User:Steven J. Koch/MTC Assignment 3
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This homework assignment is due March 1, 2011. This assignment is based on an example from Evan Evans in 2009.
How to turn in
- Your answer needs to somehow be publicly available, such as by:
- Writing your answers by hand and uploading a photograph
- Composing your answers online, such as on the wiki
- Creating a PDF on your own and uploading the PDF
Assignment
For this problem, we are considering adding NaCl to water and modeling it as NaCl "molecules" that dissolve and can dissociate into Na+ and Cl- monomers. As a reference point, we will consider 0.5 moles of NaCl added to 55.6 moles of water. The equilibrium constant for the dissociation is defined as x_{AB} / (x_{A} * x_{B}) as shown in top left of photo.
- Part A
- Use the ideal chemical potentials for the monomers Na+, Cl-, and NaCl. Recall that at chemical equilibrium the chemical potential of NaCl "molecules" is the sum of chemical potential of Na+ and Cl- monomers. See definition of delta_mu^{0} in the picture. Putting all of this together, write K_{eq} as a function of delta_mu^{0} and kT. For kT = 4.1 pN-nm and the coloumb energy as defined in the photo (about 3.6 kT), what is our specific numeric K_{eq}?
- Part B
- As shown in the top right of the photo, we get information from stoichiometry and conservation of mass. When one NaCl dissociates, one Na+ and one Cl- each are gained. Thus the number of Na+ = number of Cl- and we can define N_{I} = x_{I}*N_{T}.
- Use these relations to first find the mole fraction of AB (NaCl) as a function of the mole fraction of each ionic species.
- Next, use the relation for K_{eq} to write X_{I} as a function of Keq.
- You will solve a quadratic equation for this step
- Part C
- Use excel, Matlab, LabVIEW, or something besides a simple calculator for this step That way you will be able to play around with different values easily.
- For N_{AB}^{0} = 0.5 mols and N_{water}^{0} = 55.6 mols, and the equilibrium constant using 2 angstrom separation as in part A, calculated the following with real numbers:
- x_{AB}, x_{I}, x_{w}, N_{T}, mu_{AB}
- Do the numbers seem sensible?
- Part D
- Osmotic pressure relates to the chemical potential of water as shown in the above photo. Recall that if you calculated the chemical potential of a water molecule raised to a height difference to produce this pressure, it would be equal to the chemical potential difference due to the decrease in mole fraction of water. Use the familiar molecular volume of water, and the chemical potential of water calculated from the mole fraction in part C to calculate the osmotic pressure under the conditions given.
- Also calculate the pressure if dissociation were impossible. (Fudge some factor in your spreadsheet / program to make it very energetically costly). Next calculate if dissociation were very easy. (Again, by fudging factors numerically.)
- Part E
- Reflect. The point of this exercise is to get acquainted with everything. Fiddle with your numbers and assumptions and see how things behave. Does your answer in part D seem high or low?