- #1
fluidistic
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Homework Statement
Hi guys, I'm stuck on 2 parts for now of the following problem that appeared in an exam I totally bombed:
Given the equation ##s=s_0+R\ln \left [ C(v-b)\left (u + \frac{a}{v} \right )^{5/2} \right ]##,
1)Calculate ##u(v,T)##
2)Calculate f in its natural variables (the molar Helmholtz equation).
3)Find ##P(v,T)##
4)Calculate ##C_p## and ##C_v##
5)Show that this system doesn't satisfy the criteria of stability when ##RT > \frac{2a(v-b)^2}{v^3}##.
6)Give a way to obtain ##v_l## and ##v_s##, the liquid and solid molar volumes. Don't do the algebra but give a way that ensures to find them, with as much details as possible.
Homework Equations
(1)##\frac{1}{T} = \left ( \frac{\partial s}{\partial u} \right ) _v##
(2)##F=U-TS \Rightarrow f=u-Ts##
(3)##P=-\left ( \frac{\partial f}{\partial v} \right ) _T##
(4)##C_v= \left ( \frac{\partial u}{\partial T} \right ) _v##
(5)##-\left ( \frac{\partial P}{\partial v}\right ) T >0## (stability criterium)
The Attempt at a Solution
For part 1), I used eq. (1) to reach that ##u(v,T)=\frac{5RT}{2}-\frac{a}{v}##.
For part 2), I used eq. (2) to reach ##f(v,T)=\frac{5RT}{2}-\frac{a}{v}-Ts_0-RT \ln \left [ C(v-b) \left ( \frac{5RT}{2} \right ) ^{5/2} \right ]##.
For part 3), I used eq. (3) to reach that ##P(v,T)=\frac{RT}{C(v-b)}-\frac{a}{v^2}##.
For part 4), I used eq. (4) to reach that ##C_v=\frac{5R}{2}##.
However I don't really know how to find ##C_P##.
Here's a try: ##C_P =C_v+T \left ( \frac{\partial s}{\partial v} \right ) _T \left ( \frac{\partial v}{\partial T} \right ) _P##. Then I applied Maxwell's equation on ##\left ( \frac{\partial s}{\partial v} \right ) _T## to obtain ##C_P =C_v+T \left ( \frac{\partial p}{\partial T} \right ) _v \left ( \frac{\partial v}{\partial T} \right ) _P##.
Then a cyclic relation to obtain ##C_P =C_v-T \left ( \frac{\partial p}{\partial v} \right ) _T \left ( \frac{\partial v}{\partial T} \right ) ^2 _P##, that I rewrote as ##C_P=C_v- \frac{T \left ( \frac{\partial P}{\partial v} \right ) _T }{\left ( \frac{\partial T}{\partial v} \right ) ^2 _P}##.
Where I calculated ##\left ( \frac{\partial P}{\partial v} \right ) _T## thanks to eq. (3) and from eq. (3) I also isolated T(v,P) and then performed the partial derivative of it with respect to v by keeping P constant. And then I tried to see if some terms would cancel out but couldn't find any. So I got an enormous expression for ##C_P##, I can't believe ##C_v## to be so simple but ##C_P## being that horrible. I must have used a bad approach?
For part 5), I've used eq. (5) and found out the desired expression.
I'll think about part 6. The explanation I gave was wrong because I assumed I could find P(T) on the coexistence liquid-solid. But I should have said how I could obtain such a function and it's true, I have no idea. So I'll think about another way.
So for now, I'd like some help for part 4) on how to get ##C_P##. Thank you very much for any help.