# Derivation of Thermodynamic Relationships

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.