Thermodynamics: T-S slope at constant volume

[2DGamer]

1. Prove that the slope of an isochoric process in a T-S diagram is T/Cv where T is the temperature and Cv is the heat capacity at constant volume



2. dQ = T * dS (I understand this)
dQ = Cv * dT for isochoric processes (I don't understand this)



3. Since the two equations share dQ I set them equal to each other:
Cv * dT = T * dS
Then I just rewrote the equation as:
T/Cv = dT/dS
And dT/dS describes the slope of the line which is T/Cv
My main problem is that I don't understand why dQ = Cv*dT. The book just gave it to us, but I'm thinking that I need to be able to derive it or the problem is just too easy.

 

[Mapes]

dQ = Cv * dT for isochoric processes (I don't understand this)

For isochoric (constant-volume, dV=0) processes, no work is done (work being P\,dV), so any energy change dU in the system must be in the form of heat transfer q. Additionally, we can use the differential energy equation

dU(=q)=T\,dS-P\,dV=T\,dS=T\left(\frac{\partial S}{\partial T}\right)_V\,dT=C_V\,dT

which uses the definition of C_V. Does this make sense?