Gas constant

From the ideal gas law PV = nRT we get:

${\displaystyle R={\frac {PV}{nT}}}$

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:

${\displaystyle R={\frac {{\dfrac {\mathrm {force} }{\mathrm {area} }}\times \mathrm {volume} }{\mathrm {amount} \times \mathrm {temperature} }}}$

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

${\displaystyle R={\frac {{\dfrac {\mathrm {force} }{(\mathrm {length} )^{2}}}\times (\mathrm {length} )^{3}}{\mathrm {amount} \times \mathrm {temperature} }}={\frac {\mathrm {force} \times \mathrm {length} }{\mathrm {amount} \times \mathrm {temperature} }}}$

Since force × length = work:

${\displaystyle R={\frac {\mathrm {work} }{\mathrm {amount} \times \mathrm {temperature} }}}$

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:

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

Therefore, we can write R as:

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

And so, in terms of SI base units:

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

The Boltzmann constant k_B (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

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

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

${\displaystyle PV=Nk_{\rm {B}}T,}$

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

${\displaystyle P=\rho _{\rm {N}}k_{\rm {B}}T,}$

where ρ_N = N/V is the number density.

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

${\displaystyle c_{\mathrm {a} }(0,T)={\sqrt {\frac {\gamma _{0}RT}{A_{\mathrm {r} }(\mathrm {Ar} )M_{\mathrm {u} }}}},}$

where:

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

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

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.

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

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.

${\displaystyle R_{\rm {specific}}={\frac {k_{\rm {B}}}{m}}}$

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.

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

where c_p is the specific heat capacity for a constant pressure and c_v 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 R_air = P₀/(ρ₀T₀) = 287.052874247 J·kg⁻¹·K⁻¹. Then the molar mass of air is computed by M₀ = R/R_air = 28.964917 g/mol.[11]

The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R^∗ as:[12][13]

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

Note the use of kilomoles, 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.[13] 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 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.

• 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