# Chemical potential of water using the Van der Waals model

1. May 21, 2015

### It's me

1. The problem statement, all variables and given/known data

Obtain the chemical potential of water as a function of temperature and volume using the Van der Waals model.

2. Relevant equations

μ=∂U∂N

3. The attempt at a solution

I don't really understand how to do this at all. Any help would be greatly appreciated.

2. May 21, 2015

### Staff: Mentor

For a pure substance, how is the chemical potential related to the gibbs free energy per mole?

Chet

3. May 21, 2015

### It's me

By this relationship: $$\mu= \frac{G}{n}$$

4. May 21, 2015

### Staff: Mentor

So, if you could calculate the gibbs free energy per mole as a function of temperature and volume for a van der walls gas, you would have your answer. Suppose you took the starting state of g = 0 as water vapor at 25 C and the corresponding equilibrium vapor pressure (i.e., in the ideal gas region). Could you determine g at the same pressure and a higher temperature T (i.e., within the ideal gas region)?

Chet

5. May 21, 2015

### It's me

I'm sorry I don't understand how I could determine that.

6. May 21, 2015

### Staff: Mentor

Well, you need to go back to your textbook and find out how to determine that change in free energy with temperature at constant pressure.

Chet

7. May 22, 2015

### It's me

It is this relation? $$dG=-SdT+\mu dn$$

8. May 22, 2015

### Staff: Mentor

No. The number of moles should also be held constant.

Chet

9. May 23, 2015

### Staff: Mentor

Can you express S as a function of G, H, and T? If so, substitute it into your equation for dG.

Chet

10. May 23, 2015

### It's me

$$G=H-ST$$ $$S=\frac{H-G}{T}$$ $$dG=-SdT$$ $$\rightarrow dG=-(\frac{H-G}{T})dT$$

11. May 23, 2015

### Staff: Mentor

Good. So, if we rearrange this, we get:
$$\frac{d(G/T)}{dT}=-\frac{H}{T^2}$$
Do you know how to get H as a function of T for a gas in the ideal gas region? Once you know that, you can integrate this equation to get G as a function of T at constant (low) pressure in the ideal gas region. Can you figure out what to do next?

Chet