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Chapter 8: Q. 8.10 (page 337)

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 kT/u0 ranging from 1to 7. On the same graph, plot the data for nitrogen given in Problem 1.17, choosing the parameters r0 and u0so as to obtain a good fit.

Short Answer

Expert verified

The value of the parameter u0=0.008eV

And r0=3.0脳10-10m

The data for nitrogen is fitted with the plot after the second virial coefficient for gas of molecules is calculated and plotted.

Step by step solution

01

Given information

A gas of molecules interacting via the Lennard-Jones potential, for values of kT/u0ranging from 1 to 7.

02

Explanation

Compute the second virial coefficient's expressionB as:
B(T)=-2蟺鈭�0鈭�e-u(r)dT-1r2dr

Where, u(r)denotes Lennard Jones potential,

Tdenotes temperature of gas

kindicates Boltzmann constant.

Calculate the expression of Lennard Jones potential as:
u(r)=u0r0r12-2r0r6
Note that, u0and, r0 are constants.

Then rindicates the radial distance.

03

Explanation

Construct a graph to plot the second virial coefficient Balong yaxis and kTu0along xaxis:

04

Explanation

Construct a graph to fit the above plot of nitrogen data from problem 1.17 with u0set at 0.008eVand r0set at 3.0脳10-10m:

Fitted nitrogen data are represented by faded dots.
In order to fit Nitrogen data with the plot, the second virial coefficient for gas molecules is calculated.

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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 kT/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?

In this problem you will use the mean field approximation to analyse the behaviour of the Ising model near the critical point.

(a) Prove that, when x鈮�1,tanhx鈮�x-13x3

(b) Use the result of part (a) to find an expression for the magnetisation of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M鈭�Tc-T尾炉,where尾(not to be confused with 1/kT) is a critical exponent, analogous to the f defined for a fluid in Problem 5.55. Onsager's exact solution shows that 尾=1/8in two dimensions, while experiments and more sophisticated approximations show that 尾鈮�1/3in three dimensions. The mean field approximation, however, predicts a larger value.

(c) The magnetic susceptibility 蠂is defined as 蠂鈮�(鈭�M/鈭�B)T. The behaviour of this quantity near the critical point is conventionally written as 蠂鈭�T-Tc-纬 , where y is another critical exponent. Find the value of in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Te. (The exact value of y in two dimensions turns out to be 7/4, while in three dimensions 纬鈮�1.24.)

Show that the Lennard-Jones potential reaches its minimum value at r=r0, and that its value at this minimum is -u0. At what value of rdoes the potential equal zero?

The Ising model can be used to simulate other systems besides ferromagnets; examples include anti ferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of -u0to the energy for each pair of neighbouring sites that are both occupied.

(a) Write down a formula for the grand partition function for this system, as a function of u0, T, and p.

(b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements u0鈫�4系and 渭鈫�2渭BB-8系. (Note that is the chemical potential of the gas while uB is the magnetic moment of a dipole in the magnet.)

(c) Discuss the implications. Which states of the magnet correspond to low density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the P-T plane?

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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