Equation of state for a real gas

[WWCY]

Homework Statement

The equation of state for some gas is $P(V-b) = RT)$ (V is the volume occupied by a mole of gas), show that the internal energy of this gas is a function of temperature only.

My work leads to the wrong answer, some assistance in pointing out my mistake is greatly appreciated!

Relevant Equations

Real gas equations

My work was as follows:

The first law states $dU = TdS - PdV$, and thus
$$p =- (\partial U/\partial V)| _S$$
$$U = -RT \ln(V-b) + f(S)$$
To determine $f(S)$, I reasoned that in the ideal gas limit of $b = 0$, $U$ should take the form of the ideal gas' molar internal energy $\frac{3}{2} RT$. So I set $b = 0$ and compare the expressions for energy in the ideal gas limit to obtain
$$f(S) = \frac{3}{2}RT + RT \ln (V)$$
However, inserting this into the internal energy expression for the real gas gives
$$U = \frac{3}{2}RT + RT \ln \big(\frac{V}{V-b} \big)$$
which doesn't seem to be independent of V.

What have I done incorrectly?

 

[Lord Jestocost]

Maybe, it’s better to use the Thermodynamic Equation of State:

$\left(\frac{\partial U}{\partial V}\right) _{T} = T\left(\frac{\partial p}{\partial T}\right) _{V}-p$

 

[WWCY]

Hi, thank you for your response.

Maybe, it’s better to use the Thermodynamic Equation of State:

$\left(\frac{\partial U}{\partial V}\right) _{T} = T\left(\frac{\partial p}{\partial T}\right) _{V}-p$

I don't quite get what you mean by this, and I still don't get why my method was wrong. Do you mind elaborating? Thank you.Edit: And also, why does the "correct" method involve performing an integral using $p = -\Big(\frac{\partial F}{\partial V} \Big)_T$ to obtain $F$, rather than using the first law to obtain $U$ directly?

 

[Chestermiller]

Your initial integration is incorrect because, at constant S, T is a function of V.

 

[WWCY]

Thanks for your reply,

Your initial integration is incorrect because, at constant S, T is a function of V.

In this case, how do I discern that constant entropy processes mean that T is a function of V?

Also, is there a way to find out in general whether one thermodynamic variable varies with another variable when some thermodynamic variable is kept constant?

And finally, is the reason why $p = -\Big(\frac{\partial F}{\partial V} \Big)_T$ was correct due to the fact that we have already defined $T$ to be independent of changes with $V$?

Thank you for your patience.

 

[Chestermiller]

Thanks for your reply,
In this case, how do I discern that constant entropy processes mean that T is a function of V?

Thank you for your patience.

$$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV\tag{1}$$so, at constant entropy (dS=0), $$\left(\frac{\partial T}{\partial V}\right)_S=-\frac{\left(\frac{\partial S}{\partial V}\right)_T}{\left(\frac{\partial S}{\partial T}\right)_V}$$

Also, is there a way to find out in general whether one thermodynamic variable varies with another variable when some thermodynamic variable is kept constant?

And finally, is the reason why $p = -\Big(\frac{\partial F}{\partial V} \Big)_T$ was correct due to the fact that we have already defined $T$ to be independent of changes with $V$?

The best way to solve this problem is to substitute Eqn. 1 into your original equation for dU. This leads to the relationship that Lord Jestocost posted.

 

[WWCY]

$$dS=\left(\frac{\partial S}{\partial T}\right)_VdT+\left(\frac{\partial S}{\partial V}\right)_TdV\tag{1}$$so, at constant entropy (dS=0), $$\left(\frac{\partial T}{\partial V}\right)_S=-\frac{\left(\frac{\partial S}{\partial V}\right)_T}{\left(\frac{\partial S}{\partial T}\right)_V}$$

The best way to solve this problem is to substitute Eqn. 1 into your original equation for dU. This leads to the relationship that Lord Jestocost posted.

Ah I see it now, thank you both!

Just a quick question on the method I initially posted about... If for some reason I need to work with those types of integrals, does that mean I have to be extra careful about whether or not the state variables in my integrands are functions of other state variables? (e.g: $T$ was a function of $V$ with $S$ kept constant here).Also, is this reasoning correct?

And finally, is the reason why $p = -\Big(\frac{\partial F}{\partial V} \Big)_T$ was correct due to the fact that we have already defined $T$ to be independent of changes with $V$?

 

[Chestermiller]

Just a quick question on the method I initially posted about... If for some reason I need to work with those types of integrals, does that mean I have to be extra careful about whether or not the state variables in my integrands are functions of other state variables? (e.g: $T$ was a function of $V$ with $S$ kept constant here).

Sure.

Also, is this reasoning correct?

To see how pressure is equal to minus partial F wrt V at constant T, see my response in your other thread:
https://www.physicsforums.com/threads/thermodynamics-partial-derivatives.969924/#post-6160852