View source for Gas constant

{{2019 SI redefinition}}
{| class="wikitable" style="margin: 0 0 0 0.5em; float: right;"
! Values of ''R''{{physconst|R|ref=only}}
! Units
|-
| colspan="2" |'''[[International System of Units|SI units]]'''
|-
| {{val|8.31446261815324}}
| [[joule|J]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| 8.314 462 618 153 24
| [[cubic meter|m<sup>3</sup>]]⋅[[pascal (unit)|Pa]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
|8.314 462 618 153 24
|[[Kilogram|kg]]⋅[[Metre|m]]<sup>2</sup>·[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>[[second|s]]<sup>−2</sup>
|-
|8314.462 618 153 24
|[[litre|L]]⋅[[Pascal (unit)|Pa]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| 8.314 462 618 153 24{{e|-2}}
| [[litre|L]]⋅[[bar (unit)|bar]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| 8.314 462 618 153 24{{e|7}}
| [[erg (unit)|erg]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| colspan="2" |[[US Customary units|'''US Customary Units''']]
|-
|0.730 240 507 295 273
|[[Atmosphere (unit)|atm]]⋅[[Foot (unit)|ft]]<sup>3</sup>⋅[[Mole (unit)|lbmol]]<sup>-1</sup>[[Rankine scale|°R]]<sup>-1</sup>
|-
|10.731 557 089 016
|[[Pounds per square inch|psi]]⋅[[Foot (unit)|ft]]<sup>3</sup>⋅[[Mole (unit)|lbmol]]<sup>-1</sup>[[Rankine scale|°R]]<sup>-1</sup>
|-
|1.985 875 279 009
|[[British thermal unit|BTU]]⋅[[Mole (unit)|lbmol]]<sup>-1</sup>[[Rankine scale|°R]]<sup>-1</sup>
|-
| colspan="2" |'''Other Common Units'''
|-
|297.049 031 214
|[[Inch of water|in. H<sub>2</sub>O]]⋅[[Foot (unit)|ft]]<sup>3</sup>[[Mole (unit)|lbmol]]<sup>-1</sup>[[Rankine scale|°R]]<sup>-1</sup>
|-
|554.984 319 180
|[[torr]]⋅[[Foot (unit)|ft]]<sup>3</sup>[[Mole (unit)|lbmol]]<sup>-1</sup>[[Rankine scale|°R]]<sup>-1</sup>
|-
|0.082 057 366 080 960
|[[litre|L]]⋅[[atmosphere (unit)|atm]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| 62.636 598 221 529
|[[litre|L]]⋅[[Torr]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|1.98720425864083|e=-3|end=...}}
| [[calorie|kcal]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|8.20573660809596|e=-5|end=...}}
| [[cubic meter|m<sup>3</sup>]]⋅[[atmosphere (unit)|atm]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
|}
The '''gas constant''' is also known as the '''molar''', '''universal''', or '''ideal gas constant''', denoted by the symbol {{math|''R''}} or {{math|{{overline|''R''}}}} and is equivalent to the [[Boltzmann constant]], but expressed in units of [[energy]] per [[temperature|temperature increment]] per ''[[mole (unit)|mole]]'', i.e. the pressure–volume product, rather than energy per temperature increment per ''particle''. The constant is also a combination of the constants from [[Boyle's law]], [[Charles's law]], [[Avogadro's law]], and [[Gay-Lussac's law]].  It is a [[physical constant]] that is featured in many fundamental equations in the physical  sciences, such as the [[ideal gas law]] and the [[Nernst equation]].

Physically, the gas constant is the [[constant of proportionality]] that relates the energy scale in physics to the temperature scale, when a mole of particles at the stated temperature is being considered. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of the energy and temperature scales, plus similar historical setting of the value of the [[mole (unit)|molar scale]] used for the counting of particles. The last factor is not a consideration in the value of the [[Boltzmann constant]], which does a similar job of equating linear energy and temperature scales.

The gas constant R is defined as the [[Avogadro constant]] ''N''<sub>A</sub> multiplied by the [[Boltzman constant|Boltzmann constant]] ''k'':
:<math>R = N_{\rm A} k_,\,</math>

Since the [[2019 redefinition of SI base units]], which came into effect on 20 May 2019, both ''N''<sub>A</sub> and ''k'' are defined with exact numerical values when expressed in SI units.<ref>{{Cite web|url=https://www.bipm.org/utils/en/pdf/CIPM/CIPM2017-EN.pdf?page=23|title=Proceedings of the 106th meeting|date=16-20 October 2017}}</ref> As a consequence, the value of the gas constant is exactly {{val|8.31446261815324|u=J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>}}.

