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I know gas will increase in entropy as it goes toward equilibrium, but what if there are multiple gases of different densities?

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Got an answer from here: https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node54.html

Summary: Consider an insulated rigid container of gas separated into two halves by a heat conducting partition so the temperature of the gas in each part is the same. One side contains air, the other side another gas, say argon, both regarded as ideal gases. The mass of gas in each side is such that the pressure is also the same.

The entropy of this system is the sum of the entropies of the two parts: $ S_\textrm{system} = S_\textrm{air} + S_\textrm{argon}$ . Suppose the partition is taken away so the gases are free to diffuse throughout the volume. For an ideal gas, the energy is not a function of volume, and, for each gas, there is no change in temperature.

The entropy change of each gas is thus the same as that for a reversible isothermal expansion from the initial specific volume $ v_i$ to the final specific volume, $ v_f$ .

The entropy change of the system is

$\displaystyle \Delta S_\textrm{system} = \Delta S_\textrm{air} + \Delta S_\text... ...\textrm{argon}\ln\left(\frac{v_{f,\textrm{argon}}}{v_{i,\textrm{argon}}}\right)$

Conclusion: This equation states that there is an entropy increase due to the increased volume that each gas is able to access.

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