed recent recognition as a source of dietary calcium, but are also a practical industrial source.<ref>{{cite web |url=http://www.webmd.com/drugs/drug-16642-Natural+Oyster+Shell+Calcium+Oral.aspx?drugid=16642&drugname=Natural+Oyster+Shell+Calcium+Oral| title=WebMD: Oyster shell calcium |publisher=WebMD| accessdate=25 April 2012}}</ref><ref>{{cite web |title=Oyster Shell Calcium Carbonate|publisher=Caltron Clays &amp Chemicals|url=http://caltronclays.in/Oyster_CC.html}}</ref> Dark green vegetables such as broccoli and kale contain dietarily significant amounts of calcium carbonate, however, they are not practical as an industrial source.<ref>{{cite journal|year=1993 |title=Absorbability of Calcium from Brassica Vegetables: Broccoli, Bok Choy, and Kale |journal=Journal of Food Science |volume=58 |issue=6 |pages=1378–1380|doi=10.1111/j.1365-2621.1993.tb06187.x|last1=Heaney|first1=R.P.|last2=Weaver|first2=C.M.|last3=Hinders|first3=SM.|last4=Martin|first4=B.|last5=Packard|first5=P.T.}}</ref>
 
  
−
===Extraterrestrial===
+
====Derivation====
−
Beyond Earth, strong evidence suggests the presence of calcium carbonate on [[Mars]]. Signs of calcium carbonate have been detected at more than one location (notably at [[Gusev crater|Gusev]] and [[Huygens (crater)|Huygens]] craters). This provides some evidence for the past presence of liquid water.<ref>{{cite journal
+
The work done in accelerating a particle with mass ''m'' during the infinitesimal time interval ''dt'' is given by the dot product of ''force'' '''F''' and the infinitesimal ''displacement ''d'''x'''''
−
| last1=Boynton |first1=WV
+
:<math>\mathbf{F} \cdot d \mathbf{x} = \mathbf{F} \cdot \mathbf{v} d t = \frac{d \mathbf{p}}{d t} \cdot \mathbf{v} d t = \mathbf{v} \cdot d \mathbf{p} = \mathbf{v} \cdot d (m \mathbf{v})\,,</math>
−
| last2=Ming |first2=DW
+
where we have assumed the relationship '''p'''&nbsp;=&nbsp;''m''&nbsp;'''v''' and the validity of [[Newton's Second Law]]. (However, also see the special relativistic derivation [[Kinetic energy#Relativistic kinetic energy of rigid bodies|below]].)
−
| last3=Kounaves |first3=SP
 
−
| last4=Young |first4=SM
 
−
| last5=Arvidson |first5=RE
 
−
| last6=Hecht |first6=MH
 
−
| last7=Hoffman |first7=J
 
−
| last8=Niles |first8=PB
 
−
| last9=Hamara |first9=DK
 
−
| last10=Quinn
 
−
| first10=R. C.
 
−
| last11=Smith
 
−
| first11=P. H.
 
−
| last12=Sutter
 
−
| first12=B
 
−
| last13=Catling
 
−
| first13=D. C.
 
−
| last14=Morris
 
−
| first14=R. V.
 
−
| title=Evidence for Calcium Carbonate at the Mars Phoenix Landing Site
 
−
| url=http://planetary.chem.tufts.edu/Boynton%20etal%20Science%202009v325p61.pdf
 
−
| journal=Science |volume=325 |issue=5936 |pages= 61–64
 
−
| year=2009 |pmid=19574384 |bibcode=2009Sci...325...61B
 
−
| display-authors=3
 
−
| doi=10.1126/science.1172768
 
−
| doi-broken-date=2017-01-31
 
−
}}</ref><ref name=Clark2007>
 
−
{{cite journal
 
−
| author1=Clark
 
−
| year=2007
 
−
| title=Evidence for montmorillonite or its compositional equivalent in Columbia Hills, Mars
 
−
| journal=[[Journal of Geophysical Research]]
 
−
| volume=112 |pages=E06S01
 
−
| doi=10.1029/2006JE002756
 
−
| last2=Arvidson
 
−
| first2=R. E.
 
−
| last3=Gellert
 
−
| first3=R.
 
−
| last4=Morris
 
−
| first4=R. V.
 
−
| last5=Ming
 
−
| first5=D. W.
 
−
| last6=Richter
 
−
| first6=L.
 
−
| last7=Ruff
 
−
| first7=S. W.
 
−
| last8=Michalski
 
−
| first8=J. R.
 
−
| last9=Farrand
 
−
| first9=W. H.
 
−
| last10=Yen
 
−
| first10=A.
 
−
| last11=Herkenhoff
 
−
| first11=K. E.
 
−
| last12=Li
 
−
| first12=R.
 
−
| last13=Squyres
 
−
| first13=S. W.
 
−
| last14=Schröder
 
−
| first14=C.
 
−
| last15=Klingelhöfer
 
−
| first15=G.
 
−
| last16=Bell
 
−
| first16=J. F.
 
−
| bibcode = 2007JGRE..112.6S01C
 
−
| displayauthors=3
 
−
| url=http://dspace.stir.ac.uk/bitstream/1893/17119/1/Clark2007_Evidence_for_montmorillonite_or_its_compositional_equivalent_in_Columbia_Hills_Mars.pdf
 
−
}}</ref>
 
  
−
==Geology==
+
Applying the [[product rule]] we see that:
−
Carbonate is found frequently in geologic settings and constitutes an enormous [[carbon cycle|carbon reservoir]]. Calcium carbonate occurs as [[aragonite]], [[calcite]] and [[dolomite]]. The [[carbonate mineral]]s form the rock types: [[limestone]], [[chalk]], [[marble]], [[travertine]], [[tufa]], and others.
+
:<math>  d(\mathbf{v} \cdot \mathbf{v}) = (d \mathbf{v}) \cdot \mathbf{v} + \mathbf{v} \cdot (d \mathbf{v}) =  2(\mathbf{v} \cdot d\mathbf{v}).</math>
  
−
In warm, clear tropical waters [[coral]]s are more abundant than towards the poles where the waters are cold. Calcium carbonate contributors, including [[plankton]] (such as [[coccolith]]s and planktic [[foraminifera]]), [[coralline algae]], [[sea sponge|sponges]], [[brachiopod]]s, [[echinoderm]]s, [[bryozoa]] and [[Mollusc shell|mollusks]], are typically found in shallow water environments where sunlight and filterable food are more abundant. Cold-water carbonates do exist at higher latitudes but have a very slow growth rate. The [[calcification]] processes are changed by [[ocean acidification]].
+
Therefore, (assuming constant mass so that ''dm''=0), we have,
 +
:<math> \mathbf{v} \cdot d (m \mathbf{v}) = \frac{m}{2} d (\mathbf{v} \cdot \mathbf{v}) = \frac{m}{2} d v^2  = d \left(\frac{m v^2}{2}\right). </math>
  
−
Where the [[oceanic crust]] is [[Subduction|subducted]] under a [[continental plate]] sediments will be carried down to warmer zones in the [[asthenosphere]] and [[lithosphere]]. Under these conditions calcium carbonate decomposes to produce [[carbon dioxide]] which, along with other gases, give rise to explosive [[volcano|volcanic eruptions]].
+
Since this is a [[total differential]] (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by the force to bring the object from rest to velocity ''v'' is equal to the work necessary to do the reverse:
 +
:<math> E_\text{k} = \int_0^t \mathbf{F} \cdot d \mathbf{x} = \int_0^t \mathbf{v} \cdot d (m \mathbf{v}) = \int_0^v d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}. </math>
  
−
===Carbonate compensation depth===
+
This equation states that the kinetic energy (''E''<sub>k</sub>) is equal to the [[integral]] of the [[dot product]] of the [[velocity]] ('''v''') of a body and the [[infinitesimal]] change of the body's [[momentum]] ('''p'''). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).
−
The [[carbonate compensation depth]] (CCD) is the point in the ocean where the rate of precipitation of calcium carbonate is balanced by the rate of dissolution due to the conditions present. Deep in the ocean, the temperature drops and pressure increases. Calcium carbonate is unusual in that its solubility increases with decreasing temperature. Increasing pressure also increases the solubility of calcium carbonate. The carbonate compensation depth can range from 4–6&nbsp;km below sea level.
 
  
−
===Role in taphonomy===
+
===Rotating bodies===
−
Calcium carbonate can [[taphonomy|preserve fossils]] through [[permineralization]]. Most of the vertebrate fossils of the [[Two Medicine Formation]]—a [[geologic formation]] known for its [[duck-billed dinosaur]] eggs—are preserved by CaCO<sub>3</sub> permineralization.<ref name="twoturn" /> This type of preservation conserves high levels of detail, even down to the microscopic level. However, it also leaves specimens vulnerable to [[weathering]] when exposed to the surface.<ref name="twoturn">Trexler, D. (2001) [https://books.google.com/books?id=mgc6CS4EUPsC&pg=PA98 "Two Medicine Formation, Montana: geology and fauna"], pp. 298–309 in ''Mesozoic Vertebrate Life'', Tanke, D. H., and Carpenter, K. (eds), Indiana University Press. {{ISBN|0-253-33907-3}}</ref>
+
If a rigid body Q is rotating about any line through the center of mass then it has [[rotational energy|''rotational kinetic energy'']] (<math>E_\text{r}\,</math>) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:
  
−
[[Trilobite]] populations were once thought to have composed the majority of aquatic life during the [[Cambrian]], due to the fact that their calcium carbonate-rich shells were more easily preserved than those of other species,<ref>{{Cite book|url=https://www.nap.edu/catalog/11630/out-of-thin-air-dinosaurs-birds-and-earths-ancient-atmosphere|title=Out of Thin Air: Dinosaurs, Birds, and Earth's Ancient Atmosphere|last=Ward|first=Peter|date=|publisher=|year=|isbn=9780309666121|location=|pages=|language=en|doi=10.17226/11630}}</ref> which had purely chitinous shells.
+
:<math> E_\text{r} = \int_Q \frac{v^2 dm}{2} = \int_Q \frac{(r \omega)^2 dm}{2} = \frac{\omega^2}{2} \int_Q {r^2}dm = \frac{\omega^2}{2} I = \begin{matrix} \frac{1}{2} \end{matrix} I \omega^2 </math>
  
−
==Uses==
+
where:
 +
* ω is the body's [[angular velocity]]
 +
* ''r'' is the distance of any mass ''dm'' from that line
 +
* <math>I\,</math> is the body's [[moment of inertia]], equal to <math>\int_Q {r^2}dm</math>.
  
