# determining an expression for an entropy equation

by Benzoate
Tags: determining, entropy, equation, expression
 P: 569 1. The problem statement, all variables and given/known data Calculate the entropy of mixing for a system of two monatomic ideal gases, A and B ,whose relative proportion is arbitrary. Let N be the total number of molecules and let x be the fraction of these that are of species B. You should find delta(S) mixing=-Nk[x ln x +(1-x) ln (1-x) 2. Relevant equations delta (S(total))=delta(S(A)) + delta(S(B))=2Nk ln 2 S=Nk[ln((V/N)(((4*pi*m*U)/3Nh^2)^(3/2))+2.5] 3. The attempt at a solution according to my thermal physics text, delta(S(A))=Nk ln 2 . The problem says that in species B , x is just a fraction of N. Then , I think I would have to conclude that delta(S(B))=x/N*(k)*ln(2). so would my expressison be :delta(S(mixing))=delta(S(A))+delta(S(B))=Nk ln 2+ xk/N*(ln(2))
 PF Patron HW Helper Sci Advisor P: 2,532 You can't apply $\Delta S_A=Nk\ln 2$ to the general problem; that's the increase in entropy for a single gas expanding into twice its original volume. If $x$ can vary, there's no reason to assume the volume doubles. Also, remember that as $x$ increases, there are no longer $N$ molecules of gas A but rather $(1-x)N$. One common way to show your desired relation is to assume that each gas expands from its original volume into the total volume and to use the Maxwell relation $$\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V=\frac{nR}{V}\quad$$ $$dS=\frac{nR}{V}\,dV$$ to calculate the change in entropy.

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