Calculating Properties with ##S##, ##V##, and ##N##

[Like Tony Stark]

Homework Statement

The fundamental equation of a gas is $A=-aVT^{\frac{5}{2}} e^{\frac{\mu}{RT}}$. Determine $\alpha$, $\kappa_T$ and $c_P$, and then find the fundamental equation in energetic representation: $U(S; V; N)$.

Relevant Equations

$U$: internal energy; $T$: temperature; $\mu$: chemical potential; $R$: ideal gas constant; $V$: volume; $N$: number of moles; $\alpha$: coefficient of thermal expansion; $c_P$: heat capacity at constant pressure; $\kappa_T$: compressibility at constant temperature.

$\alpha=\frac{1}{V} \frac{\partial V}{\partial T}$; $c_P=\frac{T}{N} \frac{\partial S}{\partial T}$ at constant $P$; $\kappa_T=\frac{–1}{V} \frac{\partial V}{\partial P}$ at constant $T$

Hi

All the expressions for calculating the properties are given in terms of $S$, $V$ and $N$. Should I find the energetic representation and then apply the formulas, or is there another way?

Then, for finding the energetic representation, I know that
$A=U–TS–\mu N$
But I want all these variables to be written in terms of $S$, $V$ and $N$. How can I do that? I also know that I can differentiate to obtain the equations of state, but these ones will be written in terms of $T$, $V$ and $\mu$ too.

 

[Twigg]

Is this a stat mech class or a purely thermodynamics class? In other words, do you talk about the grand canonical ensemble at all? We just need a little more context to be helpful to you.

Edit: I thought the OP made a minus sign error, but I just read the OP's equation wrong. Deleted that correction from my post.

 

[Like Tony Stark]

Is this a stat mech class or a purely thermodynamics class? In other words, do you talk about the grand canonical ensemble at all? We just need a little more context to be helpful to you.

Edit: I thought the OP made a minus sign error, but I just read the OP's equation wrong. Deleted that correction from my post.

Hello! It's from a purely thermodynamics class. The reference book in my course is Callen's Thermodynamics.

 

[Twigg]

For $\alpha$:
The starting point is to remember that $A = -PV$. In other words, $V = -\frac{A}{P}$. You know that $\alpha = \frac{1}{V} \frac{\partial V}{\partial T}$ for constant P. To take this derivative, hold $P$ constant and only evaluate the derivative on $A$. Does that make sense?

For $c_p$:
You need to calculate the entropy S vs temperature T to use the expression you gave. Recall that $dA = -PdV - SdT - Nd\mu$. So how do you find S from A? Hint: it's a derivative of the form $\pm \frac{\partial A}{\partial X}$ for some X and some sign (plus or minus).

For $\kappa_T$:
Start with $dA = -PdV - SdT - Nd\mu$ and set $dT \rightarrow 0$ since you're interested in a constant temperature process. Divide through by $dP$ and make some substitutions to solve for $\frac{\partial V}{\partial P}$.

These kinds of problems really just rely on careful attention to detail and creativity. You'll find there's a lot of ways to derive $V=V$ doing these. Just be patient, and try different things. As a general procedure, notice that you can get a lot done by starting with $dA = -PdV - SdT - Nd\mu$ and making the right substitutions to solve for some partial derivative or another.

Let us know if the last part is still unclear to you after working these properties out. (And no sweat if it is! These kind of algebra problems can be a pain.)

 

[Like Tony Stark]

Thanks for your answer!
Let's see if I've understood...
So, for $\alpha$ I have to calculate $\frac{\partial V}{\partial T}=\frac{\partial}{\partial T}$
$\frac{-aVT^{5/2}e^{\frac{\mu}{RT}}}{P}$, for constant $P$

Then, for $c_P$, I have to calculate $\frac{\partial^2 A}{\partial T^2}$ for constant $P$; and for $\kappa_T$ I have $\frac{\partial A}{\partial P} + N \frac{\partial \mu}{\partial P}=-P \frac{\partial V}{\partial P}$, for constant $T$

 

[Twigg]

For $\alpha$ that looks good

For $c_p$, yep!

And for $\kappa_T$ yep that looks good. There might be a cleaner way out there but that's the first step of what I got.