Amagat's law

Amagat's law or the Law of Partial Volumes of 1880 describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) gases. Of use in chemistry and thermodynamics, Amagat's law states that the extensive volume V = N·v 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]

$N\cdot v(T,p)=\sum_{i=1}^K N_i\cdot 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 $V_1, V_2, \dots, V_n$ are considered to be the partial volumes of components in the gaseous mixture, then the total volume $V$ would be represented as:

$V = V_1 + V_2 + V_3 + \dots + V_n = \sum_{i} V_i$

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:

$B(T) = X_1B_1 + X_2B_2 + X_1X_2B_{1,2}\$

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

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

and

$B_{1,2} = (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.

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.