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Chemical potential of water using the Van der Waals model

  1. May 21, 2015 #1
    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. jcsd
  3. May 21, 2015 #2
    For a pure substance, how is the chemical potential related to the gibbs free energy per mole?

    Chet
     
  4. May 21, 2015 #3
    By this relationship: $$\mu= \frac{G}{n}$$
     
  5. May 21, 2015 #4
    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
     
  6. May 21, 2015 #5
    I'm sorry I don't understand how I could determine that.
     
  7. May 21, 2015 #6
    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
     
  8. May 22, 2015 #7
    It is this relation? $$dG=-SdT+\mu dn$$
     
  9. May 22, 2015 #8
    No. The number of moles should also be held constant.

    Chet
     
  10. May 23, 2015 #9
    Can you express S as a function of G, H, and T? If so, substitute it into your equation for dG.

    Chet
     
  11. May 23, 2015 #10
    $$G=H-ST$$ $$S=\frac{H-G}{T}$$ $$dG=-SdT$$ $$\rightarrow dG=-(\frac{H-G}{T})dT$$
     
  12. May 23, 2015 #11
    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
     
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