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 $k T / u_{0}$ 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

Expert verified

The value of the parameter $u_{0} = 0.008 eV$

And $r_{0} = 3.0 \times 10^{- 10} m$

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 $k T / u_{0}$ ranging from $1$ to $7$ .

Explanation

Compute the second virial coefficient's expression $B$ as:
$B (T) = - 2 π \int_{0}^{\infty} (e^{- \frac{u (r)}{d T}} - 1) r^{2} d r$

Where, $u (r)$ denotes Lennard Jones potential,

$T$ denotes temperature of gas

$k$ indicates Boltzmann constant.

Calculate the expression of Lennard Jones potential as:

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

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 $(\frac{k T}{u_{0}})$ along $x$ axis:

Explanation

Construct a graph to fit the above plot of nitrogen data from problem 1.17 with $u_{0}$ set at $0.008 eV$ and $r_{0}$ set at $3.0 \times 10^{- 10} m$ :

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.

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