Question

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 $kT/u_0$ ranging from $1$to $7$. On the same graph, plot the data for nitrogen given in Problem 1.17, choosing the parameters $r_0$ and $u_0$so as to obtain a good fit.

Short answer

The value of the parameter $u_0=0.008\mathrm{eV}$

And $r_0=3.0\times {10}^{-10}\mathrm{m}$

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

Given information

A gas of molecules interacting via the Lennard-Jones potential, for values of $kT/u_0$ranging from $1$ to $7$.

Explanation

Compute the second virial coefficient's expression$B$ as:
$B\left(T\right)=-2\pi {\int }_0^{\infty }\left(e^{-\frac{u\left(r\right)}{dT}}-1\right)r^{2}dr$

Where, $u\left(r\right)$denotes Lennard Jones potential,

$T$denotes temperature of gas

$k$indicates Boltzmann constant.

Calculate the expression of Lennard Jones potential as:
$u\left(r\right)=u_0\left({\left(\frac{r_0}{r}\right)}^{12}-2{\left(\frac{r_0}{r}\right)}^6\right)$
Note that, $u_0$and, $r_0$ are constants.

Then $r$indicates the radial distance.

Explanation

Construct a graph to plot the second virial coefficient $B$along $y$axis and $\left(\frac{kT}{u_0}\right)$along $x$axis:

Explanation

Construct a graph to fit the above plot of nitrogen data from problem 1.17 with $u_0$set at $0.008\mathrm{eV}$and $r_0$set at $3.0\times {10}^{-10}\mathrm{m}$:

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.