Some have suggested that it might be appropriate to name the symbol ''R'' the '''Regnault constant''' in honour of the [[French people|French]] [[chemist]] [[Henri Victor Regnault]], whose accurate experimental data were used to calculate the early value of the constant; however, the origin of the letter ''R'' to represent of the constant is elusive.<ref name="Jensen">{{cite journal
|title=The Universal Gas Constant ''R''
|last=Jensen
|first=William B.
|journal= J. Chem. Educ.
|volume= 80
|issue= 7
|date=July 2003
|pages=731
|doi=10.1021/ed080p731|bibcode = 2003JChEd..80..731J }}</ref><ref name="JensenReprint">{{cite web
|url= http://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/100.%20Gas%20Constant.pdf
|title=Ask the Historian: The Universal Gas Constant&nbsp;— Why is it represented by the letter ''R''?}}</ref>

The gas constant occurs in the [[ideal gas law]], as follows:
:<math>PV = nRT = m R_{\rm specific} T</math>
where ''P'' is the absolute [[pressure]] (SI unit pascals), ''V'' is the volume of gas (SI unit cubic metres), ''n'' is the [[amount of substance|amount of gas]] (SI unit moles), ''m'' is the [[mass]] (SI unit kilograms) contained in ''V'',  and ''T'' is the [[thermodynamic temperature]] (SI unit kelvins). ''R''<sub>specific</sub> is the mass-specific gas constant. The gas constant is expressed in the same physical units as molar [[entropy]] and molar [[heat capacity]].

==Dimensions==
From the ideal gas law ''PV'' = ''nRT'' we get:
:<math>R = \frac{PV}{nT}</math>
where ''P'' is pressure, ''V'' is volume, ''n'' is number of moles of a given substance, and ''T'' is [[temperature]].

As pressure is defined as force per unit area, the gas equation can also be written as:
:<math>R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} }
                { \mathrm{amount} \times \mathrm{temperature} }
</math>

Area and volume are (length)<sup>2</sup> and (length)<sup>3</sup> respectively. Therefore:
:<math>R = \frac{ \dfrac{\mathrm{force} }{ (\mathrm{length})^2} \times (\mathrm{length})^3 }
                { \mathrm{amount} \times \mathrm{temperature} }
         = \frac{ \mathrm{force} \times \mathrm{length} }
                { \mathrm{amount} \times \mathrm{temperature} }
</math>

Since force × length = work:
:<math>R = \frac{ \mathrm{work} }
           { \mathrm{amount} \times \mathrm{temperature} }
</math>

The physical significance of ''R'' is work per degree per mole. It may be expressed in any set of units representing work or energy (such as [[joule]]s), units representing degrees of temperature on an absolute scale (such as [[Kelvin]] or [[Rankine scale|Rankine]]), and any system of units designating a mole or a similar pure number that allows an equation of macroscopic mass and fundamental particle numbers in a system, such as an ideal gas (see [[Avogadro's number]]).

Instead of a mole the constant can be expressed by considering the [[normal cubic meter]].

Otherwise, we can also say that:

:<math>\mathrm{force} = \frac{ \mathrm{mass} \times \mathrm{length} }
                { (\mathrm{time})^2 }
</math>

Therefore, we can write ''R'' as:

:<math>R = \frac{ \mathrm{mass} \times \mathrm{length}^2 }
                { \mathrm{amount} \times \mathrm{temperature} \times (\mathrm{time})^2 }
</math>

And so, in [[SI base units]]:

:''R'' = {{physconst|R|unit=no|after=&nbsp;kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅K<sup>−1</sup>⋅mol<sup>−1</sup>|ref=no}}.

==Relationship with the Boltzmann constant==
The [[Boltzmann constant]] ''k''<sub>B</sub> (often abbreviated ''k'') may be used in place of the gas constant by working in pure particle count, ''N'', rather than amount of substance, ''n'', since
:<math>R = N_{\rm A} k_{\rm B},\,</math>
where ''N''<sub>A</sub> is the [[Avogadro constant]].
For example, the [[ideal gas law]] in terms of Boltzmann's constant is
:<math>PV = k_{\rm B} N T.</math>
where ''N'' is the number of particles (molecules in this case), or to generalize to an inhomogeneous system the local form holds:
:<math>P = k_{\rm B} n T.</math>
where ''n'' is the [[number density]].

:<math>\textrm{Boltzmann's}\ \textrm{constant} = \frac{ \textrm{Planck}\ \textrm{mass} \times (\textrm{Planck}\ \textrm{length})^2 }
                { \textrm{Planck}\ \textrm{temperature} \times (\textrm{Planck}\ \textrm{time})^2 }
</math>

==Measurement==
As of 2006, the most precise measurement of ''R'' is obtained by measuring the [[speed of sound]]&nbsp;''c''<sub>a</sub>(''P'',&nbsp;''T'') in [[argon]] at the temperature&nbsp;''T'' of the [[triple point of water]]  at different [[pressure]]s&nbsp;''P'', and [[extrapolation|extrapolating]] to the zero-pressure limit&nbsp;''c''<sub>a</sub>(0,&nbsp;''T''). The value of ''R'' is then obtained from the relation
:<math>c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}},</math>
where:
*''γ''<sub>0</sub> is the [[heat capacity ratio]] ({{frac|5|3}} for monatomic gases such as argon);
*''T'' is the temperature, ''T''<sub>TPW</sub> = 273.16&nbsp;K by definition of the kelvin;
*''A''<sub>r</sub>(Ar) is the relative atomic mass of argon and ''M''<sub>u</sub>&nbsp;=&nbsp;{{val|e=-3|u=kg⋅mol<sup>−1</sup>}}.

