Editing Gas constant

{{short description|Physical constant  equivalent to the Boltzmann constant, but in different units}}
{| class="wikitable" style="margin: 0 0 0 0.5em; float: right;"
! Value of {{math|''R''}}{{physconst|R|ref=only}}
! Unit
|-
| colspan="2" |'''[[International System of Units|SI units]]'''
|-
| {{val|8.31446261815324}}
| [[joule|J]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|8.31446261815324}}
| [[cubic metre|m<sup>3</sup>]]⋅[[pascal (unit)|Pa]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|8.31446261815324}}
| [[Kilogram|kg]]⋅[[Metre|m]]<sup>2</sup>⋅[[second|s]]<sup>−2</sup>⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| colspan="2" |'''Other common units'''
|-
| {{val|8314.46261815324}}
| [[litre|L]]⋅[[Pascal (unit)|Pa]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|8.31446261815324}}
| [[litre|L]]⋅[[pascal (unit)|kPa]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|0.0831446261815324}}
| [[litre|L]]⋅[[bar (unit)|bar]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|8.31446261815324|e=7}}
| [[erg (unit)|erg]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|0.730240507295273}}
| [[Atmosphere (unit)|atm]]⋅[[Foot (unit)|ft]]<sup>3</sup>⋅[[Mole (unit)#Similar units|lbmol]]<sup>−1</sup>⋅[[Rankine scale|°R]]<sup>−1</sup>
|-
| {{val|10.731577089016}}
| [[Pounds per square inch|psi]]⋅[[Foot (unit)|ft]]<sup>3</sup>⋅[[Mole (unit)#Similar units|lbmol]]<sup>−1</sup>⋅[[Rankine scale|°R]]<sup>−1</sup>
|-
| {{val|1.985875279009}}
| [[British thermal unit|BTU]]⋅[[Mole (unit)#Similar units|lbmol]]<sup>−1</sup>⋅[[Rankine scale|°R]]<sup>−1</sup>
|-
| {{val|297.031214}}
| [[Inch of water|inH<sub>2</sub>O]]⋅[[Foot (unit)|ft]]<sup>3</sup>⋅[[Mole (unit)#Similar units|lbmol]]<sup>−1</sup>⋅[[Rankine scale|°R]]<sup>−1</sup>
|-
| {{val|554.984319180}}
| [[torr]]⋅[[Foot (unit)|ft]]<sup>3</sup>⋅[[Mole (unit)#Similar units|lbmol]]<sup>−1</sup>⋅[[Rankine scale|°R]]<sup>−1</sup>
|-
| {{val|0.082057366080960}}
| [[litre|L]]⋅[[atmosphere (unit)|atm]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|62.363598221529}}
| [[litre|L]]⋅[[torr]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|1.98720425864083|end=...}}
| [[calorie|cal]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
| {{val|8.20573660809596|e=-5|end=...}}
| [[cubic metre|m<sup>3</sup>]]⋅[[atmosphere (unit)|atm]]⋅[[kelvin|K]]<sup>−1</sup>⋅[[mole (unit)|mol]]<sup>−1</sup>
|-
|}
[[File:Heating-gas-at-constant-pressure-and-constant-volume.svg|thumb|Heating gas at constant pressure and constant-volume]]
The '''molar gas constant''' (also known as the '''gas constant''', '''universal gas constant''', or '''ideal gas constant''') is denoted by the symbol {{math|''R''}} or {{math|{{overline|''R''}}}}. It is the molar equivalent to the [[Boltzmann constant]], expressed in units of [[energy]] per [[temperature|temperature increment]] per [[amount of substance]], 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]], the [[Arrhenius equation]], and the [[Nernst equation]].

The gas constant is the [[constant of proportionality]] that relates the energy scale in physics to the temperature scale and the scale used for [[amount of substance]].  Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of units of energy, temperature and amount of substance. The [[Boltzmann constant]] and the [[Avogadro constant]] were similarly determined, which separately relate energy to temperature and particle count to amount of substance.

The gas constant ''R'' is defined as the [[Avogadro constant]] ''N''<sub>A</sub> multiplied by the [[Boltzmann constant]] ''k'' (or ''k''<sub>B</sub>):
: <math>R = N_\text{A} k</math>
:: = {{physconst|NA|ref=no}} × {{physconst|k|ref=no}}
:: = {{val|8.31446261815324|u=J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>}}.

