Introduction

The pressure equation of state for a dilute gas can be written as

PV/Nk T = 1 + ρB_{2}(T) + ρ^{2}
B_{3}(T) + ρ^{3}
B_{4}(T) + …,

where P is the pressure, V is the volume, N is the number of particles, T is the temperature, and k is Boltzmann's constant. The density ρ = N/V, and B_{n} is the nth virial coefficient.

The second virial coefficient B_{2} is given by the integral

,

where β = 1/kT and u(r) is the interparticle potential. Except for simple forms of u(r), the integral must be done numerically.

The program computes the integral for the Lennard-Jones potential at various temperatures. Simpson's rule is used. The program uses units such that the Lennard-Jones parameters ε and σ are set equal to unity.

Problems

- Describe the qualitative temperature dependence of B
_{2}. Is it positive or negative at low/high temperatures? Is there a temperature at which B_{2}equals zero? - The Lennard-Jones potential is a reasonable approximation to the interparticle potential for liquid Argon with ε/k = 119.5 K and σ = 3.76 × 10
^{-10}m. Given the numerical results you found in Problem 1, what is the temperature at which B_{2}= 0? This temperature is called the Boyle temperature. - Simple arguments given in standard textbooks on statistical mechanics show that B
_{2}has the approximate form, B_{2}= b - a/kT. Interpret b and a as fit parameters and determine approximate values for a and b from your results for B_{2}.

Java Classes

- SecondVirialApp

Updated 28 December 2009.