Molecular Simulation/Dilute gases

Ideal Gas Law
Gasses are often described using ideal gas law, which relates the pressure of a gas to its density with a simple expression.
 * $$\frac{P}{k_B T} = \rho $$

Where $$\rho$$ is the density of the gas, $$P$$ is the pressure of the gas, $$k_B$$ is the Boltzmann constant, equal to $$1.38064852 \times 10^{-23} m^2 kg s^{-2} K^{-1}$$, and $$T$$ is the temperature in Kelvin. The expression is derived from approximating a gas as point masses that have only kinetic energy and experience perfectly elastic collisions. Unfortunately, this theory breaks down at densities where intermolecular forces become significant, that is, when the potential energy is non-zero. Ideal gas law is suitable for only very dilute gasses.

Virial Theorem
To more accurately describe the properties of dilute gasses the Virial equation of state is used. Virial theorem accounts for the effects of intermolecular forces, through an expansion into higher order functions of the density. Mathematically, this is described using infinite power series, where $$B_2(T)$$ and $$B_3(T)$$ are the second and third Virial coefficients.
 * $$\frac{P}{k_B T} = \rho + B_2(T)\rho^2 +B_3(T)\rho^3 + ...$$

At low densities the deviations from ideal gas behavior can be sufficiently described in the second Virial coefficient, $$B_2(T)$$.
 * $$B_2(T) = -\frac{1}{2V}(Z_2-Z_1^2) $$

Where $$V$$ is the volume of a gas and $$Z$$ is the configurational integral. The configurational integral for the second Virial coefficient is the contribution of every possible pair of positions weighted over its Boltzmann distribution. For the third Virial coefficient the configurational integral would be the contribution of three interacting particles. By extension, the configurational integral for any nth Virial coefficient would be the contribution of n particles interacting. For this reason higher Virial coefficients are quite complicated to derive, fortunately they are only necessary for describing gasses at pressures above 10atm. The configurational integral for two interacting particles is as follows:
 * $$Z = \int\int e^{\frac{-u(r_1,r_2)}{k_BT}} dr_1dr_2$$

Where $$u$$ is the potential energy of the interaction of a single pair of particles, and $$r_1$$ and $$r_2$$ are the positions of particles 1 and 2. The second virial coefficient can be written in terms of pairwise intermolecular interaction potential, $$u_r$$, if the position of particle 2 is defined relative to the position of particle 1. The distance, $$r$$, would then be the distance between the two interacting particles. The equation derived from this modification is as follows:
 * $$B_2 = -2\pi\int\limits_{0}^{\infty} \left[e^{\frac{-u(r)}{k_BT}}-1 \right] r^2dr$$

A different $$B_2$$ is derived for the hard sphere potential and the Lennard-Jones potential.

Hard Sphere Potential
The hard sphere model approximates particles as hard spheres that cannot overlap, if the spheres are not overlapping then the potential energy is zero and if they are overlapping then the potential energy is infinitely high. This approximation represents the very strong short range Pauli repulsion forces. The equation for the potential energy is as follows:
 * $$\mathcal{V}(r) = \begin{cases} \mathcal{V} = \infty & r < r_{cut} \\ \mathcal{V} = 0 & r\geq r_{cut} \end{cases}$$

Where $$\mathcal{V}$$ is the potential energy and $$r_{cut}$$ is the radius of the hard sphere. Integrating the configurational integral for the hard sphere potential gives, $$B_2(T)=\frac{2\pi}{3}r^3_{cut}$$, as the second Virial coefficient. This model is crude and only accounts for repulsive forces, a slightly more accurate model would be the Lennard-Jones potential model.

Lennard-Jones Potential
The Lennard-Jones potential is a combination of a polynomial repulsion term, $$\upsilon(r) = \frac{C_{12}}{r^{12}}$$, and a London dispersion attractive term, $$\upsilon(r) = \frac{C_{6}}{r^6}$$.
 * $$\upsilon(r) = \frac{C_{12}}{r^{12}} - \frac{C_{6}}{r^6}$$

The $$C_{12}$$ and $$C_6$$ terms can be expanded, and internal energy, $$u(r)$$, can be expressed as:
 * $$u(r)= 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right]$$

Where $$\varepsilon$$ is the potential well depth, $$\sigma$$ is the intercept, and $$r$$ is the distance between the particles. The second Virial coefficient derived from the Lennard-Jones potential has no analytical solution and must be solved numerically.
 * $$B_2 = \int\limits_{0}^{\infty} \left[ e^{-\frac{4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right]}{k_BT}} - 1 \right] r^2 dr$$

The Lennard-Jones model is a more accurate than the hard sphere model as it accounts for attractive interactions and the repulsive term is more realistic than hard sphere repulsion. That being said, it is still limited in the fact that only London dispersion attractive interactions are considered, making the model applicable to only nobel gasses.