I am currently working through some problems in thermodynamics. I am given the following relationship: $$TdS = C_vdT + l_v dV = C_pdT + l_p dP$$ where $l_p$ and $l_v$ are some functions of state variables and the number of particles is constant. I want to show two relations: $$\begin{align} C_p-C_v =& \ l_v\frac{\partial V}{\partial T} \\ l_v = T\frac{\partial P}{\partial T}\end{align}$$

My approach/work goes as follows: $$\begin{align} T\frac{\partial S}{\partial T} =& \ C_v \frac{\partial T}{\partial T} + l_v\frac{\partial V}{\partial T} \\ =& \ C_v + l_v\frac{\partial V}{\partial T} \end{align}$$ All at constant pressure. Now the using the second equation, also at constant pressure: $$T\frac{\partial S}{\partial T} = C_p + l_p\frac{\partial P}{\partial T} = C_p$$ Now substracting the equations from eachother we obtain: $$0 = C_p - C_v - l_v\frac{\partial V}{\partial T} \implies C_p-C_v = \ l_v\frac{\partial V}{\partial T} $$ Now for the second relation I proceeded as follows using the Maxwell relation $\left(\frac{\partial T}{\partial p}\right)_V = \left(\frac{\partial V}{\partial S}\right)_T$: $$\begin{align} T\left(\frac{\partial S}{\partial S}\right)_T =& \ C_v\left(\frac{\partial T}{\partial S}\right)_T\ + \ l_v \left(\frac{\partial V}{\partial S}\right)_T \\ \\ T =& \ l_v \left(\frac{\partial T}{\partial P}\right)_V\end{align} $$ My question is simply if these derivations make sense i.e. are correct? I sometimes seem to struggle with seeing when terms cancel due to holding a specific variable constant.

Any feedback would be appreciated!


They seem good to me! The first equality holds directly by $$C_vdT + l_v dV = C_pdT + l_p dP$$ finding the variation with respect to temperature in a transformation holing pressure constant $$C_v\left(\frac{\partial T}{\partial T}\right)_P+l_v\left(\frac{\partial V}{\partial T}\right)_P = C_p\left(\frac{\partial T}{\partial T}\right)_P+l_p\color{red}{\left(\frac{\partial P}{\partial T}\right)_P}$$ The last quantity, as you said, is zero since you're maintaining the pressure constant, there's nothing more to say about it.

If you think at a surface in $(P,V,T)$ space you can move along the $P$ axis maintaining $V,T$ constant or move along the $V$ axis and maintaining $P,T$ constant and so on. So by this if you have a surface in this space you can see how the surface changes with respect to some variables moving in such a way that $P$ (considering it as a simple variable of said surface), for example, remains constant.

Physically speaking you're just doing some thermodynamical transformation maintaining constant some variable while changing another and measuring how a third variable changes under this transformation, which is totally reasonable.

  • 1
    $\begingroup$ Thanks for the additional explanation, really helps to clarify some points! $\endgroup$
    Jun 18 '20 at 12:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.