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For each of the diagrams shown in equation 8.20, write down the corresponding formula in terms of functions, and explain why the symmetry factor gives the correct overall coefficient.
Problem 8.10. Use a computer to calculate and plot the second virial coefficient for a gas of molecules interacting via the Lennard-Jones potential, for values of ranging from to . On the same graph, plot the data for nitrogen given in Problem 1.17, choosing the parameters and so as to obtain a good fit.
Consider a gas of "hard spheres," which do not interact at all unless their separation distance is less than , in which case their interaction energy is infinite. Sketch the Mayer -function for this gas, and compute the second virial coefficient. Discuss the result briefly.
Consider a gas of molecules whose interaction energy u is infinite for and negative for , with a minimum value of . Suppose further that , so you can approximate the Boltzmann factor forusing . Show that under these conditions the second virial coefficient has the form , the same as what you found for a van der Waals gas in Problem 1.17. Write the van der Waals constants and in terms of and , and discuss the results briefl
Problem 8.13. Use the cluster expansion to write the total energy of a monatomic nonideal gas in terms of a sum of diagrams. Keeping only the first diagram, show that the energy is approximatelyUse a computer to evaluate this integral numerically, as a function of , for the Lennard-Jones potential. Plot the temperature-dependent part of the correction term, and explain the shape of the graph physically. Discuss the correction to the heat capacity at constant volume, and compute this correction numerically for argon at room temperature and atmospheric pressure
Use the cluster expansion to write the total energy of a monatomic nonideal gas in terms of a sum of diagrams. Keeping only the first diagram, show that the energy is approximatelyUse a computer to evaluate this integral numerically, as a function of T, for the Lennard-Jones potential. Plot the temperature-dependent part of the correction term, and explain the shape of the graph physically. Discuss the correction to the heat capacity at constant volume, and compute this correction numerically for argon at room temperature and atmospheric pressure.
In this section I've formulated the cluster expansion for a gas with a fixed number of particles, using the "canonical" formalism of Chapter 6. A somewhat cleaner approach, however, is to use the "grand canonical" formalism introduced in Section 7.1, in which we allow the system to exchange particles with a much larger reservoir.
For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four "neighbors"-above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ) for the particular state of the square lattice shown in Figure 8.4?
Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer?
Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is . Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of . Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down?
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