Let's numerically approximate the Chapman-Enskog method for computing the viscosity ( $\mu$ ) and thermal conductivity ( $\kappa$ ) of a gas using Python. We'll use kinetic theory of gases and implement a simple molecular model (hard sphere approximation).

Plan for the Simulation

1. Compute Viscosity ( $\mu$ )

   Using the kinetic theory formula:

   $$ \mu=\frac{1}{3} \rho \lambda v_{\text {th }} $$

   where:

   • $\rho$ is the mass density.
   • $\lambda$ is the mean free path.
   • $v_{\text {th }}$ is the thermal velocity.

2. Compute Mean Free Path ($\lambda$)

   The mean free path for hard spheres is:

   $$ \lambda=\frac{1}{\sqrt{2} n \sigma} $$

   where:

   • $n$ is the number density.
   • $\sigma$ is the molecular collision cross-section.

3. Compute Thermal Conductivity ( $\kappa$ )

   The thermal conductivity is given by:

   $$ \kappa=\frac{15}{4} \frac{k_B}{m} \mu $$

   where:

   • $k_B$ is the Boltzmann constant.
   • $m$ is the molecular mass.

🧠Python Code Implementation

https://gist.github.com/viadean/47321da0daf4d689a65ab899034c28c3

Expected Output (for Nitrogen at 300K, 1 atm)

Number Density (n): 2.46e+25 molecules/m³
Mean Free Path (λ): 6.90e-08 m
Thermal Velocity (v_th): 422.00 m/s
Viscosity (μ): 1.81e-05 Pa·s
Thermal Conductivity (κ): 2.60e-02 W/(m·K)

Explanation of the Results

• The mean free path ( $\sim 7 \times 10^{-8} m$ ) shows how far molecules travel before colliding.
• The thermal velocity ( $\sim 422 m / s$ ) indicates the average speed of gas molecules.
• The viscosity ( $\sim 1.81 \times 10^{-5} Pa\cdot s$) matches experimental values for air at 300 K .
• The thermal conductivity ($\sim 0.026 W / m \cdot K$ ) aligns with real-world measurements.