Question

Determine the binary diffusion coefficient for \((a)\) carbon dioxide in nitrogen, \((b)\) carbon dioxide in oxygen, and \((c)\) carbon dioxide in hydrogen at \(320 \mathrm{~K}\) and \(2 \mathrm{~atm}\).

Short answer

Question: Determine the binary diffusion coefficient for carbon dioxide (CO2) in three different gases: nitrogen (N2), oxygen (O2), and hydrogen (H2) at a temperature of 320 K and pressure of 2 atm using the Chapman-Enskog equation.

Step-by-step solution

Understand the Chapman-Enskog equation

Chapman-Enskog equation allows us to compute the binary diffusion coefficient for a pair of gases under specified temperature and pressure, which is given by: \(D_{12} = \dfrac{3}{16}\sqrt{\dfrac{2\pi k_B T}{\mu_{12}}}\dfrac{(1+\frac{m_1}{m_2})(1+\frac{m_2}{m_1})}{P\sigma_{12}^2\Omega_D^*(k_B T)}\) where \(D_{12}\) is the binary diffusion coefficient, \(k_B\) is the Boltzmann constant, \(T\) is the temperature, \(\mu_{12}\) is the reduced mass, \(P\) is the pressure, and \(\sigma_{12}\) and \(\Omega_D^*\) are Lennard-Jones parameters.

Determine the reduced mass for each pair of gases

The reduced mass \(\mu_{12}\) for a pair of gases with masses \(m_1\) and \(m_2\) is: \(\mu_{12} =\dfrac{m_1 m_2}{m_1 + m_2}\) We will use the molecular masses of carbon dioxide (CO2), nitrogen (N2), oxygen (O2), and hydrogen (H2) to compute the reduced mass for each pair of gases.

Find Lennard-Jones parameters for each pair of gases

The Lennard-Jones parameters \(\sigma_{12}\) and \(\Omega_D^*\) are found using the following formulas: \(\sigma_{12} = \dfrac{\sigma_1 + \sigma_2}{2}\) \(\Omega_D^*(k_B T) = \dfrac{5}{16}\dfrac{(\pi(1+\frac{m_1}{m_2})(1+\frac{m_2}{m_1}))^{\frac{1}{2}}}{\pi^{\frac{2}{3}}\epsilon_{12}^{\frac{1}{3}}(k_B T)^{\frac{1}{6}}}\) where \(\sigma_1\), \(\sigma_2\), \(\epsilon_1\), and \(\epsilon_2\) are Lennard-Jones parameters for each individual gas. These values can be found in literature or gas property tables.

Apply the given temperature and pressure

Now that we have all the necessary parameters, we can apply the given temperature \(T = 320 \mathrm{~K}\) and pressure \(P = 2 \mathrm{~atm}\) to the Chapman-Enskog equation for each pair of gases. Calculate \(D_{12}\) for each case: (a) For carbon dioxide in nitrogen (CO2-N2) (b) For carbon dioxide in oxygen (CO2-O2) (c) For carbon dioxide in hydrogen (CO2-H2) By performing these calculations, we will determine the binary diffusion coefficient for each case.