A qns on Thermodynamics Relations

Homework Statement

We propose to study simple processes of liquid alcohol. At T_1, the molar volume of alcohol is V_1 and its molar heat capacity at constant pressure is C_p,m. We assume that its isobaric coefficent of thermal expansion a, and the isothermal compressibilty coefficient B are constant.

a. Find the molar heat capacity (C_v,m) at constant volume and T_1, and the variation of the pressure with temperature at constant volume.

b. starting from an initial state (T_1, V_1), alcohol receives an amt. of heat at constant pressure P, and its final temp. is T_2. Evaluate the work received by alcohol during the process.

c. Find a formal expression for the rate of change with temperature of the internal energy of the liquid at constant pressure.

Homework Equations

isothermal coefficient of thermal expansion, a = 1/V * (dV/dT)_p
isothermal compressibilty coefficient B, -1/V * (dV/dP)_T

C_p = C_v + TVa^2/B ------ eqn (1)

The Attempt at a Solution

For qns a, i have no idea how to go about finding C_v,m. So i simply used the direct relatonship in eqn (1) to find C_v,m which i highly doubt its correct.
for the second part of the qns,
I used P=P(T,V)
-> dP = MdT + NdV
doing some algebric manipulation: dV = (1/N * dP) - (M/N * dT)
by comparing coefficients from the derived eqn of V=V(T,P)...
i got my final dP = (a/B * dT) - (1/VB *dV)
at constant volume: (dP/dt)_V = a/B

is this correct?

For qns b, here's what i attempted:

(T_1, V_1) ----> (T_2, V_2) at constant P.
Using U=U(T,P) thermodynamics relationship and letting dP=0
i get dU = (C_p - PVa)*dT
by integrating from T_1 to T_2, i will get the change in U.
then to find the Q received, i used H=H(T,P)
where in the end i get dH=Q=C_p * dT
again i integrate to find Q.
then finally i use first law U= Q+W to find W.
is this correct?

For qns c,

i used U=U(T,P)
at constant P, i get dU=(C_P-PVa)*dT

im not sure if this is correct or not...

thanks a lot for helping me out