−
===Industrial applications===
+
(In this equation the moment of [[inertia]] must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).
  
−
The main use of calcium carbonate is in the construction industry, either as a building material or limestone aggregate for road building or as an ingredient of cement or as the starting material for the preparation of builder's lime by burning in a kiln. However, because of weathering mainly caused by [[acid rain]],<ref>{{cite web|title = Effects of Acid Rain|publisher = US Environmental Protection Agency|accessdate = 14 March 2015|url = http://www.epa.gov/acidrain/effects/materials.html}}</ref> calcium carbonate (in limestone form) is no longer used for building purposes on its own, but only as a raw/primary substance for building materials.
+
===Kinetic energy of systems===
 +
A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the [[Solar System]] the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains.
  
−
Calcium carbonate is also used in the purification of [[iron]] from [[iron ore]] in a [[blast furnace]]. The carbonate is calcined ''in situ'' to give calcium oxide, which forms a slag with various impurities present, and separates from the purified iron.<ref>{{cite web|title = Blast Furnace|publisher = Science Aid|accessdate = 30 December 2007|url = http://www.scienceaid.co.uk/chemistry/industrial/blastfurnace.html}}</ref>
+
A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's [[center of momentum]]) may have various kinds of [[internal energy]] at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However all internal energies of all types contribute to body's mass, inertia, and total energy.
  
−
In the [[oil industry]], calcium carbonate is added to [[drilling fluid]]s as a formation-bridging and filtercake-sealing agent; it is also a weighting material which increases the density of drilling fluids to control the downhole pressure. Calcium carbonate is added to swimming pools, as a [[pH]] corrector for maintaining [[alkalinity]] and offsetting the acidic properties of the disinfectant agent.{{citation needed|date=June 2015}}
+
===Frame of reference===
  
−
It is also used as a raw material in the refining of sugar from [[sugar beet]]; It is calcined in a kiln with anthracite to produce calcium oxide and carbon dioxide. This burnt lime is then slaked in sweet water to produce a calcium hydroxide suspension for the precipitation of impurities in raw juice during [[carbonatation]].<ref>{{cite book|last1=McGinnis|first1=R.A.|title=Beet-Sugar Technology|publisher=Beet Sugar Development Foundation|page=178|edition=2nd}}</ref>
+
The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable [[inertial frame of reference]]. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy.<ref>{{cite book
 +
|title=Introduction to the theory of relativity
 +
|first1=Francis Weston
 +
|last1=Sears
 +
|first2=Robert W.
 +
|last2=Brehme
 +
|publisher=Addison-Wesley
 +
|year=1968
 +
|page=127
 +
}}, [https://books.google.com/books?id=cpzvAAAAMAAJ&dq=%22in+its+own+rest+frame%22+%22kinetic+energy%22&q=%22in+its+own+rest+frame%22 Snippet view of page 127]
 +
</ref> By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case, the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's [[invariant mass]], which is independent of the reference frame.
  
−
Calcium carbonate has traditionally been a major component of blackboard chalk. However, modern manufactured chalk is mostly [[gypsum]], hydrated [[calcium sulfate]] CaSO<sub>4</sub>·2H<sub>2</sub>O. Calcium carbonate is a main source for growing [[Seacrete]], or [[Biorock]]. Precipitated calcium carbonate (PCC), pre-dispersed in slurry form, is a common filler material for latex gloves with the aim of achieving maximum saving in material and production costs.<ref name=precaco3>{{cite web|title=Precipitated Calcium Carbonate uses |url=http://www.aristocratholding.com/calris-5.html |deadurl=yes |archiveurl=https://web.archive.org/web/20140725032803/http://www.aristocratholding.com/calris-5.html |archivedate=25 July 2014 }}</ref>
+
The total kinetic energy of a system depends on the [[inertial frame of reference]]: it is the sum of the total kinetic energy in a [[center of momentum frame]] and the kinetic energy the total mass would have if it were concentrated in the [[center of mass]].
  
−
Fine ground calcium carbonate (GCC) is an essential ingredient in the microporous film used in [[diapers]] and some building films as the pores are nucleated around the calcium carbonate particles during the manufacture of the film by biaxial stretching. GCC or PCC is used as a filler in paper because they are cheaper than wood fiber. In terms of market volume, GCC are the most important types of fillers currently used.<ref>[http://www.ceresana.com/en/market-studies/additives/fillers/ Market Study Fillers, 2nd ed., published by Ceresana, September 2011]</ref> Printing and writing paper can contain 10–20% calcium carbonate. In North America, calcium carbonate has begun to replace [[Kaolinite|kaolin]] in the production of glossy paper. Europe has been practicing this as alkaline [[papermaking]] or acid-free papermaking for some decades. PCC used for paper filling and paper coatings is precipitated and prepared in a variety of shapes and sizes having characteristic narrow particle size distributions and equivalent spherical diameters of 0.4 to 3 micrometres.{{citation needed|date=June 2015}}
+
This may be simply shown: let <math>\textstyle\mathbf{V}</math> be the relative velocity of the center of mass frame ''i'' in the frame ''k''.
 +
Since <math>\textstyle v^2 = (v_i + V)^2 = (\mathbf{v}_i + \mathbf{V}) \cdot (\mathbf{v}_i + \mathbf{V}) = \mathbf{v}_i \cdot \mathbf{v}_i + 2 \mathbf{v}_i \cdot \mathbf{V} + \mathbf{V} \cdot \mathbf{V} = v_i^2 + 2 \mathbf{v}_i \cdot \mathbf{V} + V^2</math>,
  
−
Calcium carbonate is widely used as an extender in paints,<ref name = reade>{{cite web|title = Calcium Carbonate Powder|publisher = Reade Advanced Materials |date=4 February 2006|accessdate = 30 December 2007|url = http://www.reade.com/Products/Minerals_and_Ores/calcium_carbonate.html}}</ref> in particular matte emulsion paint where typically 30% by weight of the paint is either chalk or marble. It is also a popular filler in plastics.<ref name = reade/> Some typical examples include around 15 to 20% loading of chalk in [[Polyvinyl chloride|unplasticized polyvinyl chloride]] (uPVC) drain pipe, 5 to 15% loading of stearate coated chalk or marble in uPVC window profile. [[Polyvinyl chloride|PVC]] cables can use calcium carbonate at loadings of up to 70 phr (parts per hundred parts of resin) to improve mechanical properties (tensile strength and elongation) and electrical properties (volume resistivity).{{citation needed|date=June 2015}} [[Polypropylene]] compounds are often filled with calcium carbonate to increase rigidity, a requirement that becomes important at high use temperatures.<ref name= Imerys>{{cite web|url=http://www.imerys-perfmins.com/calcium-carbonate/eu/calcium-carbonate-plastic.htm |title=Calcium carbonate in plastic applications |accessdate=1 August 2008 |publisher=Imerys Performance Minerals}}</ref> Here the percentage is often 20–40%. It also routinely used as a filler in [[Thermosetting plastic|thermosetting resins]] (sheet and bulk molding compounds)<ref name = Imerys/> and has also been mixed with [[acrylonitrile butadiene styrene|ABS]], and other ingredients, to form some types of compression molded "clay" poker chips.{{citation needed|date=June 2015}} Precipitated calcium carbonate, made by dropping [[calcium oxide]] into water, is used by itself or with additives as a white paint, known as [[whitewashing]].{{citation needed|date=June 2015}}
+
:<math>E_\text{k} = \int \frac{v^2}{2} dm = \int \frac{v_i^2}{2} dm + \mathbf{V} \cdot \int \mathbf{v}_i dm + \frac{V^2}{2} \int dm. </math>
  
−
Calcium carbonate is added to a wide range of trade and [[do it yourself]] adhesives, sealants, and decorating fillers.<ref name = reade/> Ceramic tile adhesives typically contain 70 to 80% limestone. Decorating crack fillers contain similar levels of marble or dolomite. It is also mixed with putty in setting [[stained glass]] windows, and as a resist to prevent glass from sticking to kiln shelves when firing glazes and paints at high temperature.{{citation needed|date=June 2015}}
+
However, let <math> \int \frac{v_i^2}{2} dm = E_i </math> the kinetic energy in the center of mass frame, <math> \int \mathbf{v}_i dm </math> would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: <math> \int dm = M </math>. Substituting, we get:<ref>[http://www.phy.duke.edu/~rgb/Class/intro_physics_1/intro_physics_1/node64.html Physics notes - Kinetic energy in the CM frame] {{webarchive|url=https://web.archive.org/web/20070611231147/http://www.phy.duke.edu/~rgb/Class/intro_physics_1/intro_physics_1/node64.html |date=2007-06-11 }}. [[Duke University|Duke]].edu. Accessed 2007-11-24.</ref>
  
−
In [[ceramics (art)|ceramics]]/glazing applications, calcium carbonate is known as ''whiting'',<ref name = reade/> and is a common ingredient for many glazes in its white powdered form. When a glaze containing this material is fired in a kiln, the whiting acts as a [[Ceramic flux|flux]] material in the glaze. Ground calcium carbonate is an [[abrasive]] (both as scouring powder and as an ingredient of household scouring creams), in particular in its calcite form, which has the relatively low hardness level of 3 on the [[Mohs scale of mineral hardness]], and will therefore not scratch [[glass]] and most other [[ceramic]]s, [[Vitreous enamel|enamel]], [[bronze]], [[iron]], and [[steel]], and have a moderate effect on softer metals like [[aluminium]] and [[copper]]. A paste made from calcium carbonate and [[deionized water]] can be used to clean [[tarnish]] on [[silver]].<ref name="Make it Shine">{{cite web|title = Ohio Historical Society Blog: Make It Shine|publisher = Ohio Historical Society |url = http://ohiohistory.wordpress.com/2011/06/02/making-it-shine/}}</ref>
+
:<math> E_\text{k} = E_i + \frac{M V^2}{2}. </math>
  
