Chemical potential of water using the Van der Waals model

[It's me]

Homework Statement

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

Homework Equations

μ=∂U∂N

The Attempt at a Solution

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

 

[Chestermiller]

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

Chet

 

[It's me]

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

 

[Chestermiller]

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

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

 

[It's me]

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

 

[Chestermiller]

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

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

 

[It's me]

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

 

[Chestermiller]

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

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

Chet

 

[Chestermiller]

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

Chet

 

[It's me]

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

 

[Chestermiller]

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

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