How Do You Solve These Thermodynamics Problems?

[ChronicQuantumAddict]

1. Show that for an ideal gas

a).$$f = c_v(T-T_0)-c_vT\ln(T/T_0)-RT\ln(v/v_0)-s_0T$$
b).$$g = c_p(T-T_0)-c_pT\ln(T/T_0)+RT\ln(P/P_0)-S_0T$$​

totally lost here, where do i begin?
2. The Helmholtz function of a certain gas is:

$$F = -\frac{n^2a}{V} - nRT \ln(V-nb) + J(T)$$,​

where J is a function of T only. Derive an expression for the pressure.
For this one i need the answer verified, i think i have it right. We know that $$-P = (\frac{(\delta(F))}{(\delta(V))})_T$$, and thus all i have to do is differentiate the negative of the above function for F with respect to V, and the $$J(T)$$ function disappears.
3. The Gibbs function of a certain gas is:

$$G = nRT\ln(P) + A + BP + \frac{CP^2}{2} + \frac{DP^3}{3}$$(​

where A,B,C, and D are constants. Find the equation of state of the gas.
Now, the answer in the book is:

$$nRT + BP + CP^2 + DP^3$$​

the only way i see to get this is to differentiate the expression for with respect to P, giving:

$$\frac{nRT}{P} + B + CP + DP^2$$​

and then multiplying this result through by P, or in other words, the equation of state is given by:

$$P(\frac{\delta(G)}{\delta(P)})=eq of state$$​

but how or why, or is this even correct, please help with these 3 problems, thanks.

 

[ChronicQuantumAddict]

any takers? anyone?

 

[siddharth]

For the first question, what's f and g?EDIT:
Sorry, I am unable to help in this one. I would like to know how you obtain this result as well.

 

[ChronicQuantumAddict]

f is the specific helmholtz function, and g is the specific gibbs free energy function

 

[ChronicQuantumAddict]

am i correct for part 2? and so far all i have seen is what i showed there for part 3, i can't seem to understand if that is even correct, let alone how that is arrived at. thanks

 

[ChronicQuantumAddict]

anyone someone?

 

[Physics Monkey]

For question one, write out the full Helmholtz free energy for the ideal gas and then divide by the total mass of the gas to obtain the specific Helmholtz free energy (I assume this is the definition). Rewrite the terms that appear in terms of the specific heat at constant volume. $$v$$ is the molar volume and $$s_0$$ will be related to the specific entropy at $$T = T_0$$ and $$v = v_0$$. Once you've done the Helmholtz free energy, I think you should be able to adapt your method to do the Gibbs function as well.

For question 2, you are right on track. The pressure in the Helmholtz representation is given by $$P = -\left(\frac{\partial F}{\partial V} \right)_T$$. The equation of state you get is called the van der Waals equation of state so you can check yourself online.

For question 3, in the Gibbs representation, the volume is given as $$V = \left( \frac{\partial G}{\partial P} \right)_T$$. The resulting equation for V is your equation of state.