How do you calculate mixing Gas entropy?

Answer 1

The entropy of mixing is the change in theconfiguration entropy, an extensivethermodynamic quantity, when two differentchemical substances or components are mixed and the volume available for each substance to explore is changed. The name entropy of mixing is misleading, since it is not the intermingling of the particles that creates the entropy change, but rather the change in the available volume per particle.[1] This entropy change is positive when there is more uncertainty about thespatial locations of the different kinds ofmolecules. We assume that the mixing process has reached thermodynamic equilibrium so that the mixture is uniform and homogeneous. If the substances being mixed are initially at different temperatures and pressures, there will, of course, be an additional entropy increase in the mixed substance due to these differences being equilibrated, but if the substances being mixed are initially at the same temperature and pressure, the entropy increase will be entirely due to the entropy of mixing.

The entropy of mixing may be calculated by Gibbs' Theorem which states that when two different substances mix, the entropy increase upon mixing is equal to the entropy increase that would occur if the two substances were to expand alone into the mixing volume. (In this sense, then the term "entropy of mixing" is a misnomer, since the entropy increase is not due to any "mixing" effect.) Nevertheless, the two substances must be different for the entropy of mixing to exist. This is the Gibbs paradoxwhich states that if the two substances are identical, there will be no entropy change, yet the slightest detectable difference between the two will yield a considerable entropy change, and this is just the entropy of mixing. In other words, the entropy of mixing is not a continuous function of the degree of difference between the two substances.

For the mixing of two ideal gases upon removal of a dividing partition, the entropy of mixing is given by:(1)[tex]\Delta S = n1R\ln((V1+V2)/V1) + n2R\ln((V1+V2)/V2)[/tex]

where is the gas constant, n1 and n2 are the number of moles of the respective gases and V1, V2 are their respective initial volumes. After the removal of the partition, each gas particle may explore a larger volume, which causes the entropy change. Note that this equation is only valid if both compartments have the same initial pressure.

Note that the mixing involves no heat flow (just the irreversible process of mixing). However, the change in entropy is defined as the integral of dQ/T over the reversible path between the initial and final states. The reversible path between these two states is a quasi-static isothermal expansion. Such a path DOES involve heat flow into the gas: dQ = PdV = nRTdV/V where T is constant (dU = 0). The above equation (1) for entropy is determined by taking the integral of dQ/T over such a path.