Gas Constant

The gas constant, symbol $R$, is a universal physical constant that connects thermodynamic temperature with energy per unit amount of substance. It is best known through the ideal gas law, $pV=nRT$, but also appears throughout thermodynamics and physical chemistry. Its exact value in the International System of Units is $8.31446261815324\ \mathrm{J\,mol^{-1}\,K^{-1}}$. Unlike a gas-specific material property, $R$ is the same for every chemical species. (physics.nist.gov)

Definition and exact value

The gas constant is the product of the Avogadro constant, $N_{\mathrm A}$, and the Boltzmann constant, $k_{\mathrm B}$:

The Boltzmann constant expresses the relationship between temperature and energy on a per-particle basis; multiplication by the number of particles per mole gives the corresponding molar constant. In the International System of Units (SI), the defining values are exactly

Their product therefore gives an exact value of $R$, rather than an experimentally adjusted value with an associated uncertainty. This status dates from the revised SI definitions that took effect on May 20, 2019. Exactness of the constant does not eliminate uncertainty in measurements made using it. (bipm.org)

Units and numerical forms

The SI unit of $R$ is the joule per mole per kelvin. Because a pressure–volume product has the dimensions of energy, the same unit can be written as $\mathrm{Pa\,m^3\,mol^{-1}\,K^{-1}}$. Common alternative numerical forms, obtained by unit conversion, include

and

These are not different physical constants: their numerical differences reflect different units. A calculation must use a value compatible with the units of pressure, volume, energy, and amount of substance. Rounded forms such as $8.314$ are adequate for many calculations but do not represent the complete exact SI value. (iupac.org)

Role in the ideal gas law

The ideal gas equation of state is

where $p$ is absolute pressure, $V$ is volume, $n$ is amount of substance, and $T$ is thermodynamic temperature. Temperature must be expressed on an absolute scale, conventionally in kelvins, and pressure must be measured relative to vacuum rather than as gauge pressure. At fixed $n$ and $T$, the equation predicts an inverse relationship between pressure and volume; at fixed $n$ and $p$, volume is proportional to temperature. (goldbook.iupac.org)

Dividing by $n$ gives the molar-volume relationship $V_{\mathrm m}=RT/p$. As an illustrative calculation, one mole of ideal gas at $273.15\ \mathrm K$ and $100\,000\ \mathrm{Pa}$ occupies approximately $22.711\ \mathrm L$. At the same temperature but $101\,325\ \mathrm{Pa}$, it occupies approximately $22.414\ \mathrm L$. Thus a quoted gas volume requires an explicit temperature and pressure; a “standard” molar volume is not an additional universal constant. (physics.nist.gov)

Universal and specific gas constants

Engineering equations frequently use the specific gas constant, $R_{\mathrm s}$, defined by

where $M$ is the molar mass of the gas. If $M$ is in kilograms per mole, $R_{\mathrm s}$ has units of joules per kilogram per kelvin. Substituting $n=m/M$, with $m$ denoting mass, yields

where $\rho=m/V$ is density. Unlike $R$, the specific constant depends on the gas or mixture composition. Some engineering texts denote it simply by $R$, so the units and definition distinguish it from the universal molar constant. (www1.grc.nasa.gov)

Thermodynamic and chemical applications

For an ideal gas, the molar heat capacities at constant pressure and constant volume satisfy

This follows from the definition of enthalpy, $H=U+pV$, and the ideal gas law. The difference reflects the expansion work associated with heating at constant pressure. It does not mean that $R$ itself is the heat capacity of every gas: individual heat capacities depend on molecular structure and temperature. Here $U$ denotes internal energy. (www1.grc.nasa.gov)

In chemical equilibrium, the standard molar reaction Gibbs energy and the dimensionless equilibrium constant are related by

The factor $RT$ also appears in expressions for chemical potential, connecting molar energy with logarithms of activity. These uses extend beyond gases to solutions and other phases. (iupac.org)

In chemical kinetics, the Arrhenius equation takes the form

where $E_{\mathrm a}$ is molar activation energy and $A$ is the pre-exponential factor. The ratio $E_{\mathrm a}/RT$ is dimensionless; using a per-particle activation energy instead requires $k_{\mathrm B}T$. (goldbook.iupac.org)

Real gases and departures from ideality

For a real gas, departures from the ideal equation can be represented by the compressibility factor,

An ideal gas has $Z=1$. A value different from unity indicates a departure from ideal behavior, not a change in the universal gas constant. The same $R$ remains in the equation; the factor $Z$ describes the non-ideal pressure–volume–temperature relationship. (goldbook.iupac.org)