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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 8: Q. 8.9 (page 338)

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?

Short Answer

Expert verified

At, $r = \frac{r_{0}}{2^{\frac{1}{6}}}$ the value of Lennard-Jones potential become 0.

Step by step solution

01

Step 1. Given information

Lennard- Jones potential=

$u (r) = u_{0} [{(\frac{r_{0}}{r})}^{12} - 2 {(\frac{r_{0}}{r})}^{6}]$

02

Step 2. Differentiating both sides with respect to r

We get,

$\frac{d u}{d r} = u_{0} [r_{0}^{12} 脳 - 12 \frac{1}{r^{13}} - 2 r_{0}^{6} 脳 - 6 \frac{1}{r^{7}}]$

= u_{0} [- 12 \frac{r_{0}^{12}}{r^{13}} + 12 \frac{r_{0}^{6}}{r^{7}}]

For extreme points,

$\frac{d u}{d r} = 0$

$鈭� u_{0} [- 12 \frac{r_{0}^{12}}{r^{13}} + 12 \frac{r_{0}^{6}}{r^{7}}] = 0$

$\frac{12 r_{0}^{6} u_{0}}{r^{13}} [r^{6} - r_{0}^{6}] = 0$

$r^{6} - r_{0}^{6} = 0$

$r = r_{0}$

Thus the Lennard-Jones potential has extremum point at $r = r_{0}$

03

Step 3. To check that the potential at r=r0 is maximum or minimum take double derivative

We get,

\frac{d^{2} u}{d r^{2}} = u_{0} [- 12 脳 \frac{- 13 r_{0}^{12}}{r^{14}} + 12 脳 \frac{- 7 r_{0}^{6}}{r^{8}}]

$= 12 u_{0} [\frac{13 r_{0}^{12}}{r^{14}} - \frac{7 r_{0}^{6}}{r^{8}}]$

${\frac{d^{2} u}{d r^{2}}|}_{r = s_{0}} = 12 u_{0} [\frac{13 r_{0}^{12}}{r_{0}^{14}} - \frac{7 r_{0}^{6}}{r_{0}^{8}}]$

$= 12 u_{0} [\frac{13}{r_{0}^{2}} - \frac{7}{r_{0}^{2}}]$

$= \frac{12 u_{0}}{r_{0}^{2}} [6]$

$= \frac{72 u_{0}}{r_{0}^{2}} > 0$

${鈭� \frac{d^{2} u}{d r^{2}}|}_{r = r_{0}} > 0$

So the Lennard- Jones potential has minimum value at $r^{'} = r_{0}$

${u (r)|}_{r = r_{0}} = u_{0} {[{(\frac{r_{0}}{r})}^{12} - 2 {(\frac{r_{0}}{r})}^{6}]}_{r = r_{0}}$

$= u_{0} [{(\frac{r_{0}}{r})}^{12} - 2 {(\frac{r_{0}}{r})}^{6}]$

$= u_{0} [1 - 2]$

${u (r)|}_{r = r_{0}} = - u_{0}$

04

Step 4. If the potential becomes 0.

$u (r) = 0$

$u_{0} [{(\frac{r_{0}^{*}}{r})}^{12} - 2 {(\frac{r_{0}}{r})}^{6}] = 0$

$\frac{u_{0}}{r^{12}} [r_{0}^{12} - 2 r_{0}^{6} r^{6}] = 0$

$r_{0}^{12} - 2 r_{0}^{6} r^{6} = 0$

$r^{6} = \frac{r_{0}^{12}}{2 r_{0}^{6}} = \frac{r_{0}^{6}}{2}$

$r^{6} = \frac{r_{0}^{12}}{2 r_{0}^{6}}$

$r = \frac{1}{2^{\frac{1}{6}}} r_{0}$

Thus the Lennard-Jones potential becomes zero at $r = \frac{1}{2^{\frac{1}{6}}} r_{0}$

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Most popular questions from this chapter

Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialise subroutine compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 x 5 lattice for T values from 4 down to l in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10x 10 lattice and for a 20 x 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.)

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.

(a) Write down a formula for the grand partition function (Z) of a weakly interacting gas in thermal and diffusive equilibrium with a reservoir at fixed T and碌. Express Z as a sum over all possible particle numbers N, with each term involving the ordinary partition function Z(N).

(b) Use equations 8.6 and 8.20 to express Z(N) as a sum of diagrams, then carry out the sum over N, diagram by diagram. Express the result as a sum of similar diagrams, but with a new rule 1 that associates the expression (>./vQ) J d3ri with each dot, where >. = e13碌,. Now, with the awkward factors of N(N - 1) 路路路 taken care of, you should find that the sum of all diagrams organizes itself into exponential form, resulting in the formula

Note that the exponent contains all connected diagrams, including those that can be disconnected by removal of a single line.

(c) Using the properties of the grand partition function (see Problem 7.7), find diagrammatic expressions for the average number of particles and the pressure of this gas.

(d) Keeping only the first diagram in each sum, express N(碌) and P(碌) in terms of an integral of the Mayer /-function. Eliminate 碌 to obtain the same result for the pressure (and the second virial coefficient) as derived in the text.

(e) Repeat part (d) keeping the three-dot diagrams as well, to obtain an expression for the third virial coefficient in terms of an integral of /-functions. You should find that the A-shaped diagram cancels, leaving only the triangle diagram to contribute to C(T).

Modify the ising program to simulate a one-dimensional Ising model.

(a) For a lattice size of 100, observe the sequence of states generated at various temperatures and discuss the results. According to the exact solution (for an infinite lattice), we expect this system to magnetise only as the temperature goes to zero; is the behaviour of your program consistent with this prediction? How does the typical cluster size depend on temperature?

(b) Modify your program to compute the average energy as in Problem 8.27. Plot the energy and heat capacity vs. temperature and compare to the exact result for an infinite lattice.

(c) Modify your program to compute the magnetisation as in Problem 8.28. Determine the most likely magnetisation for various temperatures and sketch a graph of this quantity. Discuss.

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.

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?

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