Thermodynamics and Hooke's Law

[anothersivil]

Homework Statement

*1* The Gibbs function G(T; P) of a certain gas is:
G = nRT ln P + A + BP + (1/2)*CP^2 + (1/3)DP^3

where A, B, C and D are constants. Find the equation of state of the gas.

Homework Equations

G = nRT ln P + A + BP + (1/2)*CP^2 + (1/3)*DP^3

The Attempt at a Solution

I think I have a solution for this, but it seemed to be too easy.

The definition for Gibbs free energy is defined to be:
1) G = U - TS + PV and 2) dG = -SdT + vdP

solving 2) for dG/dP yields: dG/dP = -SdT/dP + V 3)

I thought to take 3) at constant temperature yielding: dG/dP = V
Now, taking dG/dP at constant T from the given formula yields:

dG/dP =nRT/P + B + CP + DP^2 4)

Setting 3) = 4) and solving for PV yields:

PV = nRT + PB + CP^2 + DP^3 That just seemed too easy for me >.>
The second part of this homework is:

Homework Statement

Consider a spring which follows Hookes law; namely the displacement x from equilibrium
position is proportional to the tension X when it is pulled at a constant temperature. The spring constant is temperature dependent, k = k(T). Determine the free energy F, the internal energy U, and the entropy S, as a function of T and x. Neglect the thermal expansion of spring. Use F0(T) ´ F(T; x = 0); U0(T) ´ U(T; x = 0); S0(T) ´ S(T; x = 0) where necessary.

Homework Equations

k = k(T)
F[0](T) = F(T; x = 0)
U[0](T) = U(T; x = 0)
S[0](T) =S(T; x = 0)

where the brackets denote subscripts

The Attempt at a Solution

The force of the spring can be obtained by: F = -k(T) * x
And the work: W = int(-k(T) * x) dx = -(1/2)*k(T)*x^2

and dW = k(T) *xdx

Using the first law of thermodynamics: dU = dQ - dW, so

dU = dQ + k(T)*xdx

 

[Mapes]

The elongation of the spring would add another term to your energy equation:

$$U=TS-PV+Fx=TS-PV+k(T)x^2$$

$$F=U-TS=-PV+k(T)x^2$$

$$S=-\left(\frac{\partial F}{\partial T}\right)_V=-\frac{\partial k(T)}{\partial T} x^2$$

Does this help?

 

[anothersivil]

That makes sense to me, but what about those 3 initial conditions? This just seems too easy to me >.>

F[0](T) = F(T; x = 0)
U[0](T) = U(T; x = 0)
S[0](T) =S(T; x = 0)

What do I do with those?

 

[Mapes]

I cheated a bit by ignoring the temperature dependence of PV and by setting my energy baseline at $\mu\,N$. The initial conditions make these constants go away anyway. Wouldn't you end up with, for example,

$$S=S_0-\frac{\partial k(T)}{\partial T} x^2$$

I'll leave the rest for you.

 

[anothersivil]

Thanks a bunch! This helps alot! :P