−
===Health and dietary applications===
+
Thus the kinetic energy of a system is lowest to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the [[center of mass frame]] or any other [[center of momentum frame]]). In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the [[center of momentum frame]] is a quantity that is invariant (all observers see it to be the same).
−
[[File:500 mg calcium supplements with vitamin D.jpg|thumb|500-milligram calcium supplements made from calcium carbonate]]
 
−
Calcium carbonate is widely used medicinally as an inexpensive dietary calcium supplement for [[antacid|gastric antacid]]<ref name = medline>{{cite web|work = Medline Plus|publisher = [[National Institutes of Health]]|title = Calcium Carbonate |date=1 October 2005|accessdate = 30 December 2007|url = https://www.nlm.nih.gov/medlineplus/druginfo/medmaster/a601032.html |archiveurl = https://web.archive.org/web/20071017031324/http://www.nlm.nih.gov/medlineplus/druginfo/medmaster/a601032.html <!-- Bot retrieved archive --> |archivedate = 17 October 2007}}</ref> (e.g., [[Tums]]). It may be used as a [[phosphate binder]] for the treatment of [[hyperphosphatemia]] (primarily in patients with [[chronic renal failure]]). It is also used in the pharmaceutical industry as an inert [[Excipient|filler]] for [[Tablet (pharmacy)|tablets]] and other [[pharmaceuticals]].<ref>{{cite book|author1=Lieberman, Herbert A. |author2=Lachman, Leon |author3=Schwartz, Joseph B. |title = Pharmaceutical Dosage Forms: Tablets|year = 1990|isbn = 0-8247-8044-2|page=153|publisher = Dekker|location = New York}}</ref>
 
  
−
Calcium carbonate is used in the production of calcium oxide as well as toothpaste and has seen a resurgence as a food preservative and color retainer, when used in or with products such as organic apples.<ref>[http://chemistry.about.com/od/foodcookingchemistry/a/cadditives.htm Food Additives – Names Starting with C]. Chemistry.about.com (10 April 2012). Retrieved 2012-05-24.</ref>
+
===Rotation in systems===
 +
It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass ([[rotational energy]]):
  
−
Excess calcium from supplements, fortified food and high-calcium diets, can cause [[milk-alkali syndrome]], which has serious toxicity and can be fatal. In 1915, Bertram Sippy introduced the "Sippy regimen" of hourly ingestion of milk and cream, and the gradual addition of eggs and cooked cereal, for 10 days, combined with alkaline powders, which provided symptomatic relief for peptic ulcer disease. Over the next several decades, the Sippy regimen resulted in [[renal failure]], [[alkalosis]], and [[hypercalcaemia]], mostly in men with peptic ulcer disease. These adverse effects were reversed when the regimen stopped, but it was fatal in some patients with protracted vomiting. Milk-alkali syndrome declined in men after effective treatments for [[peptic ulcer]] disease arose. During the past 15 years, it has been reported in women taking calcium supplements above the recommended range of 1.2 to 1.5&nbsp;g daily, for prevention and treatment of osteoporosis, and is exacerbated by [[dehydration]]. Calcium has been added to over-the-counter products, which contributes to inadvertent excessive intake. Excessive calcium intake can lead to [[hypercalcemia]], complications of which include vomiting, abdominal pain and altered mental status.<ref>{{cite journal|title=Clinical problem-solving, back to basics|author=Gabriely, Ilan |journal=New England Journal of Medicine|year=2008|volume=358|pmid=18450607|doi=10.1056/NEJMcps0706188|issue=18|last2=Leu|first2=James P.|last3=Barzel|first3=Uriel S.|pages=1952–6}}</ref>
+
:<math> E_\text{k} = E_t + E_\text{r} \, </math>
  
−
As a [[food additive]] it is designated E170,<ref>{{cite web|title=Food-Info.net : E-numbers : E170 Calcium carbonate|url=http://www.food-info.net/uk/e/e170.htm}} 080419 food-info.net</ref> and it has an INS number of 170. Used as an acidity regulator, anticaking agent, stabiliser or colour it is approved for usage in the EU,<ref>UK Food Standards Agency: {{cite web |url=http://www.food.gov.uk/safereating/chemsafe/additivesbranch/enumberlist |title=Current EU approved additives and their E Numbers |accessdate=27 October 2011}}</ref> USA<ref>US [[Food and Drug Administration]]: {{cite web|url=http://www.fda.gov/Food/FoodIngredientsPackaging/FoodAdditives/FoodAdditiveListings/ucm091048.htm |title=Listing of Food Additives Status Part I |accessdate=27 October 2011 |deadurl=yes |archiveurl=https://web.archive.org/web/20130314104055/http://www.fda.gov/Food/FoodIngredientsPackaging/FoodAdditives/FoodAdditiveListings/ucm091048.htm |archivedate=14 March 2013 |df=dmy }}</ref> and [[Australia]] and [[New Zealand]].<ref>Australia New Zealand Food Standards Code{{cite web |url=http://www.comlaw.gov.au/Details/F2011C00827 |title=Standard 1.2.4 – Labelling of ingredients |accessdate=27 October 2011}}</ref> It is used in some [[soy milk]] and [[almond milk]] products as a source of dietary calcium; one study suggests that calcium carbonate might be as [[bioavailable]] as the calcium in cow's milk.<ref>{{Cite journal
+
where:
−
| pmid = 16177199
+
:''E''<sub>k</sub> is the total kinetic energy
−
| year = 2005
+
:''E''<sub>t</sub> is the translational kinetic energy
−
| author1 = Zhao
+
:''E''<sub>r</sub> is the ''rotational energy'' or ''angular kinetic energy'' in the rest frame
−
| first1 = Y
 
−
| title = Calcium bioavailability of calcium carbonate fortified soymilk is equivalent to cow's milk in young women
 
−
| journal = The Journal of Nutrition
 
−
| volume = 135
 
−
| issue = 10
 
−
| pages = 2379–82
 
−
| last2 = Martin
 
−
| first2 = B. R.
 
−
| last3 = Weaver
 
−
| first3 = C. M.
 
−
}}</ref> Calcium carbonate is also used as a [[firming agent]] in many canned or bottled vegetable products.
 
  
−
===Agricultural use===
+
Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.
−
[[Agricultural lime]], powdered chalk or limestone, is used as a cheap method for neutralising acidic soil, making it suitable for planting.<ref name="Oates2008">{{cite book|first=J. A. H.|last=Oates|title=Lime and Limestone: Chemistry and Technology, Production and Uses|url=https://books.google.com/books?id=MVoEMNI5Vb0C&pg=PA111|date=11 July 2008|publisher=John Wiley & Sons|isbn=978-3-527-61201-7|pages=111–3}}</ref>
 
  
−
===Household use===
+
==Relativistic kinetic energy of rigid bodies==
−
Calcium carbonate is a key ingredient in many household cleaning powders like [[Comet (cleanser)]] and is used as a scrubbing agent.
+
{{See also|Mass in special relativity|Tests of relativistic energy and momentum}}
  
−
===Environmental applications===
+
If a body's speed is a significant fraction of the [[speed of light]], it is necessary to use relativistic mechanics to calculate its kinetic energy. In [[special relativity]] theory, the expression for linear momentum is modified.
  
−
In 1989, a researcher, Ken Simmons, introduced CaCO<sub>3</sub> into the Whetstone Brook in [[Massachusetts]].<ref>{{cite news|agency = [[Associated Press]]|title =
+
With ''m'' being an object's [[rest mass]], '''v''' and ''v'' its velocity and speed, and ''c'' the speed of light in vacuum, we use the expression for linear momentum <math>\mathbf{p}=m\gamma \mathbf{v}</math>, where <math>\gamma = 1/\sqrt{1-v^2/c^2}</math>.
−
Limestone Dispenser Fights Acid Rain in Stream |date=13 June 1989|url = https://query.nytimes.com/gst/fullpage.html?res=950DEFD9173FF930A25755C0A96F948260|work = The New York Times}}</ref> His hope was that the calcium carbonate would counter the acid in the stream from acid rain and save the trout that had ceased to spawn. Although his experiment was a success, it did increase the amount of aluminium ions in the area of the brook that was not treated with the limestone. This shows that CaCO<sub>3</sub> can be added to neutralize the effects of acid rain in [[river]] ecosystems. Currently calcium carbonate is used to neutralize acidic conditions in both soil and water.<ref name=env>{{cite web|title=Environmental Uses for Calcium Carbonate|url=http://www.congcal.com/markets/environmental/|publisher=Congcal|accessdate=5 August 2013}}</ref><ref>{{cite journal|author = Schreiber, R. K. |title = Cooperative federal-state liming research on surface waters impacted by acidic deposition|year = 1988|journal =Water, Air, & Soil Pollution|volume = 41|issue = 1|pages = 53–73|doi=10.1007/BF00160344|url=https://link.springer.com/article/10.1007%2FBF00160344|doi-broken-date = 2017-01-31}}</ref><ref>{{cite web|title = Effects of low pH and high aluminum on Atlantic salmon smolts in Eastern Maine and liming project feasibility analysis|year = 2006|author1=Kircheis, Dan |author2=Dill, Richard |publisher = National Marine Fisheries Service and Maine Atlantic Salmon Commission|url = http://www.mainesalmonrivers.org/pages/Liming%20Project%20Rpt.pdf|format = reprinted at Downeast Salmon Federation}}</ref> Since the 1970s, such ''liming'' has been practiced on a large scale in Sweden to mitigate acidification and several thousand lakes and streams are limed repeatedly.<ref>{{Cite journal |doi= 10.1007/s10933-006-9014-9 |title= Liming placed in a long-term perspective: A paleolimnological study of 12 lakes in the Swedish liming program |journal= Journal of Paleolimnology |volume= 37 |issue= 2 |pages= 247–258 |year= 2006 |last1= Guhrén |first1= M. |last2= Bigler |first2= C. |last3= Renberg |first3= I. |bibcode= 2007JPall..37..247G }}</ref>
 
