# Thermodynamics Question, entropy Problem

1. Jan 19, 2014

### Matt atkinson

1. The problem statement, all variables and given/known data
Considering entropy as a function of temperature and volume and using the Maxwell relation;
$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V$$
Show that the entropy of a vessel is given by;
$$S= R \left(\frac{3}{2}lnT+lnV+const\right)$$
Where R is the gas constant.
2. Relevant equations
Given in the question.

3. The attempt at a solution
So I made and attempt because S(T,V) you can write it as;
$$dS=\left(\frac{\partial S}{\partial V}\right)_T dV +\left(\frac{\partial S}{\partial T}\right)_V dT$$
Then substituted the maxwell relation;
$$dS=\left(\frac{\partial p}{\partial T}\right)_V dV + \left(\frac{\partial S}{\partial T}\right)_V dT$$
But from there I'm not sure where to go, i tried a few different things like dividing by dV but I'm drawing a blank, a nudge in the right direction would be appreciated.

2. Jan 19, 2014

### hilbert2

Looks like you are not giving us all the information needed to solve the problem. How exactly was the problem worded in your homework/textbook? I also find it confusing that in the formula for entropy there are logarithms of dimensional quantities. Usually we only calculate logarithms (or exponentials) of dimensionless numbers.

3. Jan 19, 2014

### Matt atkinson

This is the Question part (d), I managed to find a solution but I didnt use the maxwel relation above, I also found it fairly strange but It must be possible it was on a list of practice examples I found.

4. Jan 19, 2014

### BruceW

"ideal monatomic gas" is a crucial part of the question. what equations are there for ideal gas, that might be useful?

edit: p.s. sorry for barging in.

5. Jan 19, 2014

### hilbert2

The problem is related to the so-called Sackur-Tetrode equation, which gives the entropy of an ideal monoatomic gas as a function of internal energy, volume, and number of atoms. See http://hyperphysics.phy-astr.gsu.edu/hbase/therm/entropgas.html . You just have to find out how to derive that from the Maxwell equation. Googling with cleverly chosen keywords might help.

6. Jan 19, 2014

### Matt atkinson

Like I said I did manage to find a Solution using $$dQ=dU+pdV$$, $$pV=nRT$$ and $$dU=\frac{3}{2}nRdT$$ but Doing that I didn't use the Maxwell relation as the question states, and I'm not sure how to use it to find the solution.

7. Jan 19, 2014

### Matt atkinson

Oh I will, Thank you, sorry i posted that reply before seeing your second hilbert.

8. Jan 19, 2014

### Matt atkinson

Thanks alot guys I figured it out now wiht your help, I was just going blank.