Since the [[2019 revision of the SI]], both ''N''<sub>A</sub> and ''k'' are defined with exact numerical values when expressed in SI units.<ref name="SI2019">
{{cite book
 | last1 = Newell
 | first1 = David B.
 | last2 = Tiesinga
 | first2 = Eite
 | year = 2019
 | title = The International System of Units (SI)
 | series = NIST Special Publication 330
 | publisher = National Institute of Standards and Technology
 | location = Gaithersburg, Maryland
 | url = https://www.nist.gov/si-redefinition/meet-constants
 | doi = 10.6028/nist.sp.330-2019
 | s2cid = 242934226
}}</ref> As a consequence, the SI value of the molar gas constant is exact.

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 the constant is elusive. The universal gas constant was apparently introduced independently by [[August Friedrich Horstmann]] (1873)<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 
 |author1-link=William B. Jensen}}</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> and [[Dmitri Mendeleev]] who reported it first on 12 September 1874.<ref name="Mendeleev2">
{{cite journal
 |title=An exert from the Proceedings of the Chemical Society's Meeting on Sept. 12, 1874
 |last=Mendeleev |first=Dmitri I.
 |journal= Journal of Russian Chemical-Physical Society, Chemical Part
 |volume= VI
 |issue= 7
 |date=September 12, 1874
 |pages=208–209
}}</ref> Using his extensive measurements of the properties of gases,<ref name="Mendeleev3">
{{cite book
 |title=On the elasticity of gases [Объ упругости газовъ]
 |last=Mendeleev |first=Dmitri I.
 |date=1875
 | publisher = A. M. Kotomin, St.-Petersburg
}}</ref><ref>[http://gallica.bnf.fr/ark:/12148/bpt6k95208b/f12.image.r=mendeleev.langEN D. Mendeleev. On the elasticity of gases. 1875 (in Russian)] {{free access}}</ref>
Mendeleev also calculated it with high precision, within 0.3% of its modern value.<ref name="Mendeleev">
{{cite journal
 |title=Mendeleef's researches on Mariotte's law 1
 |last=Mendeleev |first=Dmitri I.
 |journal= Nature
 |volume= 15
 |issue= 388
 |date=March 22, 1877
 |pages=498–500
 |doi=10.1038/015498a0 |doi-access=free
 |bibcode=1877Natur..15..498D
}} {{free access}}</ref>

The gas constant occurs in the ideal gas law:
<math display="block">PV = nRT = m R_\text{specific} T,</math>
where ''P'' is the absolute [[pressure]], ''V'' is the volume of gas, ''n'' is the [[amount of substance]], ''m'' is the [[mass]], and ''T'' is the [[thermodynamic temperature]]. ''R''<sub>specific</sub> is the mass-specific gas constant. The gas constant is expressed in the same unit as [[molar heat]].

== 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 area of measurement, the gas equation can also be written as
: <math>R = \frac{ \dfrac{\text{force}}{\text{area}} \times \text{volume} }
                 { \text{amount} \times \text{temperature} }.
</math>

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

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

The physical significance of ''R'' is work per mole per kelvin. It may be expressed in any set of units representing work or energy (such as [[joule]]s), units representing 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 constant]]).

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

Otherwise, we can also say that
: <math>\text{force} = \frac{ \text{mass} \times \text{length} }
                            { (\text{time})^2 }.
</math>

Therefore, we can write ''R'' as
: <math>R = \frac{ \text{mass} \times \text{length}^2 }
                 { \text{amount} \times \text{temperature} \times (\text{time})^2 }.
</math>

And so, in terms of [[SI base units]],
: ''R'' = {{physconst|R|unit=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> (alternatively ''k'') may be used in place of the molar gas constant by working in pure particle count, ''N'', rather than amount of substance, ''n'', since
: <math>R = N_\text{A} k_\text{B},</math>
where ''N''<sub>A</sub> is the [[Avogadro constant]].
For example, the [[ideal gas law]] in terms of the Boltzmann constant is
: <math>pV = Nk_\text{B} 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 = n k_\text{B} T,</math>
where ''n'' = ''N''/''V'' is the [[number density]].
Finally, by defining the [[kinetic energy]] associated to the temperature,
: <math>T := k_\text{B} T,</math>
the equation becomes simply
: <math>p = n T,</math>
which is the form usually encountered in statistical mechanics and other branches of theoretical physics.