  
−
Calcium carbonate is also used in [[flue gas desulfurisation]] applications eliminating harmful SO<sub>2</sub> and NO<sub>2</sub> emissions from coal and other fossil fuels burnt in large fossil fuel power stations.<ref name=env/>
+
[[Integration by parts|Integrating by parts]] yields
 +
:<math>E_\text{k} = \int \mathbf{v} \cdot d \mathbf{p}= \int \mathbf{v} \cdot d (m \gamma \mathbf{v}) = m \gamma \mathbf{v} \cdot \mathbf{v} - \int m \gamma \mathbf{v} \cdot d \mathbf{v} = m \gamma v^2 - \frac{m}{2} \int \gamma d (v^2)</math>
 +
Since <math>\gamma = (1 - v^2/c^2)^{-1/2}\!</math>,
 +
:<math>\begin{align}
 +
E_\text{k} &= m \gamma v^2 - \frac{- m c^2}{2} \int \gamma d (1 - v^2/c^2) \\
 +
    &= m \gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} - E_0
 +
\end{align}</math>
 +
<math>E_0</math> is a [[constant of integration]] for the [[indefinite integral]].
 +
Simplifying the expression we obtain
 +
:<math>\begin{align}
 +
E_\text{k} &= m \gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\
 +
    &= m \gamma (v^2 + c^2 - v^2) - E_0 \\
 +
    &= m \gamma c^2 - E_0
 +
\end{align}</math>
 +
<math>E_0</math> is found by observing that when <math>\mathbf{v }= 0 , \ \gamma = 1\!</math> and <math> E_\text{k} = 0 \!</math>, giving
 +
:<math>E_0 = m c^2 \,</math>
 +
resulting in the formula
 +
:<math>E_\text{k} = m \gamma c^2 - m c^2 = \frac{m c^2}{\sqrt{1 - v^2/c^2}} - m c^2</math>
  
−
==Calcination equilibrium==
+
This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary.
−
[[Calcination]] of [[limestone]] using [[charcoal]] fires to produce [[calcium oxide|quicklime]] has been practiced since antiquity by cultures all over the world. The temperature at which limestone yields calcium oxide is usually given as 825&nbsp;°C, but stating an absolute threshold is misleading. Calcium carbonate exists in equilibrium with calcium oxide and [[carbon dioxide]] at any temperature. At each temperature there is a [[partial pressure]] of carbon dioxide that is in equilibrium with calcium carbonate. At room temperature the equilibrium overwhelmingly favors calcium carbonate, because the equilibrium CO<sub>2</sub> pressure is only a tiny fraction of the partial CO<sub>2</sub> pressure in air, which is about 0.035 kPa.
 
  
−
At temperatures above 550&nbsp;°C the equilibrium CO<sub>2</sub> pressure begins to exceed the CO<sub>2</sub> pressure in air. So above 550&nbsp;°C, calcium carbonate begins to outgas CO<sub>2</sub> into air. However, in a charcoal fired kiln, the concentration of CO<sub>2</sub> will be much higher than it is in air. Indeed, if all the [[oxygen]] in the kiln is consumed in the fire, then the partial pressure of CO<sub>2</sub> in the kiln can be as high as 20 kPa.<ref name="solvaypcc2007">{{cite web|title = Solvay Precipitated Calcium Carbonate: Production|publisher = Solvay S. A. |date=9 March 2007|accessdate = 30 December 2007|url = http://www.solvaypcc.com/safety_environment/0,0,1000044-_EN,00.html}}</ref>
+
The mathematical by-product of this calculation is the [[mass-energy equivalence]] formula—the body at rest must have energy content
  
−
The table shows that this partial pressure is not achieved until the temperature is nearly 800&nbsp;°C. For the outgassing of CO<sub>2</sub> from calcium carbonate to happen at an economically useful rate, the equilibrium pressure must significantly exceed the ambient pressure of CO<sub>2</sub>. And for it to happen rapidly, the equilibrium pressure must exceed total atmospheric pressure of 101 kPa, which happens at 898&nbsp;°C.{{clear right}}
+
:<math>E_\text{rest} = E_0 = m c^2 \!</math>
  
−
{| class="wikitable"
+
At a low speed (<math>v</math><<<math>c</math>), the relativistic kinetic energy is approximated well by the classical kinetic energy. This is done by [[binomial approximation]] or by taking the first two terms of the [[Taylor expansion]] for the reciprocal square root:
−
|+ {{chembox header}} |Equilibrium pressure of CO<sub>2</sub> over CaCO<sub>3</sub> (P) vs. temperature (T).<ref name=crc>{{RubberBible86th}}</ref>
 
−
|-
 
−
|'''P (kPa)'''||0.055||0.13||0.31||1.80||5.9||9.3||14||24||34||51||72 ||80||91||101||179||901||3961
 
−
|-
 
−
|'''T (°C)'''||550||587||605||680||727||748||777||800||830||852||871||881||891||898||937||1082||1241
 
−
|}
 
  
−
==Solubility==
+
:<math>E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2\right) - m c^2 = \frac{1}{2} m v^2</math>
  
−
===With varying CO<sub>2</sub> pressure===
+
So, the total energy <math>E_k</math> can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds.
−
[[File:CanarySpring.jpg|thumb|right|[[Travertine]] calcium carbonate deposits from a [[hot spring]]]]
 
−
Calcium carbonate is poorly soluble in pure water (47&nbsp;mg/L at normal atmospheric CO<sub>2</sub> partial pressure as shown below).
 
  
−
The equilibrium of its solution is given by the equation (with dissolved calcium carbonate on the right):
+
When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the Taylor series approximation
−
:{| width="500"
 
−
| style="width:50%; height:30px;"| CaCO<sub>3</sub> {{eqm}} Ca<sup>2+</sup> + CO<sub>3</sub><sup>2−</sup>
 
−
| ''K''<sub>sp</sub> = 3.7×10<sup>−9</sup> to 8.7×10<sup>−9</sup> at 25&nbsp;°C
 
−
|}
 
  
−
where the [[solubility product]] for [Ca<sup>2+</sup>] [CO<sub>3</sub><sup>2−</sup>] is given as anywhere from ''K''<sub>sp</sub> = 3.7×10<sup>−9</sup> to ''K''<sub>sp</sub> = 8.7×10<sup>−9</sup> at 25&nbsp;°C, depending upon the data source.<ref name = crc/><ref>{{cite web|title = Selected Solubility Products and Formation Constants at 25 °C|publisher = [[California State University, Dominguez Hills]]|url = http://www.csudh.edu/oliver/chemdata/data-ksp.htm}}</ref> What the equation means is that the product of molar concentration of calcium ions ([[mole (unit)|moles]] of dissolved Ca<sup>2+</sup> per liter of solution) with the molar concentration of dissolved CO<sub>3</sub><sup>2−</sup> cannot exceed the value of ''K''<sub>sp</sub>. This seemingly simple solubility equation, however, must be taken along with the more complicated equilibrium of [[carbon dioxide]] with [[water]] (see [[carbonic acid]]). Some of the CO<sub>3</sub><sup>2−</sup> combines with H<sup>+</sup> in the solution according to:
+
:<math> E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2  + \frac{3}{8} v^4/c^4\right) - m c^2 = \frac{1}{2} m v^2 + \frac{3}{8} m v^4/c^2</math>
  
−
:{| width="500"
+
is small for low speeds. For example, for a speed of {{convert|10|km/s|mph|abbr=on}} the correction to the Newtonian kinetic energy is 0.0417&nbsp;J/kg (on a Newtonian kinetic energy of 50&nbsp;MJ/kg) and for a speed of 100&nbsp;km/s it is 417&nbsp;J/kg (on a Newtonian kinetic energy of 5&nbsp;GJ/kg).
−
| style="width:50%; height:25px;"| HCO<sub>3</sub><sup>−</sup> {{eqm}} H<sup>+</sup> + CO<sub>3</sub><sup>2−</sup> &nbsp;&nbsp;
 
−
| ''K''<sub>a2</sub> = 5.61×10<sup>−11</sup> at 25&nbsp;°C
 
−
|}
 
  
−
HCO<sub>3</sub><sup>−</sup> is known as the [[bicarbonate]] ion. [[Calcium bicarbonate]] is many times more soluble in water than calcium carbonate—indeed it exists ''only'' in solution.
+
The relativistic relation between kinetic energy and momentum is given by
  
−
Some of the HCO<sub>3</sub><sup>−</sup> combines with H<sup>+</sup> in solution according to:
+
:<math>E_\text{k} = \sqrt{p^2 c^2 + m^2 c^4} - m c^2</math>
  
−
:{| width="500"
+
This can also be expanded as a [[Taylor series]], the first term of which is the simple expression from Newtonian mechanics:<ref>{{cite web
−
| style="width:50%; height:25px;"|H<sub>2</sub>CO<sub>3</sub> {{eqm}} H<sup>+</sup> + HCO<sub>3</sub><sup>−</sup> &nbsp;&nbsp;
+
|url=http://farside.ph.utexas.edu/teaching/qmech/Quantum/node107.html
−
| ''K''<sub>a1</sub> = 2.5×10<sup>−4</sup> at 25&nbsp;°C
+
|title=Fine Structure of Hydrogen
−
|}
+
|first=Richard
 +
|last=Fitzpatrick
 +
|date=20 July 2010
 +
|work=Quantum Mechanics
 +
|accessdate=20 August 2016
 +
}}</ref>
 +
:<math> E_\text{k} \approx \frac{p^2}{2 m} -  \frac{p^4}{8 m^3 c^2} . </math>
 +
This suggests that the formulae for energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity.
  
−
Some of the H<sub>2</sub>CO<sub>3</sub> breaks up into water and dissolved carbon dioxide according to:
+
===General relativity===
 +
{{see also|Schwarzschild geodesics}}
 +
Using the convention that
 +
:<math>g_{\alpha \beta} \, u^{\alpha} \, u^{\beta} \, = \, - c^2 </math>
  
−
:{| width="500"
+
where the [[four-velocity]] of a particle is
−
| style="width:50%; height:25px;"| H<sub>2</sub>O + CO<sub>2</sub>(dissolved) {{eqm}} H<sub>2</sub>CO<sub>3</sub> &nbsp;&nbsp;
+
:<math>u^{\alpha} \, = \, \frac{d x^{\alpha}}{d \tau} </math>
−
| ''K''<sub>h</sub> = 1.70×10<sup>−3</sup> at 25&nbsp;°C
 
−
|}
 
  
−
And dissolved carbon dioxide is in equilibrium with atmospheric carbon dioxide according to:
+
and <math>\tau \,</math> is the [[proper time]] of the particle, there is also an expression for the kinetic energy of the particle in [[general relativity]].
  
