Amagat's law

Amagat's law or the Law of Partial Volumes describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) gases. Of use in chemistry and thermodynamics.

Overview

Amagat's law states that the extensive volume V = Nv of a gas mixture is equal to the sum of volumes V_i of the K component gases, if the temperature T and the pressure p remain the same:[1][2]

${\displaystyle N\,v(T,p)=\sum _{i=1}^{K}N_{i}\,v_{i}(T,p).}$

This is the experimental expression of volume as an extensive quantity. It is named after Emile Amagat.

According to Amagat's law of partial volume, the total volume of a non-reacting mixture of gases at constant temperature and pressure should be equal to the sum of the individual partial volumes of the constituent gases. So if ${\displaystyle V_{1},V_{2},\dots ,V_{n}}$ are considered to be the partial volumes of components in the gaseous mixture, then the total volume ${\displaystyle V}$ would be represented as:

${\displaystyle V=V_{1}+V_{2}+V_{3}+\dots +V_{n}=\sum _{i}V_{i}}$

Both Amagat's and Dalton's laws predict the properties of gas mixtures. Their predictions are the same for ideal gases. However, for real (non-ideal) gases, the results differ.[3] Dalton's law of partial pressures assumes that the gases in the mixture are non-interacting (with each other) and each gas independently applies its own pressure, the sum of which is the total pressure. Amagat's law assumes that the volumes of the component gases (again at the same temperature and pressure) are additive; the interactions of the different gases are the same as the average interactions of the components.

The interactions can be interpreted in terms of a second virial coefficient, B(T), for the mixture. For two components, the second virial coefficient for the mixture can be expressed as:

${\displaystyle B(T)=X_{1}B_{1}+X_{2}B_{2}+X_{1}X_{2}B_{1,2}\ }$

where the subscripts refer to components 1 and 2, the Xs are the mole fractions, and the Bs are the second virial coefficients. The cross term, B_1,2, of the mixture is given by:

${\displaystyle B_{1,2}=0\ }$ (Dalton's law)

and

${\displaystyle B_{1,2}={\frac {B_{1}+B_{2}}{2}}\ }$ (Amagat's law).

When the volumes of each component gas (same temperature and pressure) are very similar, then Amagat's law becomes mathematically equivalent to Vegard's law for solid mixtures.

Ideal gas mixture

When Amagat's law is valid and the gas mixture is made of ideal gases:

${\displaystyle {\frac {V_{i}}{V}}={\frac {\frac {n_{i}RT}{p}}{\frac {nRT}{p}}}={\frac {n_{i}}{n}}=x_{i}}$

where:

• ${\displaystyle p}$ is the pressure of the gas mixture,
• ${\displaystyle V_{i}={\frac {n_{i}RT}{p}}}$ is the volume of the i-component of the gas mixture,
• ${\displaystyle V=\sum V_{i}}$ is the total volume of the gas mixture,
• ${\displaystyle n_{i}}$ is the amount of substance of i-component of the gas mixture (in mol),
• ${\displaystyle n=\sum n_{i}}$ is the total amount of substance of gas mixture (in mol),
• ${\displaystyle R}$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
• ${\displaystyle T}$ is the absolute temperature of the gas mixture (in K),
• ${\displaystyle x_{i}={\frac {n_{i}}{n}}}$ is the mole fraction of the i-component of the gas mixture.

It follows that the mole fraction and volume fraction are the same. This is true also for other equation of state.

References

1. Amagat's law of additive volumes
2. Bejan, A. (2006). Advanced Engineering Thermodynamics (3rd ed.). John Wiley & Sons. ISBN 0471677639.
3. Noggle, J. H. (1996). Physical Chemistry (3rd ed.). New York: Harper Collins. ISBN 0673523411.