== Measurement and replacement with defined value ==
As of 2006, the most precise measurement of ''R'' had been obtained by measuring the [[speed of sound]] ''c''<sub>a</sub>(''P'',&nbsp;''T'') in [[argon]] at the temperature ''T'' of the [[triple point of water]]  at different [[pressure]]s ''P'', and [[extrapolation|extrapolating]] to the zero-pressure limit ''c''<sub>a</sub>(0,&nbsp;''T''). The value of ''R'' is then obtained from the relation
: <math>c_\text{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\text{r}(\text{Ar}) M_\text{u}}},</math>
where
: ''γ''<sub>0</sub> is the [[heat capacity ratio]] (5/3 for monatomic gases such as argon);
: ''T'' is the temperature, ''T''<sub>TPW</sub> = 273.16&nbsp;K by the definition of the kelvin at that time;
: ''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>}} as defined at the time.

However, following the [[2019 revision of the SI]], ''R'' now has an exact value defined in terms of other exactly defined physical constants.

== Specific gas constant ==
{| class="wikitable" style="float: right;"
! ''R''<sub>specific</sub><br />for dry air<ref>Based on a mean molar mass for [[Atmosphere of Earth#Composition|dry air]] of 28.964917&nbsp;g/mol.</ref>
! Unit
|-
| 287.052874
| J⋅kg<sup>−1</sup>⋅K<sup>−1</sup>
|-
| 53.3523
| ft⋅[[Pound-force|lbf]]⋅[[Pound (mass)|lb]]<sup>−1</sup>⋅°R<sup>−1</sup>
|-
| 1,716.46
| ft⋅[[Pound-force|lbf]]⋅[[slug (unit)|slug]]<sup>−1</sup>⋅°R<sup>−1</sup>
|}

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_\text{specific} = \frac{R}{M}.</math>

Just as the molar 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_\text{specific} = \frac{k_\text{B}}{m}.</math>

Another important relationship comes from thermodynamics. [[Mayer's relation]] relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas:
: <math> R_\text{specific} = c_p - c_V,</math>
where ''c<sub>p</sub>'' is the [[specific heat capacity]] for a constant pressure, and ''c<sub>V</sub>'' is the specific heat capacity 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 unit of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.<ref name="Moran2018">
{{cite book
 | last1 = Moran | first1 = Michael J.
 | last2 = Shapiro | first2 = Howard N.
 | last3 = Boettner | first3 = Daisie D.
 | last4 = Bailey | first4 = Margaret B.
 | title = Fundamentals of Engineering Thermodynamics
 | publisher = Wiley
 | location = Hoboken, New Jersey
 | edition = 9th
 | url = https://www.wiley.com/en-us/Fundamentals+of+Engineering+Thermodynamics%2C+9th+Edition-p-9781119391388
 | year = 2018
}}</ref>

In case of air, using the perfect gas law and the [[standard sea-level conditions]] (SSL) (air density ''ρ''<sub>0</sub> = 1.225&nbsp;kg/m<sup>3</sup>, temperature ''T''<sub>0</sub> = 288.15&nbsp;[[Kelvin|K]] and pressure ''p''<sub>0</sub> = {{val|101325|ul=Pa}}), we have that ''R''<sub>air</sub> = ''P''<sub>0</sub>/(''ρ''<sub>0</sub>''T''<sub>0</sub>) = {{val|287.052874247|u=J·kg<sup>−1</sup>·K<sup>−1</sup>}}. Then the molar mass of air is computed by ''M''<sub>0</sub> = ''R''/''R''<sub>air</sub> = {{val|28.964917|u=g/mol}}.<ref name="ICAO manual">{{cite book |title=Manual of the US Standard Atmosphere |date=1962 |publisher=National Aeronautics and Space Administration |pages=7–11 |edition=3 |url=https://ntrs.nasa.gov/api/citations/19630003300/downloads/19630003300.pdf}}</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 |access-date=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>}} = {{val|8.31432||u=J⋅K<sup>−1</sup>⋅mol<sup>−1</sup>}}.

Note the use of the kilomole, with the resulting factor of {{val|1000}} 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;kilometres (the equivalent of a difference of only 17.4&nbsp;centimetres or 6.8&nbsp;inches) and 0.292&nbsp;Pa at 20&nbsp;km (the equivalent of a difference of only 33.8&nbsp;cm or 13.2&nbsp;in).

Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value.

== 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] {{Webarchive|url=https://web.archive.org/web/20120715222930/http://calculator.tutorvista.com/chemistry/567/ideal-gas-law-calculator.html |date=2012-07-15 }}'' – 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]]
[[Category:Statistical mechanics]]
[[Category:Thermodynamics]]
[[Category:Molar quantities]]