−
:{| width="500"
+
If the particle has momentum
−
| style="width:45%;"|<math chem>\frac{P_{\ce{CO2}}}{[\ce{CO2}]}\ =\ k_\ce{H}</math>
+
:<math>p_{\beta} \, = \, m \, g_{\beta \alpha} \, u^{\alpha} </math>
−
| where ''k''<sub>H</sub> = 29.76 atm/(mol/L) at 25&nbsp;°C ([[Henry's law|Henry constant]]), <math chem>P_{\ce{CO2}}</math> being the CO<sub>2</sub> partial pressure.
 
−
|}
 
  
−
For ambient air, <math chem>P_{\ce{CO2}}</math> is around 3.5×10<sup>−4</sup> atmospheres (or equivalently 35 [[Pascal (unit)|Pa]]). The last equation above fixes the concentration of dissolved CO<sub>2</sub> as a function of <math chem>P_{\ce{CO2}}</math>, independent of the concentration of dissolved CaCO<sub>3</sub>. At atmospheric partial pressure of CO<sub>2</sub>, dissolved CO<sub>2</sub> concentration is 1.2×10<sup>−5</sup> moles/liter. The equation before that fixes the concentration of H<sub>2</sub>CO<sub>3</sub> as a function of [CO<sub>2</sub>]. For [CO<sub>2</sub>]=1.2×10<sup>−5</sup>, it results in [H<sub>2</sub>CO<sub>3</sub>]=2.0×10<sup>−8</sup> moles per liter. When [H<sub>2</sub>CO<sub>3</sub>] is known, the remaining three equations together with
+
as it passes by an observer with four-velocity ''u''<sub>obs</sub>, then the expression for total energy of the particle as observed (measured in a local inertial frame) is
−
{| class="wikitable floatright"
+
:<math>E \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} </math>
−
|+ {{chembox header}} |Calcium ion solubility as a function of [[carbon dioxide|CO<sub>2</sub>]] [[partial pressure]] at 25&nbsp;°C {{math|1=(''K''<sub>sp</sub> = 4.47×10<sup>−9</sup>)}}
 
−
|-
 
−
!<math chem>\scriptstyle P_\ce{CO2}</math> (atm)
 
−
![[pH]]
 
−
![Ca<sup>2+</sup>] (mol/L)
 
−
|-
 
−
| 10<sup>−12</sup> ||12.0||5.19 × 10<sup>−3</sup>
 
−
|-
 
−
| 10<sup>−10</sup> ||11.3||1.12 × 10<sup>−3</sup>
 
−
|-
 
−
| 10<sup>−8</sup> ||10.7||2.55 × 10<sup>−4</sup>
 
−
|-
 
−
| 10<sup>−6</sup> ||9.83||1.20 × 10<sup>−4</sup>
 
−
|-
 
−
| 10<sup>−4</sup> ||8.62||3.16 × 10<sup>−4</sup>
 
−
|-
 
−
| 3.5 × 10<sup>−4</sup>||8.27||4.70 × 10<sup>−4</sup>
 
−
|-
 
−
| 10<sup>−3</sup> ||7.96||6.62 × 10<sup>−4</sup>
 
−
|-
 
−
| 10<sup>−2</sup> ||7.30||1.42 × 10<sup>−3</sup>
 
−
|-
 
−
| 10<sup>−1</sup> ||6.63||3.05 × 10<sup>−3</sup>
 
−
|-
 
−
| 1 ||5.96||6.58 × 10<sup>−3</sup>
 
−
|-
 
−
| 10 ||5.30||1.42 × 10<sup>−2</sup>
 
−
|}
 
  
−
:{| width="450"
+
and the kinetic energy can be expressed as the total energy minus the rest energy:
−
| style="width:50%; height:25px;"| H<sub>2</sub>O {{eqm}} H<sup>+</sup> + OH<sup>−</sup>
+
:<math>E_{k} \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} \, - \, m \, c^2 \, .</math>
−
| ''K'' = 10<sup>−14</sup> at 25&nbsp;°C
 
−
|}
 
  
−
(which is true for all aqueous solutions), and the fact that the solution must be electrically neutral,
+
Consider the case of a metric that is diagonal and spatially isotropic (''g''<sub>tt</sub>,''g''<sub>ss</sub>,''g''<sub>ss</sub>,''g''<sub>ss</sub>). Since
 +
:<math>u^{\alpha} = \frac{d x^{\alpha}}{d t} \frac{d t}{d \tau} = v^{\alpha} u^{t} \,</math>
  
−
:2[Ca<sup>2+</sup>] + [H<sup>+</sup>] = [HCO<sub>3</sub><sup>−</sup>] + 2[CO<sub>3</sub><sup>2−</sup>] + [OH<sup>−</sup>]
+
where ''v''<sup>α</sup> is the ordinary velocity measured w.r.t. the coordinate system, we get
 +
:<math>-c^2 = g_{\alpha \beta} u^{\alpha} u^{\beta} = g_{t t} (u^{t})^2 + g_{s s} v^2 (u^{t})^2 \,.</math>
  
−
make it possible to solve simultaneously for the remaining five unknown concentrations (note that the above form of the neutrality equation is valid only if calcium carbonate has been put in contact with pure water or with a neutral pH solution; in the case where the initial water solvent pH is not neutral, the equation is modified).
+
Solving for ''u''<sup>t</sup> gives
 +
:<math>u^{t} = c \sqrt{\frac{-1}{g_{t t} + g_{s s} v^2}} \,.</math>
  
−
The table on the right shows the result for [Ca<sup>2+</sup>] and [H<sup>+</sup>] (in the form of pH) as a function of ambient partial pressure of CO<sub>2</sub> (''K''<sub>sp</sub> = 4.47×10<sup>−9</sup> has been taken for the calculation).
+
Thus for a stationary observer (''v''= 0)
−
* At atmospheric levels of ambient CO<sub>2</sub> the table indicates the solution will be slightly alkaline with a maximum CaCO<sub>3</sub> solubility of 47&nbsp;mg/L.
+
:<math>u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{t t}}} \,</math>
−
* As ambient CO<sub>2</sub> partial pressure is reduced below atmospheric levels, the solution becomes more and more alkaline. At extremely low <math chem>P_{\ce{CO2}}</math>, dissolved CO<sub>2</sub>, bicarbonate ion, and carbonate ion largely evaporate from the solution, leaving a highly alkaline solution of [[calcium hydroxide]], which is more soluble than CaCO<sub>3</sub>. Note that for <math chem>P_{\ce{CO2}} = 10^{-12} \mathrm{atm}</math>, the [Ca<sup>2+</sup>] [OH<sup>−</sup>]<sup>2</sup> product is still below the solubility product of Ca(OH)<sub>2</sub> (8×10<sup>−6</sup>). For still lower CO<sub>2</sub> pressure, Ca(OH)<sub>2</sub> precipitation will occur before CaCO<sub>3</sub> precipitation.
 
−
* As ambient CO<sub>2</sub> partial pressure increases to levels above atmospheric, pH drops, and much of the carbonate ion is converted to bicarbonate ion, which results in higher solubility of Ca<sup>2+</sup>.
 
  
−
The effect of the latter is especially evident in day-to-day life of people who have hard water. Water in aquifers underground can be exposed to levels of CO<sub>2</sub> much higher than atmospheric. As such water percolates through calcium carbonate rock, the CaCO<sub>3</sub> dissolves according to the second trend. When that same water then emerges from the tap, in time it comes into equilibrium with CO<sub>2</sub> levels in the air by outgassing its excess CO<sub>2</sub>. The calcium carbonate becomes less soluble as a result and the excess precipitates as lime scale. This same process is responsible for the formation of [[stalactites]] and [[stalagmite]]s in limestone caves.
+
and thus the kinetic energy takes the form
 +
:<math>E_\text{k} = - m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,.</math>
  
−
Two hydrated phases of calcium carbonate, [[monohydrocalcite]], CaCO<sub>3</sub>·H<sub>2</sub>O and [[ikaite]], CaCO<sub>3</sub>·6H<sub>2</sub>O, may [[precipitate]] from water at ambient conditions and persist as metastable phases.
+
Factoring out the rest energy gives:
 +
:<math>E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,.</math>
  
−
=== With varying pH, temperature and salinity: CaCO<sub>3</sub> scaling in swimming pools ===
+
This expression reduces to the special relativistic case for the flat-space metric where
−
[[File:CaCO3-pH.gif|thumb|alt=Effects of salinity and pH on the maximum calcium ion level before scaling is anticipated at 25 C and 1 mM bicarbonate (e.g. in swimming pools)]]
+
:<math>g_{t t} = -c^2 \,</math>
−
[[File:CaCO3-Temp.gif|thumb|alt=Effects of temperature and bicarbonate concentration on the maximum calcium ion level before scaling is anticipated at pH 7 and 5,000 ppm salinity (e.g. in swimming pools)]]
+
:<math>g_{s s} = 1 \,.</math>
−
In contrast to the open equilibrium scenario above, many swimming pools are managed by addition of [[sodium bicarbonate]] (NaHCO<sub>3</sub>) to about 2 mM as a buffer, then control of pH through use of HCl, NaHSO<sub>4</sub>, Na<sub>2</sub>CO<sub>3</sub>, NaOH or chlorine formulations that are acidic or basic. In this situation, dissolved inorganic carbon ([[total inorganic carbon]]) is far from equilibrium with atmospheric CO<sub>2</sub>. Progress towards equilibrium through outgassing of CO<sub>2</sub> is slowed by (i) the slow reaction [[Carbonic acid|H<sub>2</sub>CO<sub>3</sub>]] ⇌ CO<sub>2</sub>(aq) + H<sub>2</sub>O;<ref>{{Cite journal | doi = 10.1021/jp909019u| pmid = 20039712| title = Comprehensive Study of the Hydration and Dehydration Reactions of Carbon Dioxide in Aqueous Solution| journal = The Journal of Physical Chemistry A| volume = 114| issue = 4| pages = 1734–40| year = 2010| last1 = Wang | first1 = X. | last2 = Conway | first2 = W. | last3 = Burns | first3 = R. | last4 = McCann | first4 = N. | last5 = Maeder | first5 = M. | bibcode = 2010JPCA..114.1734W}}</ref> (ii) limited aeration in a deep water column and (iii) periodic replenishment of bicarbonate to maintain buffer capacity (often estimated through measurement of [[alkalinity|‘total alkalinity’]]).
 
