problem 1: Partial derivatives. The heat capacity at constant volume, C_{V} ≡ (∂U/∂T)_{V}, was not the same as heat capacity at constant pressure, C_{P} ≡ (∂U/∂T)_{P} .
As an illustration of why it matters which variable is held fixed when taking a derivative of a multi-variable function, let’s consider the following abstract ex. Define a = 2bc^{3} and b = c/f. Compute the partial derivatives:
(∂_{a}/∂_{b})_{c} and (∂_{a}/∂_{b})_{f}
and show that they are not equal. By what factor do they differ? Hint: To compute (∂_{a}/∂_{b})f, use a formula for a in terms of b and f but not c.
problem 2: Adiabats and Isotherms. An ideal gas with 5 quadratic degrees of freedom is expanded to three times its initial volume: V_{f} = 3V_{i}. Assume that there are N atoms, starting at T_{i}, with P_{i}. Answer the following problems with two different assumptions: (I) adiabatic compression and (II) isothermal compression: (Answers for a-e should be given in terms of P_{i}, T_{i}, N, and k_{B}.)
(a) What is the final pressure P_{f} in terms of P_{i}?
(b) What is the final temperature T_{f} in terms of T_{i}?
(c) How much work was done?
(d) How much heat was added to the system? (give a negative quantity if heat lost)
(e) How much total energy was added to the system?
(f) find out, in Joules, the amount of work done (as you found in (c) above) if the number of molecules is 100NA, the initial temperature is 25C, and the initial pressure is atmospheric pressure.
problem 3: Probability. Let’s look at the probability distributions for the sum of what you get by rolling 5-sided dice. Plot the probability distribution of the outcome of 1 die, the sum of 2 dice, and the sum of 3 dice. (Make 3 separate plots.) You can check your answer by verifying that the sum of your probability distribution is 100%. What is the general trend in shape as the number of dice increases?