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# Entropy of mixing

Entropy of mixing
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The property relationships derived also enable us to calculate the entropy of mixing of two ideal gases. Consider the entropy change when nA moles of an ideal gas A and nB moles of an ideal gas B mix together. The actual situation is like this. We have initially nA moles of an ideal gas A occupying volume VA, and nB moles of an ideal gas B occupying volume VB. Since they are ideal gases at the same temperature, number of moles is proportional to its volume. After mixing together, they become a random mixture of A and B gas, with total moles of (nA+nB) and total volume of (VA and VB).
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For the convenience, we can think of this situation as the sum of two events. Let’s split the system into two. A gas occupies volume VA and the counter part is vacuum with volume VB. B gas occupies volume VB and the counter part is vacuum with volume VA also. Now consider the isothermal expansion of each into the counter part. nA moles of A gas expand against the vacuum until the volume becomes (VA+VB). nB moles of B gas expand against the vacuum until the volume becomes (VA+VB). The initial and final states are the same with our original mixing process.
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Since entropy is a state function, the entropy change of the original mixing is the same with the entropy change of these two processes. Just add entropy changes of those two. For the calculation, we need entropy changes of isothermal expansion, (dS over dV) at constant T. To determine which property relation works for this case, consider which thermodynamic potential has V, T as variables. It is F, the Helmholtz free energy. Let’s start from A gas. For A, the mixing process is equivalent to isothermal expansion from VA to VB. To get (dS over dV) at constant T, consider dF. dF is - SdT - PdV.
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Property relation from here is (dS over dV) at constant T equals to (dP over dT) at constant V, and it is nR over V for ideal gases. Entropy change for A is the difference of entropy between inital and final states. So it is integration of (nA R over V) dV from VA to VT. The entropy change for A is thus nA R log (VT over VA). For B, the mixing process is equivalent to isothermal expansion from VB to VT. Similar calculation gives the entropy change for B as nB R log (VT over VB). We have a underlying assumption that enables us to spilt the original mixing into the individual expansion against vacuum.
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It is that they do not interact with each other since they are ideal. Again, the real situation is mixing of initial nA moles of A and nB mole of B into a random mixture. The total volume VT is VA + VB, and the total mole nT is nA + nB. And the mole fraction of A and B is the ratio of moles of A and B to the total moles, respectively. The total entropy change is the sum of entropy change of A and B since they do not interact. So the total entropy change is nA R log ((VA+VB) over VA) + nB R log ((VA+VB) over VB). Now, let’s divide both sides by (nA+nB) to give molar quantity.
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The molar entropy of mixing is total entropy change divided by the total moles. So the entropy of mixing is like this. nA over (nA+nB) times R log (VA over (VA+VB)) - nB over (nA+nB) times R times log (VB over (VA+VB)). The ratios in this equations are mole fractions. So, entropy of mixing can be summarized as - R times (xA log xA+xB log XB) for ideal mixing. Ideal mixing means that there is no interaction between the components. For multicoponents system, the generalized entropy of mixing is the summation of - R times (xi log xi) over each component. Here, we need a caution. It is a ideal mixing case. The real entropy of mixing is (entropy of mixing ideal + some excess).

In this video, we focus our attention to derive entropy of mixing for ideal gases.

Since the molecules of ideal gases do not interact, we can spilit the mixing of two ideal gases into two events: Expansion of each gas system into the final volume of the mixture. The entropy change accompanied is thus the entropy change with volume. The summation of entropy changes of the constituent ideal gas systems results in the entropy of ideal mixing.