Domain: Physical chemistry

\
 Using the expression for the second virial coefficient \( B = -2\pi N_A \int_0^\infty \left[ e^{-u(r)/kT} - 1 \right] r^2 \, dr \), where \( u(r) \) is the intermolecular potential, calculate the value of \( B \) for nitrogen gas (\( \text{N}_2 \)) at \( T = 300 \, \text{K} \) using the Lennard–Jones potential \( u(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right] \), with parameters \( \frac{\varepsilon}{k} = 95.0 \, \text{K} \) and \( \sigma = 3.75 \, \text{AA} \). Report the final value of \( B \) in units of \( \text{cm}^3 \, \text{mol}^{-1} \).

Grading rubric:
criterion: Correctly identifies the integrand term (exp(-u/kT) - 1) as the Mayer f-function.
weight: 20
criterion: Correctly states that the integral converges at the lower limit because of the repulsive potential term.
weight: 20
criterion: Correctly includes the Jacobian factor sigma-cubed in the transformed integral expression.
weight: 10
criterion: Correctly calculates the reduced inverse temperature parameter (epsilon/kT) as 0.3167.
weight: 20
criterion: Correctly calculates the dimensional pre-factor value as approximately 1.995e-4 m^3/mol (or 199.5 cm^3/mol).
weight: 10
criterion: Correctly states that the negative sign of the final result indicates the dominance of attractive forces.
weight: 10
criterion: Correctly reports the final numerical answer as -4.5 cm^3 mol^-1.
weight: 10

Answer:
-4.5 cm^3 mol^-1