  
−
In this situation, the dissociation constants for the much faster reactions H<sub>2</sub>CO<sub>3</sub> ⇌ H<sup>+</sup> + HCO<sub>3</sub><sup>‾</sup> ⇌ 2 H<sup>+</sup> + CO<sub>3</sub><sup>2−</sup> allow the prediction of concentrations of each dissolved inorganic carbon species in solution, from the added concentration of HCO<sub>3</sub><sup>−</sup> (which constitutes more than 90% of [[Bjerrum plot]] species from pH 7 to pH 8 at 25&nbsp;°C in fresh water).<ref name="Mook 2000">Mook, W. (2000) [http://www-naweb.iaea.org/napc/ih/documents/global_cycle/vol%20I/cht_i_09.pdf "Chemistry of carbonic acid in water"], pp. 143–165 in ''Environmental Isotopes in the Hydrological Cycle: Principles and Applications''. INEA/UNESCO: Paris.</ref> Addition of HCO<sub>3</sub><sup>−</sup> will increase CO<sub>3</sub><sup>2−</sup> concentration at any pH. Rearranging the equations given above, we can see that [Ca<sup>2+</sup>] = Ksp / [CO<sub>3</sub><sup>2−</sup>], and [CO<sub>3</sub><sup>2−</sup>] = K<sub>a2</sub> × [HCO<sub>3</sub><sup>−</sup>] / [H<sup>+</sup>]. Therefore, when HCO<sub>3</sub><sup>−</sup> concentration is known, the maximum concentration of Ca<sup>2+</sup> ions before scaling through CaCO<sub>3</sub> precipitation can be predicted from the formula:
+
In the Newtonian approximation to general relativity
 +
:<math>g_{t t} = - \left( c^2 + 2 \Phi \right) \,</math>
 +
:<math>g_{s s} = 1 - \frac{2 \Phi}{c^2} \,</math>
  
−
:Ca<sup>2+</sup><sub>max</sub> = (K<sub>sp</sub> / K<sub>a2</sub>) × ([H<sup>+</sup>] / [HCO<sub>3</sub><sup>−</sup>])
+
where Φ is the Newtonian [[gravitational potential]]. This means clocks run slower and measuring rods are shorter near massive bodies.
  
−
The solubility product for CaCO<sub>3</sub> (K<sub>sp</sub>) and the dissociation constants for the dissolved inorganic carbon species (including K<sub>a2</sub>) are all substantially affected by temperature and [[salinity]],<ref name="Mook 2000" /> with the overall effect that Ca<sup>2+</sup><sub>max</sub> increases from fresh to salt water, and decreases with rising temperature, pH, or added bicarbonate level, as illustrated in the accompanying graphs.
+
==Kinetic energy in quantum mechanics==
 +
{{further|Hamiltonian (quantum mechanics)}}
  
−
The trends are illustrative for pool management, but whether scaling occurs also depends on other factors including interactions with Mg<sup>2+</sup>, B(OH)<sub>4</sub><sup>−</sup> and other ions in the pool, as well as supersaturation effects.<ref>{{cite journal|author=Wojtowicz, J. A. |year=1998|title= Factors affecting precipitation of calcium carbonate|journal= Journal of the Swimming Pool and Spa Industry |volume=3 |issue=1|pages= 18–23|url=http://jspsi.poolhelp.com/ARTICLES/JSPSI_V3N1_pp18-23.pdf}}</ref><ref>{{cite journal|author=Wojtowicz, J. A. |year=1998|title= Corrections, potential errors, and significance of the saturation index|journal= Journal of the Swimming Pool and Spa Industry |volume=3 |issue=1|pages=37–40|url=http://jspsi.poolhelp.com/ARTICLES/JSPSI_V3N1_pp37-40.pdf}}</ref> Scaling is commonly observed in electrolytic chlorine generators, where there is a high pH near the cathode surface and scale deposition further increases temperature. This is one reason that some pool operators prefer borate over bicarbonate as the primary pH buffer, and avoid the use of pool chemicals containing calcium.<ref>Birch, R. G. (2013) [http://members.iinet.net.au/~jorobbirch/BABES.pdf BABES: a better method than "BBB" for pools with a salt-water chlorine generator.] iinet.net.au</ref>
+
In [[quantum mechanics]], observables like kinetic energy are represented as [[Operator (physics)|operators]]. For one particle of mass ''m'', the kinetic energy operator appears as a term in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and is defined in terms of the more fundamental momentum operator <math>\hat p</math>. The kinetic energy operator in the [[Relativistic quantum mechanics#Non-relativistic and relativistic Hamiltonians|non-relativistic]] case can be written as
  
−
===Solubility in a strong or weak acid solution===
+
:<math>\hat T = \frac{\hat p^2}{2m}.</math>
−
Solutions of [[strong acid|strong]] ([[hydrochloric acid|HCl]]), moderately strong ([[sulfamic acid|sulfamic]]) or [[weak acid|weak]] ([[acetic acid|acetic]], [[citric acid|citric]], [[sorbic acid|sorbic]], [[lactic acid|lactic]], [[phosphoric acid|phosphoric]]) acids are commercially available. They are commonly used as [[descaling agent]]s to remove [[limescale]] deposits. The maximum amount of CaCO<sub>3</sub> that can be "dissolved" by one liter of an acid solution can be calculated using the above equilibrium equations.
 
−
* In the case of a strong monoacid with decreasing acid concentration [A] = [A<sup>−</sup>], we obtain (with CaCO<sub>3</sub> molar mass = 100 g):
 
  
−
{| border="1" cellspacing="0" cellpadding="4" style="margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;" class="wikitable"
+
Notice that this can be obtained by replacing <math>p</math> by <math>\hat p</math> in the classical expression for kinetic energy in terms of [[momentum]],
−
|-
+
:<math>E_\text{k} = \frac{p^2}{2m}.</math>
−
! width="160" {{chembox header}} |[A] (mol/L)
 
−
| 1
 
−
| 10<sup>−1</sup>
 
−
| 10<sup>−2</sup>
 
−
| 10<sup>−3</sup>
 
−
| 10<sup>−4</sup>
 
−
| 10<sup>−5</sup>
 
−
| 10<sup>−6</sup>
 
−
| 10<sup>−7</sup>
 
−
| 10<sup>−10</sup>
 
−
|-
 
−
! width="160" {{chembox header}} |Initial pH
 
−
| 0.00||1.00||2.00||3.00||4.00||5.00||6.00||6.79||7.00
 
−
|-
 
−
! width="160" {{chembox header}} |Final pH
 
−
| 6.75||7.25||7.75||8.14||8.25||8.26||8.26||8.26||8.27
 
−
|-
 
−
! width="160" {{chembox header}} |Dissolved CaCO<sub>3</sub><br />(g/[[liter|L]] of acid)
 
−
| 50.0||5.00||0.514||0.0849||0.0504||0.0474||0.0471||0.0470||0.0470
 
−
|}
 
  
−
where the initial state is the acid solution with no Ca<sup>2+</sup> (not taking into account possible CO<sub>2</sub> dissolution) and the final state is the solution with saturated Ca<sup>2+</sup>. For strong acid concentrations, all species have a negligible concentration in the final state with respect to Ca<sup>2+</sup> and A<sup>−</sup> so that the neutrality equation reduces approximately to 2[Ca<sup>2+</sup>] = [A<sup>−</sup>] yielding <math>\scriptstyle[\mathrm{Ca}^{2+}] \simeq \frac{[\mathrm{A}^-]}{2}</math>. When the concentration decreases, [HCO<sub>3</sub><sup>−</sup>] becomes non-negligible so that the preceding expression is no longer valid. For vanishing acid concentrations, one can recover the final pH and the solubility of CaCO<sub>3</sub> in pure water.
+
In the [[Schrödinger picture]], <math>\hat p</math> takes the form <math>-i\hbar\nabla </math> where the derivative is taken with respect to position coordinates and hence
−
* In the case of a weak monoacid (here we take acetic acid with p''K''<sub>A</sub> = 4.76) with decreasing total acid concentration [A] = [A<sup>−</sup>]+[AH], we obtain:
 
  
−
{| border="1" cellspacing="0" cellpadding="4" style="margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;" class="wikitable"
+
:<math>\hat T = -\frac{\hbar^2}{2m}\nabla^2.</math>
−
|-
 
−
! width="160" {{chembox header}} |[A] (mol/L)
 
−
| 1
 
−
| 10<sup>−1</sup>
 
−
| 10<sup>−2</sup>
 
−
| 10<sup>−3</sup>
 
−
| 10<sup>−4</sup>
 
−
| 10<sup>−5</sup>
 
−
| 10<sup>−6</sup>
 
−
| 10<sup>−7</sup>
 
−
| 10<sup>−10</sup>
 
−
|-
 
−
! width="160" {{chembox header}} |Initial pH
 
−
| 2.38||2.88||3.39||3.91||4.47||5.15||6.02||6.79||7.00
 
−
|-
 
−
! width="160" {{chembox header}} |Final pH
 
−
| 6.75||7.25||7.75||8.14||8.25||8.26||8.26||8.26||8.27
 
−
|-
 
−
! width="160" {{chembox header}} |Dissolved CaCO<sub>3</sub><br />(g/[[liter|L]] of acid)
 
−
| 49.5||4.99||0.513||0.0848||0.0504||0.0474||0.0471||0.0470||0.0470
 
−
|}
 
−
For the same total acid concentration, the initial pH of the weak acid is less acid than the one of the strong acid; however, the maximum amount of CaCO<sub>3</sub> which can be dissolved is approximately the same. This is because in the final state, the pH is larger than the p''K''<sub>A</sub>, so that the weak acid is almost completely dissociated, yielding in the end as many H<sup>+</sup> ions as the strong acid to "dissolve" the calcium carbonate.
 
