 # 1. Distributions (A) Calculate the number of ways to divide 17 dis...

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1. Distributions (A) Calculate the number of ways to divide 17 distinguishable objects into 5 groups as follows: group A has 5 objects group B has 0 objects group C has 7 objects group D has 2 objects group E has 3 objects (B) Suppose you can’t evaluate factorials (x!) directly. Show me how you can use the simple version of Stirling’s approximation to calculate 17!. Does it work well? 2. Boltzmann distribution Consider a system of exactly 8 moles of independent but distinguishable particles. Each particle has just three levels available The lowest level has an energy of 0 and is not degenerate. The second level has an energy of  and is two-fold degenerate. The third level has an energy of 3 and is not degenerate. (A) What is the molecular partition function at absolute zero temperature? What are the occupation numbers (N0, N1, N2, N3) of the four states? What is the average energy of a particle? (B) What is the molecular partition function at infinite temperature? What are the occupation numbers (N0, N1, N2, N3) of the four states? What is the average energy of a particle? (C) What is the molecular partition function at a temperature of 2/k? What are the occupation numbers (N0, N1, N2, N3) of the four states? What is the average energy of a particle? (D) Is N0 = 3.0 mol, N1 = 2.5 mol, N2 = 1.5 mol, N3 = 1.0 mol a Boltzmann distribution? Explain. 3. Entropy OR equilibrium The following problems use related information at P = 1.000 bar and T = 1000. K. (A) Calculate the standard molar entropy, Sm o , of I2(g) using statistical mechanics.* (B) Calculate the equilibrium constant for I2(g)  2I(g) using statistical mechanics.** The following information will be helpful: I: q T /N = 1.165109 g0 E = 4 I2: D0 = 12,366 cm1 g0 E = 1 B᷉= 0.0374 cm1 v᷉= 214.5 cm1 To make your life (and mine) easier, you might want to first calculate and report some of the following quantities: I: q/N I2: , R ,  V , q T /N, q R , q V , q E * By comparing your answer to (A) with the appendix value (not at 1000 K) you might find confidence or an error in your answer. ** By comparing properties of I2 on this exam with those of Na2 from the lecture, you should be able to judge whether your calculations are reasonable for (B). (C) : You shouldn’t understand why g0 E = 4 for an iodine atom (it has to do with the spin-orbit effect which we did not cover), but you should be able to propose a different value for g0 E and justify it well. 4. Internal energy and constant-volume heat capacity Let’s consider I2 again, but this time at the more reasonable temperature of 400 K. Both the molar internal energy (Um) and the molar constant-volume heat capacity (CV,m) can be expressed in the following manner where (i) we reasonably ignore any electronic contributions, (ii) x1 and x2 are just numbers representing the amount of RT or R that is needed, and (iii) v1 and v2 represent the vibrational contributions. Um = x1RT + v1 CV,m = x2R + v2 (A) What is x1? What is x2? (B) Accurately determine v1 in units of J mol1 . (C) Accurately determine v2 in units of J K1 mol1 . Equation 15E.6 (page 638) has a typo and is just the high-temperature limit, so you’ll want to think hard and refer to your notes. Also, remember that the “V” in CV,m refers to “constant volume” not “vibrational mode” Answer (A) or (B) 5. Equilibrium The diagram below shows the energy levels (assume nondegenerate) for the reactants and products of a 1:1 ideal gas chemical reaction R(g)  P(g). I want to know how the equilibrium constant behaves with temperature. The four possibilities are listed below. (A) Is this reaction exothermic or endothermic, i.e. is rH o > 0 or rH o < 0? (B) Does entropy increase or decrease in the forward direction, i.e. is rS o > 0 or rS o < 0? (C) Explain which possibility (1-4) for the temperature dependence of the equilibrium constant is observed by considering the extreme cases: T = 0 and T = . You may consider the occupation numbers (state populations) graphically by replicating the diagram above – or you may consider the contributions to the expression of the equilibrium constant given above.

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