Which equation describes the entropy of mixing of two perfect gases?

The equation that correctly describes the entropy of mixing of two perfect gases involves the negative of the product of the number of moles, the ideal gas constant, and the sum of the mole fractions multiplied by the natural logarithm of the mole fractions of each component.

In the case of mixing two perfect gases, the entropy change can be derived from statistical mechanics and thermodynamics principles. When two gases are mixed, the total entropy increases because the number of accessible microstates increases due to the mixing. This is quantitatively expressed by the equation:

Δ_mixS = -nR(x_A ln x_A + x_B ln x_B)

Here, n is the total number of moles, R is the universal gas constant, and x_A and x_B are the mole fractions of the two gases. Each term x_A ln x_A and x_B ln x_B reflects the contribution to the entropy change from the respective components accounting for their statistical contributions to the total entropy upon mixing.

The negative sign in front indicates that as the gases mix, the change in entropy is positive, consistent with the idea that mixing results in greater disorder. This is a foundational principle in thermodynamics, relating to the second law, which states that spontaneous processes, such as mixing