−
* The calculation in the case of [[phosphoric acid]] (which is the most widely used for domestic applications) is more complicated since the concentrations of the four dissociation states corresponding to this acid must be calculated together with [HCO<sub>3</sub><sup>−</sup>], [CO<sub>3</sub><sup>2−</sup>], [Ca<sup>2+</sup>], [H<sup>+</sup>] and [OH<sup>−</sup>]. The system may be reduced to a seventh degree equation for [H<sup>+</sup>] the numerical solution of which gives
 
  
−
{| border="1" cellspacing="0" cellpadding="4" style="margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;" class="wikitable"
+
The expectation value of the electron kinetic energy, <math>\langle\hat{T}\rangle</math>, for a system of ''N'' electrons described by the [[Wave function|wavefunction]] <math>\vert\psi\rangle</math> is a sum of 1-electron operator expectation values:
−
|-
+
:<math>\langle\hat{T}\rangle = \bigg\langle\psi \bigg\vert \sum_{i=1}^N \frac{-\hbar^2}{2 m_\text{e}} \nabla^2_i \bigg\vert \psi \bigg\rangle = -\frac{\hbar^2}{2 m_\text{e}} \sum_{i=1}^N \bigg\langle\psi \bigg\vert \nabla^2_i \bigg\vert \psi \bigg\rangle</math>
−
! width="160" {{chembox header}} |[A] (mol/L)
+
where <math>m_\text{e}</math> is the mass of the electron and <math>\nabla^2_i</math> is the [[Laplacian]] operator acting upon the coordinates of the ''i''<sup>th</sup> electron and the summation runs over all electrons.
−
| 1
 
−
| 10<sup>−1</sup>
 
−
| 10<sup>−2</sup>
 
−
| 10<sup>−3</sup>
 
−
| 10<sup>−4</sup>
 
−
| 10<sup>−5</sup>
 
−
| 10<sup>−6</sup>
 
−
| 10<sup>−7</sup>
 
−
| 10<sup>−10</sup>
 
−
|-
 
−
! width="160" {{chembox header}} |Initial pH
 
−
| 1.08||1.62||2.25||3.05||4.01||5.00||5.97||6.74||7.00
 
−
|-
 
−
! width="160" {{chembox header}} |Final pH
 
−
| 6.71||7.17||7.63||8.06||8.24||8.26||8.26||8.26||8.27
 
−
|-
 
−
! width="160" {{chembox header}} |Dissolved CaCO<sub>3</sub><br />(g/[[liter|L]] of acid)
 
−
| 62.0||7.39||0.874||0.123||0.0536||0.0477||0.0471||0.0471||0.0470
 
−
|}
 
  
−
where [A] = [H<sub>3</sub>PO<sub>4</sub>] + [H<sub>2</sub>PO<sub>4</sub><sup>−</sup>] + [HPO<sub>4</sub><sup>2−</sup>] + [PO<sub>4</sub><sup>3−</sup>] is the total acid concentration. Thus phosphoric acid is more efficient than a monoacid since at the final almost neutral pH, the second dissociated state concentration [HPO<sub>4</sub><sup>2−</sup>] is not negligible (see [[phosphoric acid#pH and composition of a phosphoric acid aqueous solution|phosphoric acid]]).
+
The [[Density functional theory|density functional]] formalism of quantum mechanics requires knowledge of the electron density ''only'', i.e., it formally does not require knowledge of the wavefunction.  Given an electron density <math>\rho(\mathbf{r})</math>, the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as
 +
:<math> T[\rho]  =  \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r </math>
 +
where <math>T[\rho]</math> is known as the [[Carl Friedrich von Weizsäcker|von Weizsäcker]] kinetic energy functional.
  
 
==See also==
 
==See also==
−
{{div col|colwidth=22em}}
+
{{Portal|Energy}}
−
* [[Cuttlebone]]
+
* [[Escape velocity]]
−
* [[Cuttlefish]]
+
* [[Joule]]
−
* [[Gesso]]
+
* [[KE-Munitions]]
−
* [[Limescale]]
+
* [[Projectile#Typical projectile speeds|Kinetic energy per unit mass of projectiles]]
−
* [[Marble]]
+
* [[Projectile#Kinetic projectiles|Kinetic projectile]]
−
* [[Ocean acidification]]
+
* [[Parallel axis theorem]]
−
{{div col end}}
+
* [[Potential energy]]
 +
* [[Recoil]]
 +
 
 +
==Notes==
 +
{{reflist}}
  
 
==References==
 
==References==
−
{{reflist|30em}}
+
*{{cite web | url = http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy | title = Kinetic Energy | accessdate = 2015-07-19 | author = Physics Classroom | year = 2000 }}
 +
*[[Oxford Dictionary]] 1998
 +
*{{cite web | url = http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Coriolis.html | title = Biography of Gaspard-Gustave de Coriolis (1792-1843) | accessdate = 2006-03-03 | author = School of Mathematics and Statistics, University of St Andrews | year = 2000 }}
 +
*{{cite book | last = Serway | first = Raymond A. |author2=Jewett, John W.  | title = Physics for Scientists and Engineers | edition = 6th | publisher = Brooks/Cole | year = 2004 | isbn = 0-534-40842-7 }}
 +
*{{cite book | last = Tipler | first = Paul | title = Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics | edition = 5th | publisher = W. H. Freeman | year = 2004 | isbn = 0-7167-0809-4 }}
 +
*{{cite book | last = Tipler | first = Paul |author2=Llewellyn, Ralph  | title = Modern Physics | edition = 4th | publisher = W. H. Freeman | year = 2002 | isbn = 0-7167-4345-0 }}
  
 
==External links==
 
==External links==
−
* {{ICSC|1193|11}}
+
*{{Commonscat-inline}}
−
* {{PubChemLink|516889}}
+
*[http://www.kineticenergys.com kinetic energy] - what it is and how it works
−
* [[ATC codes]]: {{ATC|A02|AC01}} and {{ATC|A12|AA04}}
 
−
* [http://calcium-carbonate.org.uk/calcium-carbonate.asp The British Calcium Carbonate Association – What is calcium carbonate]
 
−
* [https://www.cdc.gov/niosh/npg/npgd0090.html CDC – NIOSH Pocket Guide to Chemical Hazards – Calcium Carbonate]
 
−
 
 
−
{{Calcium compounds}}
 
−
{{Antacids}}
 
−
{{Drugs for treatment of hyperkalemia and hyperphosphatemia}}
 
  
−
{{Authority control}}
+
{{Footer energy}}
  
−
{{DEFAULTSORT:Calcium Carbonate}}
+
[[Category:Kinetic energy| ]]
−
[[Category:Calcium compounds]]
+
[[Category:Dynamics (mechanics)]]
−
[[Category:Carbonates]]
+
[[Category:Forms of energy]]
−
[[Category:Limestone]]
 
−
[[Category:Phosphate binders]]
 
−
[[Category:Excipients]]
 
−
[[Category:Antacids]]
 
−
[[Category:Food stabilizers]]
 

Revision as of 17:56, 27 July 2018

Template:Infobox physical quantity Template:Classical mechanics In physics, the kinetic energy of an object is the energy that it possesses due to its motion.<ref>Template:Cite book, Chapter 1, p. 9 </ref> It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is Template:Smallmath. In relativistic mechanics, this is a good approximation only when v is much less than the speed of light.

The standard unit of kinetic energy is the joule.

History and etymology

The adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality.<ref>Template:Cite book Extract of page 93</ref>

The principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva. Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation.<ref>Template:Cite book</ref>

The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>

Overview

Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy.<ref>Template:Cite web</ref>

Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist.

The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat.

Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant.

Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an entirely circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However, it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is elliptical or hyperbolic, then throughout the orbit kinetic and potential energy are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Without loss or gain, however, the sum of the kinetic and potential energy remains constant.

Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically, and the ball it hit accelerates its speed as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, sound, binding energy (breaking bound structures).

Flywheels have been developed as a method of energy storage. This illustrates that kinetic energy is also stored in rotational motion.

Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian (classical) mechanics is suitable. However, if the speed of the object is comparable to the speed of light, relativistic effects become significant and the relativistic formula is used. If the object is on the atomic or sub-atomic scale, quantum mechanical effects are significant, and a quantum mechanical model must be employed.

Newtonian kinetic energy

Kinetic energy of rigid bodies

In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body depends on the mass of the body as well as its speed. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. In formula form:

\[E_\text{k} =\tfrac{1}{2} mv^2 \] where \(m\) is the mass and \(v\) is the speed (or the velocity) of the body. In SI units, mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules.

For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as \[E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12960 \,\text{J} = 12.96 \,\text{kJ}\]

When you throw a ball, you do work on it to give it speed as it leaves your hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e.,

\[F s =\tfrac{1}{2} mv^2\]

Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed.

The kinetic energy of an object is related to its momentum by the equation: \[E_\text{k} = \frac{p^2}{2m}\]

where: \[p\;\] is momentum \[m\;\] is mass of the body

For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass \(m\;\), whose center of mass is moving in a straight line with speed \(v\;\), as seen above is equal to

\[ E_\text{t} =\tfrac{1}{2} mv^2 \]

where: \[m\;\] is the mass of the body \[v\;\] is the speed of the center of mass of the body.

The kinetic energy of any entity depends on the reference frame in which it is measured. However the total energy of an isolated system, i.e. one in which energy can neither enter nor leave, does not change over time in the reference frame in which it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames would however disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.

Derivation

The work done in accelerating a particle with mass m during the infinitesimal time interval dt is given by the dot product of force F and the infinitesimal displacement dx \[\mathbf{F} \cdot d \mathbf{x} = \mathbf{F} \cdot \mathbf{v} d t = \frac{d \mathbf{p}}{d t} \cdot \mathbf{v} d t = \mathbf{v} \cdot d \mathbf{p} = \mathbf{v} \cdot d (m \mathbf{v})\,,\] where we have assumed the relationship p = m v and the validity of Newton's Second Law. (However, also see the special relativistic derivation below.)

