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From the ideal gas law PV = nRT we get:

R = P V n T {\displaystyle R={\frac {PV}{nT}}}

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:

R = f o r c e a r e a × v o l u m e a m o u n t × t e m p e r a t u r e {\displaystyle R={\frac {{\dfrac {\mathrm {force} }{\mathrm {area} }}\times \mathrm {volume} }{\mathrm {amount} \times \mathrm {temperature} }}}

Area and volume are (length)2 and (length)3 respectively. Therefore:

R = f o r c e ( l e n g t h ) 2 × ( l e n g t h ) 3 a m o u n t × t e m p e r a t u r e = f o r c e × l e n g t h a m o u n t × t e m p e r a t u r e {\displaystyle 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} }}}

Since force × length = work:

R = w o r k a m o u n t × t e m p e r a t u r e {\displaystyle R={\frac {\mathrm {work} }{\mathrm {amount} \times \mathrm {temperature} }}}

The physical significance of R is work per mole per degree. It may be expressed in any set of units representing work or energy (such as joules), units representing degrees of temperature on an absolute scale (such as kelvin or 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:

f o r c e = m a s s × l e n g t h ( t i m e ) 2 {\displaystyle \mathrm {force} ={\frac {\mathrm {mass} \times \mathrm {length} }{(\mathrm {time} )^{2}}}}

Therefore, we can write R as:

R = m a s s × l e n g t h 2 a m o u n t × t e m p e r a t u r e × ( t i m e ) 2 {\displaystyle R={\frac {\mathrm {mass} \times \mathrm {length} ^{2}}{\mathrm {amount} \times \mathrm {temperature} \times (\mathrm {time} )^{2}}}}

And so, in terms of SI base units:

R = 8.314462618... kg⋅m2⋅s−2⋅K−1⋅mol−1.

Relationship with the Boltzmann constant

The Boltzmann constant kB (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:

R = N A k B , {\displaystyle R=N_{\rm {A}}k_{\rm {B}},\,}

where NA is the Avogadro constant. For example, the ideal gas law in terms of the Boltzmann constant is:

p V = N k B T , {\displaystyle pV=Nk_{\rm {B}}T,}

where N is the number of particles (molecules in this case), or to generalize to an inhomogeneous system the local form holds:

p = n k B T , {\displaystyle p=nk_{\rm {B}}T,}

where n = N/V is the number density. Finally, by defining the kinetic energy associated to the temperature:

T := k B T , {\displaystyle T:=k_{\rm {B}}T,}

the equation becomes simply:

p = n T , {\displaystyle p=nT,}

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 ca(P, T) in argon at the temperature T of the triple point of water at different pressures P, and extrapolating to the zero-pressure limit ca(0, T). The value of R is then obtained from the relation:

c a ( 0 , T ) = γ 0 R T A r ( A r ) M u , {\displaystyle c_{\mathrm {a} }(0,T)={\sqrt {\frac {\gamma _{0}RT}{A_{\mathrm {r} }(\mathrm {Ar} )M_{\mathrm {u} }}}},}

where:

• γ0 is the heat capacity ratio (⁠5/3⁠ for monatomic gases such as argon);
• T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time;
• Ar(Ar) is the relative atomic mass of argon and Mu = 10−3 kg⋅mol−1 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

Rspecificfor dry air	Unit
287.052874	J⋅kg−1⋅K−1
53.3523	ft⋅lbf⋅lb−1⋅°R−1
1,716.46	ft⋅lbf⋅slug−1⋅°R−1
Based on a mean molar massfor dry air of 28.964917 g/mol.

The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture:

R s p e c i f i c = R M {\displaystyle R_{\rm {specific}}={\frac {R}{M}}}

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:

R s p e c i f i c = k B m {\displaystyle R_{\rm {specific}}={\frac {k_{\rm {B}}}{m}}}

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:

R s p e c i f i c = c p − c v   {\displaystyle R_{\rm {specific}}=c_{\rm {p}}-c_{\rm {v}}\ }

where cp is the specific heat capacity for a constant pressure and cv is the specific heat capacity for a constant volume.⁹

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 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.¹⁰

In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = 101325 Pa), we have that Rair = P0/(ρ0T0) = 287.052874247 J·kg−1·K−1. Then the molar mass of air is computed by M0 = R/Rair = 28.964917 g/mol.¹¹

U.S. Standard Atmosphere

The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as:¹²¹³

R∗ = 8.31432×103 N⋅m⋅kmol−1⋅K−1 = 8.31432 J⋅K−1⋅mol−1.

