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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!

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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.

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

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