Gas constant

The gas constant (also known as the molar gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol $R$ or $R$ . It is equivalent to the Boltzmann constant, but expressed in units of energy per temperature increment per 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, the Arrhenius equation, and the Nernst equation.

Values of R[1]	Units
SI Units
8.31446261815324	J⋅K⁻¹⋅mol⁻¹
8.31446261815324	m³⋅Pa⋅K⁻¹⋅mol⁻¹
8.31446261815324	kg⋅m²·K⁻¹⋅mol⁻¹s⁻²
8.31446261815324×10³	L⋅Pa⋅K⁻¹⋅mol⁻¹
8.31446261815324×10⁻²	L⋅bar⋅K⁻¹⋅mol⁻¹
US Customary Units
0.730240507295273	atm⋅ft³⋅lb⋅mol⁻¹°R⁻¹
10.731557089016	psi⋅ft³⋅⋅lb⋅mol⁻¹°R⁻¹
1.985875279009	BTU⋅⋅lb⋅mol⁻¹°R⁻¹
Other Common Units
297.049031214	in. H₂O⋅ft³⋅lb⋅mol⁻¹°R⁻¹
554.984319180	torr⋅ft³⋅lb⋅mol⁻¹°R⁻¹
0.082057366080960	L⋅atm⋅K⁻¹⋅mol⁻¹
62.363598221529	L⋅Torr⋅K⁻¹⋅mol⁻¹
1.98720425864083...×10⁻³	kcal⋅K⁻¹⋅mol⁻¹
8.20573660809596...×10⁻⁵	m³⋅atm⋅K⁻¹⋅mol⁻¹
8.31446261815324×10⁷	erg⋅K⁻¹⋅mol⁻¹

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 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_A multiplied by the Boltzmann constant ( $k B$ or $k$ ):

R=N_{\rm {A}}k_{,}\,

Since the 2019 redefinition of SI base units, which came into effect on 20 May 2019, both N_A and k are defined with exact numerical values when expressed in SI units.[2] As a consequence, the value of the gas constant is also exactly defined, at precisely 8.31446261815324 J⋅K⁻¹⋅mol⁻¹.

Some have suggested that it might be appropriate to name the symbol R the Regnault constant in honour of the 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.[3][4]

The gas constant occurs in the ideal gas law, as follows:

PV=nRT=mR_{\rm {specific}}T

where P is the absolute pressure (SI unit pascals), V is the volume of gas (SI unit cubic metres), n is the 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_specific 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:

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 unit area, the gas equation can also be written as:

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={\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={\frac {\mathrm {work} }{\mathrm {amount} \times \mathrm {temperature} }}

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 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 meter.

Otherwise, we can also say that:

\mathrm {force} ={\frac {\mathrm {mass} \times \mathrm {length} }{(\mathrm {time} )^{2}}}

Therefore, we can write R as:

R={\frac {\mathrm {mass} \times \mathrm {length} ^{2}}{\mathrm {amount} \times \mathrm {temperature} \times (\mathrm {time} )^{2}}}

And so, in SI base units:

R = 8.31446261815324 kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹.

Relationship with the Boltzmann constant

The Boltzmann constant k_B (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

R=N_{\rm {A}}k_{\rm {B}},\,

where N_A is the Avogadro constant. For example, the ideal gas law in terms of Boltzmann's constant is

PV=k_{\rm {B}}NT.

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

P=k_{\rm {B}}nT.

where n is the number density.

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_a(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 c_a(0, T). The value of R is then obtained from the relation

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, T_TPW = 273.16 K by definition of the kelvin;
A_r(Ar) is the relative atomic mass of argon and M_u = 10⁻³ kg⋅mol⁻¹.

However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other precisely defined physical constants.

Specific gas constant

R_specific for dry air	Units
287.058	J⋅kg⁻¹⋅K⁻¹
53.3533	ft⋅lbf⋅lb⁻¹⋅°R⁻¹
1,716.49	ft⋅lbf⋅slug⁻¹⋅°R⁻¹
Based on a mean molar mass for dry air of 28.9645 g/mol.

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

R_{\rm {specific}}={\frac {R}{M}}

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.

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 heats for a calorically perfect gas and a thermally perfect gas.

R_{\rm {specific}}=c_{\rm {p}}-c_{\rm {v}}\

where c_p is the specific heat for a constant pressure and c_v is the specific heat for a constant volume.[5]

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 units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.[6]

U.S. Standard Atmosphere

The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R^∗ as:[7][8]

R^∗ = 8.31432×10³ N⋅m⋅kmol⁻¹⋅K⁻¹.

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.[8] 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 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and an increase of 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 which gave the constant an exact value.

References

"2018 CODATA Value: molar gas constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
"Proceedings of the 106th meeting" (PDF). 16–20 October 2017.
Jensen, William B. (July 2003). "The Universal Gas Constant R". J. Chem. Educ. 80 (7): 731. Bibcode:2003JChEd..80..731J. doi:10.1021/ed080p731.
"Ask the Historian: The Universal Gas Constant — Why is it represented by the letter R?" (PDF).
Anderson, Hypersonic and High-Temperature Gas Dynamics, AIAA Education Series, 2nd Ed, 2006
Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000
"Standard Atmospheres". Retrieved 2007-01-07.
NOAA, NASA, USAF (1976). U.S. Standard Atmosphere, 1976 (PDF). U.S. Government Printing Office, Washington, D.C. NOAA-S/T 76-1562.CS1 maint: multiple names: authors list (link) Part 1, p. 3, (Linked file is 17 Meg)

External links

Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved.
Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox

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[physconst-R-1] "2018 CODATA Value: molar gas constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

[2] "Proceedings of the 106th meeting" (PDF). 16–20 October 2017.

[Jensen-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.

[JensenReprint-4] "Ask the Historian: The Universal Gas Constant — Why is it represented by the letter R?" (PDF).

[5] Anderson, Hypersonic and High-Temperature Gas Dynamics, AIAA Education Series, 2nd Ed, 2006

[6] Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000

[7] "Standard Atmospheres". Retrieved 2007-01-07.

[USSA1976-8] NOAA, NASA, USAF (1976). U.S. Standard Atmosphere, 1976 (PDF). U.S. Government Printing Office, Washington, D.C. NOAA-S/T 76-1562.CS1 maint: multiple names: authors list (link) Part 1, p. 3, (Linked file is 17 Meg)

Mole concepts
Constants	Avogadro constant Boltzmann constant Gas constant
Physical quantities	Mass concentration Molar concentration Molality Mass Volume Density Mole fraction Mass fraction Amount of substance Molar mass Atomic mass Particle number Pressure Thermodynamic temperature Molar volume Specific volume
Laws	Charles's law Boyle's law Gay-Lussac's law Combined gas law Avogadro's law Ideal gas law