Note the use of the kilomole, with the resulting factor of 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.¹⁴ This disparity is not a significant departure from accuracy, and USSA1976 uses this value of R∗ for all the calculations of the standard atmosphere. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometres (the equivalent of a difference of only 17.4 centimetres or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in).

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

External links

• Ideal gas calculator Archived 2012-07-15 at the Wayback Machine – Ideal gas calculator provides the correct information for the moles of gas involved.
• Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox

References

1. "2022 CODATA Value: molar gas constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. May 2024. Retrieved 2024-05-18. https://physics.nist.gov/cgi-bin/cuu/Value?r ↩

2. Newell, David B.; Tiesinga, Eite (2019). The International System of Units (SI). NIST Special Publication 330. Gaithersburg, Maryland: National Institute of Standards and Technology. doi:10.6028/nist.sp.330-2019. S2CID 242934226. https://www.nist.gov/si-redefinition/meet-constants ↩

3. Jensen, William B. (July 2003). "The Universal Gas Constant R". J. Chem. Educ. 80 (7): 731. Bibcode:2003JChEd..80..731J. doi:10.1021/ed080p731. /wiki/William_B._Jensen ↩

4. "Ask the Historian: The Universal Gas Constant — Why is it represented by the letter R?" (PDF). http://www.che.uc.edu/jensen/W.%20B.%20Jensen/Reprints/100.%20Gas%20Constant.pdf ↩

5. Mendeleev, Dmitri I. (September 12, 1874). "An exert from the Proceedings of the Chemical Society's Meeting on Sept. 12, 1874". Journal of Russian Chemical-Physical Society, Chemical Part. VI (7): 208–209. ↩

6. Mendeleev, Dmitri I. (1875). On the elasticity of gases [Объ упругости газовъ]. A.M. Kotomin, St.-Petersburg. ↩

7. D. Mendeleev. On the elasticity of gases. 1875 (in Russian) http://gallica.bnf.fr/ark:/12148/bpt6k95208b/f12.image.r=mendeleev.langEN ↩

8. Mendeleev, Dmitri I. (March 22, 1877). "Mendeleef's researches on Mariotte's law 1". Nature. 15 (388): 498–500. Bibcode:1877Natur..15..498D. doi:10.1038/015498a0. https://doi.org/10.1038%2F015498a0 ↩

9. Anderson, Hypersonic and High-Temperature Gas Dynamics, AIAA Education Series, 2nd Ed, 2006 ↩

10. Moran, Michael J.; Shapiro, Howard N.; Boettner, Daisie D.; Bailey, Margaret B. (2018). Fundamentals of Engineering Thermodynamics (9th ed.). Hoboken, New Jersey: Wiley. https://www.wiley.com/en-us/Fundamentals+of+Engineering+Thermodynamics%2C+9th+Edition-p-9781119391388 ↩

11. Manual of the US Standard Atmosphere (PDF) (3 ed.). National Aeronautics and Space Administration. 1962. pp. 7–11. https://ntrs.nasa.gov/api/citations/19630003300/downloads/19630003300.pdf ↩

12. "Standard Atmospheres". Retrieved 2007-01-07. http://www.sworld.com.au/steven/space/atmosphere/ ↩

13. NOAA, NASA, USAF (1976). U.S. Standard Atmosphere, 1976 (PDF). U.S. Government Printing Office, Washington, D.C. NOAA-S/T 76-1562.{{cite book}}: CS1 maint: multiple names: authors list (link) Part 1, p. 3, (Linked file is 17 Meg) https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539_1977009539.pdf ↩

14. NOAA, NASA, USAF (1976). U.S. Standard Atmosphere, 1976 (PDF). U.S. Government Printing Office, Washington, D.C. NOAA-S/T 76-1562.{{cite book}}: CS1 maint: multiple names: authors list (link) Part 1, p. 3, (Linked file is 17 Meg) https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539_1977009539.pdf ↩