Thermodynamics and Propulsion

7.1 Entropy Change in Mixing of Two Ideal Gases

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 energy of the overall system is unchanged, the two gases were at the same temperature initially, so the final temperature is the same as the initial 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, $v_f$ . For a mass $m$ of ideal gas, the entropy change is $\Delta S = mR\ln(v_f/v_i)$ . The entropy change of the system is

$$\Delta S_\textrm{system} = \Delta S_\textrm{air} + \Delta S_\text... ...\textrm{argon}\ln\left(\frac{v_{f,\textrm{argon}}}{v_{i,\textrm{argon}}}\right)$$ (7..1)

Equation (7.1) states that there is an entropy increase due to the increased volume that each gas is able to access.

Examining the mixing process on a molecular level gives additional insight. Suppose we were able to see the gas molecules in different colors, say the air molecules as white and the argon molecules as red. After we took the partition away, we would see white molecules start to move into the red region and, similarly, red molecules start to come into the white volume. As we watched, as the gases mixed, there would be more and more of the different color molecules in the regions that were initially all white and all red. If we moved further away so we could no longer pick out individual molecules, we would see the growth of pink regions spreading into the initially red and white areas. In the final state, we would expect a uniform pink gas to exist throughout the volume. There might be occasional small regions which were slightly more red or slightly more white, but these fluctuations would only last for a time on the order of several molecular collisions.

In terms of the overall spatial distribution of the molecules, we would say this final state was more random, more mixed, than the initial state in which the red and white molecules were confined to specific regions. Another way to say this is in terms of ``disorder;'' there is more disorder in the final state than in the initial state. One view of entropy is thus that increases in entropy are connected with increases in randomness or disorder. This link can be made rigorous and is extremely useful in describing systems on a microscopic basis. While we do not have scope to examine this topic in depth, the purpose of this chapter is to make plausible the link between disorder and entropy through a statistical definition of entropy.