Deriving Relationships for C_P - C_V & C_F - C_L in Stretched Wires

[Collisionman]

Homework Statement

I have been stuck on the following question for a while now, particularly part B and C, and I'd like some help with it please. I've completed part A already. The Question:

(A) Combine the following TdS equations:

$TdS = C_{v} dT + T\left(\frac{\partial P}{\partial T}\right)_{v}dV$
$TdS = C_{P} dT - T\left(\frac{\partial V}{\partial T}\right)_{p}dP$

to derive a relationship for $C_{p} - C_{v}$ as a function of volume, pressure and temperature (V, P and T, respectively) using the cyclic rule.

(B) Derive modifications to the two TdS equations, mention in part A, for application to a stretched wire under constant force (for example, supported by a weight). The force on the wire can be considered solely as a function of length, L, and temperature, T. Derive the modified equations in terms of the coefficient of linear expansion with temperature and Young's modulus.

(C) Derive a relationship for $C_{F} - C_{L}$ for a stretched wire under constant force (for example, supported by a weight). The force on the wire, F, can be considered solely as a function of the length, L, and temperature, T. Derive this relationship in terms of the coefficient of linear expansion, Young's modulus, volume and temperature. You may use the relationship that you derived in part (A).

Homework Equations

(1) Volume expansivity: $\beta = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}$
(2) Bulk modulus: $K = - V\left(\frac{\partial P}{\partial V}\right)_{T}$
(3) Young's modulus: $Y = \frac{L}{A}\left(\frac{\partial F}{\partial L}\right)_{T}$
(4) Coefficient of Linear Expansion: $\alpha = \frac{1}{L}\left(\frac{\partial L}{\partial T}\right)_{F}$

The Attempt at a Solution

I've completed part A of the question and got the following expression using the cyclic rule:

$C_{p} - C_{v} = - T\left(\frac{\partial V}{\partial T}\right)^{2}_{P}\left(\frac{\partial P}{\partial V}\right)_{T}$

I know that the above expression for $C_{p} - C_{v}$ can be written as;

$C_{p} - C_{v} = T\beta^{2}KV$

where β and K are the volume expansivity and the bulk modulus, respectively and $\beta\approx 3\alpha$.

However, I really don't know where to start with part B.

Could someone please help me with part B and C of this question? If I knew how to do part B I think I'd be able to derive a relation for part C.

I would really appreciate any help. Thank You!

 

[Collisionman]

Anyone? Any help greatly appreciated.

 

[Collisionman]

Please, anyone...