- #1
Dazed&Confused
- 191
- 3
Homework Statement
The constant-volume heat capacity of a particular simple system is [tex]
c_v = AT^3 [/tex]
where A is a constant. In addition the equation of state is known to be of the form
[tex]
(v-v_0)p = B(T)
[/tex]
where [itex]B(T)[/itex] is an unspecified function of T. Evaluate the permissible functional form of [itex]B(T)[/itex].
Homework Equations
3. The Attempt at a Solution [/B]
So we have
[tex]
{\frac{\partial S}{\partial v}}
_U = \frac{B(T)}{T(v-v_0)}[/tex]
and
[tex]
{\frac{\partial S}{\partial T}}_v = AT^2
[/tex]
I apply the first derivative to the second equation and vice versa. I equate and get
[tex]
\frac{\partial}{\partial T} \left ( \frac{B(T)}{T} \right) \frac{1}{v-v_0} = 2A T {\frac{\partial T}{\partial v}}_u
[/tex]
The rightmost term can be rewritten as
[tex]
{\frac{\partial T}{\partial v}}_u = -\frac{ {\frac{\partial u}{\partial v}}_T}{{\frac{\partial u}{\partial T}}_v} = -\frac{T \frac{\partial s}{\partial v}_T - p}{c_v}= -\frac{T \frac{\partial p}{\partial T}_v - p}{c_v}[/tex] so that
[tex]
\frac{2}{T^2} \left [ -\frac{T B'(T)}{v-v_0} + \frac{B(T)}{v-v_0} \right] = \frac{\partial}{\partial T} \left ( \frac{B(T)}{T} \right) \frac{1}{v-v_0}
[/tex]
which I solve for [itex]B(T)[/itex] and get [itex]B(T) = ET [/itex] with [itex]E[/itex] a constant. Now
[tex]
c_p = c_v + \frac{Tv\alpha^2}{\kappa_T}
[/tex]
where [itex]\alpha[/itex] is the isobaric compressability with temperature and [itex]\kappa_T[/itex] is the isothermal compressability with pressure. Thus with
[tex]
v = \frac{B(T)}{p} + v_0
[/tex]
this should equal (I think) to
[tex]
c_v + T \left(\frac{\partial V }{\partial T}_p\right)^2\left/\right. \left(\frac{\partial V}{\partial p}\right)_T =c_v+ T\frac{B'(T)^2}{B(T)}
[/tex]
which in my case would be simply [itex]c_v + E[/itex]. The answers give
[tex]
c_v + (T^3/DT + E)
[/tex].
Now when solving for their [itex]B(T)[/itex] I get a very complicated expression. I do not see where my mistake lies, except I am not 100% sure if the two partial derivatives I had at the beginning commute.