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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 8: Q. 8.13 (page 338)

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 approximately $U 鈮� \frac{3}{2} N k T + \frac{N^{2}}{V} 路 2 蟺 {鈭�}_{0}^{鈭�} r^{2} u (r) e^{- 尾 u (r)} d r$ Use 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.

Short Answer

Expert verified

Since, the energy is $U = \frac{3}{2} N k T + 2 蟺路 \frac{N^{2}}{V} 路鈭� r^{2} 路 u (r) e^{- 尾路 u (r)} d r$ . Hence, Proved.

The plot of the temperature-dependent part of the correction term is

Step by step solution

01

Given information

We have been given that $d U = - \frac{1}{Z} 路 \frac{d Z}{d 尾} = - \frac{d}{d 尾} (\frac{d Z}{Z})$

02

Simplify

There are two terms in equation

$U = U_{i d} + U_{e} = - \frac{d}{d 尾} \ln (Z_{i d}) - \frac{d}{d 尾} \ln (Z_{e})$

First term is equal to:

$U_{i d} = - \frac{d}{d 尾} \ln (Z_{i d}) = \frac{3}{2} N k T$

The second part is:

$U_{e} = - \frac{d}{d 尾} (\frac{1}{2} \frac{N^{2}}{V}) (鈭� u (r) d^{3} r)$

Now, we will use the equation

$U_{e} = - \frac{d}{d 尾} (\frac{1}{2} \frac{N^{2}}{V}) (鈭� u (r) d^{3} r) = - \frac{d}{d 尾} (\frac{1}{2} \frac{N^{2}}{V}) (4 蟺路鈭� r^{2} 路 u (r) d r)$

$= - 2 蟺路 \frac{N^{2}}{V} 路鈭� r^{2} 路 u (r) e^{- 尾路 u (r)} d r$

The energy term is given by:

$= \frac{3}{2} N k T + 2 蟺路 \frac{N^{2}}{V} 路鈭� r^{2} 路 u (r) e^{- 尾路 u (r)} d r$

The second virial coefficient is given by:

$U_{e} = - \frac{d}{d 尾} (\frac{1}{2} \frac{N^{2}}{V}) (鈭� u (r) d^{3} r) = - \frac{d}{d 尾} (\frac{1}{2} \frac{N^{2}}{V}) (4 蟺路鈭� r^{2} 路 u (r) d r)$

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Most popular questions from this chapter

Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is $卤系$ . 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 $k T / 系$ . 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?

Consider a gas of "hard spheres," which do not interact at all unless their separation distance is less than $r_{0}$ , in which case their interaction energy is infinite. Sketch the Mayer $f$ -function for this gas, and compute the second virial coefficient. Discuss the result briefly.

Show that the Lennard-Jones potential reaches its minimum value at $r = r_{0}$ , and that its value at this minimum is $- u_{0}$ . At what value of $r$ does the potential equal zero?

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 $蔚$ ) for the particular state of the $4 脳 4$ square lattice shown in Figure 8.4?

Figure 8.4. One particular state of an Ising model on a $4 脳 4$ square lattice (Problem 8.15).

Modify the ising program to compute the total magnetisation (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetisation value occurs during a run, plotting the results as a histogram. Run the program for a 5 x 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetisation value as a function of temperature. If your computer is fast enough, repeat for a 10 x 10 lattice.

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