==Specific gas constant==
{| class="wikitable" style="float: right;"
! ''R''<sub>specific</sub><br />for dry air
! Units
|-
| 287.058
| J⋅kg<sup>−1</sup>⋅K<sup>−1</sup>
|-
| 53.3533
| ft⋅[[Pound-force|lbf]]⋅[[Pound (mass)|lb]]<sup>−1</sup>⋅°R<sup>−1</sup>
|-
| 1,716.49
| ft⋅[[Pound-force|lbf]]⋅[[slug (unit)|slug]]<sup>−1</sup>⋅°R<sup>−1</sup>
|-
| colspan=2 | {{small|Based on a mean molar mass<br />for dry air of 28.9645&nbsp;g/mol.}}
|-
|}
The '''specific gas constant''' of a gas or a mixture of gases (''R''<sub>specific</sub>) is given by the molar gas constant divided by the [[molar mass]] (''M'') of the gas or mixture.
:<math> R_{\rm specific} = \frac{R}{M} </math>

Just as the ideal gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas.
:<math> R_{\rm specific} = \frac{k_{\rm B}}{m} </math>

Another important relationship comes from thermodynamics. [[Julius Robert von Mayer|Mayer]]'s relation relates the specific gas constant to the specific heats for a calorically perfect gas and a thermally perfect gas.
:<math> R_{\rm specific} = c_{\rm p} - c_{\rm v}\ </math>
where ''c''<sub>p</sub> is the [[Specific_heat_capacity|specific heat]] for a constant pressure and ''c''<sub>v</sub> is the [[Specific_heat_capacity|specific heat]] for a constant volume.<ref>Anderson, ''Hypersonic and High-Temperature Gas Dynamics'', AIAA Education Series, 2nd Ed, 2006</ref>

It is common, especially in engineering applications, to represent the specific gas constant by the symbol ''R''. In such cases, the universal gas constant is usually given a different symbol such as ''{{overline|R}}'' to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.<ref>Moran and Shapiro, ''Fundamentals of Engineering Thermodynamics'', Wiley, 4th Ed, 2000</ref>

==U.S. Standard Atmosphere==
The [[U.S. Standard Atmosphere]], 1976 (USSA1976) defines the gas constant ''R''<sup>∗</sup> as:<ref>{{cite web |url=http://www.sworld.com.au/steven/space/atmosphere/ |title=Standard Atmospheres |accessdate=2007-01-07}}</ref><ref name="USSA1976">{{Cite book
| last = NOAA, NASA, USAF
| title = U.S. Standard Atmosphere, 1976
| publisher = U.S. Government Printing Office, Washington, D.C.
| date = 1976
| url = https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539_1977009539.pdf
| id = NOAA-S/T 76-1562}} Part 1, p. 3, (Linked file is 17 Meg)</ref>

:''R''<sup>∗</sup> = {{val|8.31432|e=3|u=N⋅m⋅kmol<sup>−1</sup>⋅K<sup>−1</sup>}}.

Note the use of kilomole units resulting in the factor of 1,000 in the constant. The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant.<ref name="USSA1976"/> This disparity is not a significant departure from accuracy, and USSA1976 uses this value of ''R''<sup>∗</sup> for all the calculations of the standard atmosphere. When using the [[International Organization for Standardization|ISO]] value of ''R'', the calculated pressure increases by only 0.62&nbsp;[[pascal (unit)|pascal]] at 11&nbsp;kilometers (the equivalent of a difference of only 17.4&nbsp;centimeters or 6.8&nbsp;inches) and an increase of 0.292&nbsp;Pa at 20&nbsp;km (the equivalent of a difference of only 33.8&nbsp;cm or 13.2&nbsp;in).

==References==
<!--See [[Wikipedia:Footnotes]] for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
{{reflist}}

==External links==
* ''[http://calculator.tutorvista.com/chemistry/567/ideal-gas-law-calculator.html Ideal gas calculator]'' – Ideal gas calculator provides the correct information for the moles of gas involved.
* [http://www.engineeringtoolbox.com/individual-universal-gas-constant-d_588.html Individual Gas Constants and the Universal Gas Constant] – Engineering Toolbox
{{Mole concepts}}

{{DEFAULTSORT:Gas Constant}}
[[Category:Ideal gas]]
[[Category:Physical constants]]
[[Category:Amount of substance]]