Numerical evaluation of the second virial coefficient


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

PV/Nk T = 1 + ρB2(T) + ρ2 B3(T) + ρ3 B4(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 Bn is the nth virial coefficient.

     The second virial coefficient B2 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.


  1. Describe the qualitative temperature dependence of B2. Is it positive or negative at low/high temperatures? Is there a temperature at which B2 equals zero?
  2. 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 × 10-10m. Given the numerical results you found in Problem 1, what is the temperature at which B2 = 0? This temperature is called the Boyle temperature.
  3. Simple arguments given in standard textbooks on statistical mechanics show that B2 has the approximate form, B2 = b - a/kT. Interpret b and a as fit parameters and determine approximate values for a and b from your results for B2.

Java Classes

Updated 28 December 2009.