Tag: Solutions

Problem 7.6 Show that when a system is in thermal and di⌥usive equilibrium with a reservoir, the average number of particles in the system is where the partial derivative is taken at fixed temperature and volume. Show also that the mean square number of particles is Use these results to show that the standard deviation…

Problem 7.4 Repeat the previous problem, taking into account the two independent spin states of the electron. Now the system has two “occupied” states, one with the electron in each spin configuration. However, the chemical potential of the electron gas is also slightly different. Show that the ratio of probabilities is the same as before:…

Problem 7.3 Consider a system consisting of a single hydrogen atom/ion, which has two possible states: unoccupied (i.e., no electron present) and occupied (i.e., one electron present, in the ground state). Calculate the ratio of the probabilities of these two states, to obtain the Saha equation, already derived in Section 5.6. Treat the electrons as…

Problem 6.48 For a diatomic gas near room temperature, the internal partition function is simply the rotational partition function computed in Section 6.2, multiplied by the degeneracy of the electronic ground state. (a) Show that the entropy in this case is Calculate the entropy of a mole of oxygen at room temperature and atmospheric pressure,…

Problem 6.43 Some advanced textbooks define entropy by the formula where the sum runs over all microstates accessible to the system and is the probability of the system being in microstate (a) For an isolated system, for all accessible states Show that in this case the preceding formula reduces to our familiar definition of entropy.…

Problem 6.39 A particle near earth’s surface traveling faster than about 11 km/s has enough kinetic energy to completely escape from the earth, despite earth’s gravitational pull. Molecules in the upper atmosphere that are moving faster than this will therefore escape if they do not suffer any collisions on the way out. (a) The temperature…

Problem 6.34 Carefully plot the Maxwell speed distribution for nitrogen molecules at T = 300 K and at T = 600 K. Plot both graphs on the same axes, and label the axes with numbers. Solution: Problem 6.34 Solution (Download)

Problem 6.32 Consider a classical particle moving in a one-dimensional potential well as shown in Figure 6.10. The particle is in thermal equilibrium with a reservoir at temperature so the probabilities of its various states are determined by Boltzmann statistics. (a) Show that the average position of the particle is given by where each integral…

Problem 6.27 Use a computer to sum the exact rotational partition function (equation 6.30) numerically, and plot the result as a function of Keep enough terms in the sum to be confident that the series has converged. Show that the approximation in equation 6.31 is a bit low, and estimate by how much. Explain the…

Problem 6.24 For an molecule, the constant is approximately 0.00018 eV. Estimate the rotational partition function for an molecule at room temperature. Solution: Problem 6.24 Solution (Download)