Applying the product rule we see that: \[ d(\mathbf{v} \cdot \mathbf{v}) = (d \mathbf{v}) \cdot \mathbf{v} + \mathbf{v} \cdot (d \mathbf{v}) = 2(\mathbf{v} \cdot d\mathbf{v}).\]

Therefore, (assuming constant mass so that dm=0), we have, \[ \mathbf{v} \cdot d (m \mathbf{v}) = \frac{m}{2} d (\mathbf{v} \cdot \mathbf{v}) = \frac{m}{2} d v^2 = d \left(\frac{m v^2}{2}\right). \]

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by the force to bring the object from rest to velocity v is equal to the work necessary to do the reverse: \[ E_\text{k} = \int_0^t \mathbf{F} \cdot d \mathbf{x} = \int_0^t \mathbf{v} \cdot d (m \mathbf{v}) = \int_0^v d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}. \]

This equation states that the kinetic energy (Ek) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal change of the body's momentum (p). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

Rotating bodies

If a rigid body Q is rotating about any line through the center of mass then it has rotational kinetic energy (\(E_\text{r}\,\)) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

\[ E_\text{r} = \int_Q \frac{v^2 dm}{2} = \int_Q \frac{(r \omega)^2 dm}{2} = \frac{\omega^2}{2} \int_Q {r^2}dm = \frac{\omega^2}{2} I = \begin{matrix} \frac{1}{2} \end{matrix} I \omega^2 \]

where:

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).

Kinetic energy of systems

A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the Solar System the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains.

A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's center of momentum) may have various kinds of internal energy at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However all internal energies of all types contribute to body's mass, inertia, and total energy.

Frame of reference

The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy.<ref>Template:Cite book, Snippet view of page 127 </ref> By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case, the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame.

The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.

This may be simply shown: let \(\textstyle\mathbf{V}\) be the relative velocity of the center of mass frame i in the frame k. Since \(\textstyle v^2 = (v_i + V)^2 = (\mathbf{v}_i + \mathbf{V}) \cdot (\mathbf{v}_i + \mathbf{V}) = \mathbf{v}_i \cdot \mathbf{v}_i + 2 \mathbf{v}_i \cdot \mathbf{V} + \mathbf{V} \cdot \mathbf{V} = v_i^2 + 2 \mathbf{v}_i \cdot \mathbf{V} + V^2\),

\[E_\text{k} = \int \frac{v^2}{2} dm = \int \frac{v_i^2}{2} dm + \mathbf{V} \cdot \int \mathbf{v}_i dm + \frac{V^2}{2} \int dm. \]

However, let \( \int \frac{v_i^2}{2} dm = E_i \) the kinetic energy in the center of mass frame, \( \int \mathbf{v}_i dm \) would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass\[ \int dm = M \]. Substituting, we get:<ref>Physics notes - Kinetic energy in the CM frame Template:Webarchive. Duke.edu. Accessed 2007-11-24.</ref>

\[ E_\text{k} = E_i + \frac{M V^2}{2}. \]

Thus the kinetic energy of a system is lowest to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the center of mass frame or any other center of momentum frame). In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the center of momentum frame is a quantity that is invariant (all observers see it to be the same).

Rotation in systems

It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass (rotational energy):

\[ E_\text{k} = E_t + E_\text{r} \, \]

where:

Ek is the total kinetic energy
Et is the translational kinetic energy
Er is the rotational energy or angular kinetic energy in the rest frame

Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.

Relativistic kinetic energy of rigid bodies

Template:See also

If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy. In special relativity theory, the expression for linear momentum is modified.

With m being an object's rest mass, v and v its velocity and speed, and c the speed of light in vacuum, we use the expression for linear momentum \(\mathbf{p}=m\gamma \mathbf{v}\), where \(\gamma = 1/\sqrt{1-v^2/c^2}\).

Integrating by parts yields \[E_\text{k} = \int \mathbf{v} \cdot d \mathbf{p}= \int \mathbf{v} \cdot d (m \gamma \mathbf{v}) = m \gamma \mathbf{v} \cdot \mathbf{v} - \int m \gamma \mathbf{v} \cdot d \mathbf{v} = m \gamma v^2 - \frac{m}{2} \int \gamma d (v^2)\] Since \(\gamma = (1 - v^2/c^2)^{-1/2}\!\), \[\begin{align} E_\text{k} &= m \gamma v^2 - \frac{- m c^2}{2} \int \gamma d (1 - v^2/c^2) \\ &= m \gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} - E_0 \end{align}\] \(E_0\) is a constant of integration for the indefinite integral. Simplifying the expression we obtain \[\begin{align} E_\text{k} &= m \gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\ &= m \gamma (v^2 + c^2 - v^2) - E_0 \\ &= m \gamma c^2 - E_0 \end{align}\] \(E_0\) is found by observing that when \(\mathbf{v }= 0 , \ \gamma = 1\!\) and \( E_\text{k} = 0 \!\), giving \[E_0 = m c^2 \,\] resulting in the formula \[E_\text{k} = m \gamma c^2 - m c^2 = \frac{m c^2}{\sqrt{1 - v^2/c^2}} - m c^2\]

This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary.

The mathematical by-product of this calculation is the mass-energy equivalence formula—the body at rest must have energy content

\[E_\text{rest} = E_0 = m c^2 \!\]

At a low speed (\(v\)<<\(c\)), the relativistic kinetic energy is approximated well by the classical kinetic energy. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root:

\[E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2\right) - m c^2 = \frac{1}{2} m v^2\]

So, the total energy \(E_k\) can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds.

When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the Taylor series approximation

\[ E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} v^2/c^2 + \frac{3}{8} v^4/c^4\right) - m c^2 = \frac{1}{2} m v^2 + \frac{3}{8} m v^4/c^2\]

is small for low speeds. For example, for a speed of Template:Convert the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg).

The relativistic relation between kinetic energy and momentum is given by

\[E_\text{k} = \sqrt{p^2 c^2 + m^2 c^4} - m c^2\]

This can also be expanded as a Taylor series, the first term of which is the simple expression from Newtonian mechanics:<ref>Template:Cite web</ref> \[ E_\text{k} \approx \frac{p^2}{2 m} - \frac{p^4}{8 m^3 c^2} . \] This suggests that the formulae for energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity.

General relativity

Template:See also Using the convention that \[g_{\alpha \beta} \, u^{\alpha} \, u^{\beta} \, = \, - c^2 \]

where the four-velocity of a particle is \[u^{\alpha} \, = \, \frac{d x^{\alpha}}{d \tau} \]

and \(\tau \,\) is the proper time of the particle, there is also an expression for the kinetic energy of the particle in general relativity.

If the particle has momentum \[p_{\beta} \, = \, m \, g_{\beta \alpha} \, u^{\alpha} \]

as it passes by an observer with four-velocity uobs, then the expression for total energy of the particle as observed (measured in a local inertial frame) is \[E \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} \]

and the kinetic energy can be expressed as the total energy minus the rest energy: \[E_{k} \, = \, - \, p_{\beta} \, u_{\text{obs}}^{\beta} \, - \, m \, c^2 \, .\]

Consider the case of a metric that is diagonal and spatially isotropic (gtt,gss,gss,gss). Since \[u^{\alpha} = \frac{d x^{\alpha}}{d t} \frac{d t}{d \tau} = v^{\alpha} u^{t} \,\]

where vα is the ordinary velocity measured w.r.t. the coordinate system, we get \[-c^2 = g_{\alpha \beta} u^{\alpha} u^{\beta} = g_{t t} (u^{t})^2 + g_{s s} v^2 (u^{t})^2 \,.\]

Solving for ut gives \[u^{t} = c \sqrt{\frac{-1}{g_{t t} + g_{s s} v^2}} \,.\]

Thus for a stationary observer (v= 0) \[u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{t t}}} \,\]

and thus the kinetic energy takes the form \[E_\text{k} = - m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,.\]

Factoring out the rest energy gives: \[E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,.\]

This expression reduces to the special relativistic case for the flat-space metric where \[g_{t t} = -c^2 \,\] \[g_{s s} = 1 \,.\]

In the Newtonian approximation to general relativity \[g_{t t} = - \left( c^2 + 2 \Phi \right) \,\] \[g_{s s} = 1 - \frac{2 \Phi}{c^2} \,\]

where Φ is the Newtonian gravitational potential. This means clocks run slower and measuring rods are shorter near massive bodies.

Kinetic energy in quantum mechanics

Template:Further

In quantum mechanics, observables like kinetic energy are represented as operators. For one particle of mass m, the kinetic energy operator appears as a term in the Hamiltonian and is defined in terms of the more fundamental momentum operator \(\hat p\). The kinetic energy operator in the non-relativistic case can be written as

\[\hat T = \frac{\hat p^2}{2m}.\]

Notice that this can be obtained by replacing \(p\) by \(\hat p\) in the classical expression for kinetic energy in terms of momentum, \[E_\text{k} = \frac{p^2}{2m}.\]

In the Schrödinger picture, \(\hat p\) takes the form \(-i\hbar\nabla \) where the derivative is taken with respect to position coordinates and hence

\[\hat T = -\frac{\hbar^2}{2m}\nabla^2.\]

The expectation value of the electron kinetic energy, \(\langle\hat{T}\rangle\), for a system of N electrons described by the wavefunction \(\vert\psi\rangle\) is a sum of 1-electron operator expectation values: \[\langle\hat{T}\rangle = \bigg\langle\psi \bigg\vert \sum_{i=1}^N \frac{-\hbar^2}{2 m_\text{e}} \nabla^2_i \bigg\vert \psi \bigg\rangle = -\frac{\hbar^2}{2 m_\text{e}} \sum_{i=1}^N \bigg\langle\psi \bigg\vert \nabla^2_i \bigg\vert \psi \bigg\rangle\] where \(m_\text{e}\) is the mass of the electron and \(\nabla^2_i\) is the Laplacian operator acting upon the coordinates of the ith electron and the summation runs over all electrons.

The density functional formalism of quantum mechanics requires knowledge of the electron density only, i.e., it formally does not require knowledge of the wavefunction. Given an electron density \(\rho(\mathbf{r})\), the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as \[ T[\rho] = \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r \] where \(T[\rho]\) is known as the von Weizsäcker kinetic energy functional.

See also

Template:Portal

Notes

Template:Reflist

References

External links

Template:Footer energy

Retrieved from "https://keystagewiki.com/index.php?title=Testpage&oldid=151"