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Problem Set 1 - Problem Set 1 Chem 3900 Physical Chemistry...

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Unformatted text preview: Problem Set 1 Chem 3900: Physical Chemistry II Spring 2011 Due: Class, Friday, Feb 4th Please make sure to write your name and the recitation you’ll attend. When you print out your work, try to minimize the number of pages. You can do this by printing multiple pages per sheet and/or by using a duplex printer. Note: In this problem set we will use V to denote the molar volume instead of V . Problem 1. i) The first problem examines some of the different equations and properties of gases using the four different gases: helium, argon, carbon dioxide, and methane as examples. You might find the “NestList” command in Mathematica helpful for the iterative calculations in questions ii) and v). Graphing the integrand of B2v could also be helpful, because you see that the integrand goes to zero quickly as x increases, making it okay to approximate the infinite integral as a finite integral when using the trapezoidal approximation. Calculate the ideal gas molar volume Videal at 0oC and 100 bar. ii) For each of the four gases use the Newton‐Rapson method to calculate the molar volume at 0oC and 100 bar for both the Van der Waals and the Redlich‐Kwong equation of state. Make a nice table of the result that is easy for your TA to grade. iii) For each of the four gases calculate V ‐ Videal. What does this tell you about the intermolecular force between the gas molecules? iv) Show that another way to write the compressibility is simply Z = V/Videal and calculate this value for each gas. v) The second Virial coefficient is an important parameter that links the intermolecular potential to physical properties that are easily measured. It thus provides a nice connection between theory and experiments. Equation 16.31 gives the second virial coefficient of the Lennard‐Jones intermolecular potential in reduced units terms of an integral that needs to be evaluated numerically. Use Mathematica to first numerically calculate (using the trapezoidal approximation discusses in chapter G) then plot B*2V(T*) from T* = 1 to T* = 5 with a step size of 0.1. vi) Use the plot to find the reduced temperature (T*) where B*2V(T*) = 0. What is special about this point? vii) For each of the four gases calculate the absolute temperature T where B2V = 0. 1 Use the plot to calculate B2V at 0oC for each gas (if you are having problems with the plot you can use Figure 16.15) viii) ix) Equation 16.32 provides a nice intuitive understanding of B2V. How does your results in question viii compare to what you calculated in question iii? x) How do the deviations from ideality calculated for each gas agree with your chemical intuition? Explain. Problem 2. The second problem examines a different equation of state to exemplify some of the key derivations in chapter 16. Consider the following equation of state: exp where α and β positive constants. i) Show that this equation of state reduces to the IGES in the appropriate limits. What are the physical meanings of the parameters α and β? ii) For this equation of state, the critical molar volume is Vc = 2α. Relate Pc and Tc toα and β. (Hint: This question can be answered without taking any second derivatives.) iii) Calculate the value of the dimensionless quantity predicted by this equation of state. Compare to the values given in Table 16.5 and predicted by the two equations of state in Example 16‐4. How did I do? iv) The “Law of Corresponding States” is described in Section 16‐4. Show that this equation of state obeys this rule, by writing it in terms of the quantities PR, TR, and VR, defined above Eq. 16.19. Your final result should contain only dimensionless quantities. v) Graph isotherms at TR = 0.85, 1.0, and 1.15 on a plot of PR versus VR over the range 0.51 ≤ VR ≤ 5.0. This should be a quantitative plot, not a rough sketch. Compare your results qualitatively to the plots shown in Fig. 16.8, and comment on how these isotherms would describe the equation of state of a real material. vi) Measured values of Tc and Pc for methane are given in Table 16.5. Use these data to determine α and for methane. vii) On pages 643‐644, the pressure exerted by 1 mol of methane in 250 mL at 0oC as measured in the lab is compared to the prediction of the van der Waals equation of state. Calculate 2 this pressure for the present equation of state, and compare to the experimental value, and to the predictions of the van der Waals and ideal gas equations. viii) The Boyle temperature of methane is measured to be 642K. Calculate the Boyle temperature for methane predicted by this equation of state, and compare to the experimental value. 3 ...
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