How does the mass diffusivity of a gas mixture change with \((a)\) temperature and \((b)\) pressure?

Mass diffusivity (D) is a measure of how fast one gas diffuses into another gas. It is dependent on temperature (T), pressure (P), and the properties of the gases involved, such as their molecular weights and sizes. We will consider a binary gas mixture, which consists of two gases, A and B, diffusing into each other.

The mass diffusivity of a binary gas mixture can be calculated using the Chapman-Enskog theory, which gives the following formula: D = \frac{3}{16} \frac{(k_B T)^{3/2}}{(\pi \mu)^{1/2} \sigma^2 \Omega_D} where, - D is mass diffusivity - k_B is Boltzmann's constant - T is the temperature (in Kelvin) - \mu is the reduced mass of the gas pair, given by \frac{m_A m_B}{m_A+m_B} (m_A and m_B are the molecular masses of gases A and B) - \sigma is the collision diameter (a measure of the size of the gas molecules) - \Omega_D is a dimensionless quantity dependent on the gas properties and temperature, which accounts for the collision interactions between the two gases.

To study the effect of temperature on mass diffusivity, we need to look at how D changes with T, keeping pressure and other parameters constant. From the formula above, we can see that D is directly proportional to T^{3/2}. Therefore, as the temperature increases, the mass diffusivity also increases, which implies that gas molecules will diffuse faster at higher temperatures.

Pressure does not appear explicitly in the formula for mass diffusivity, so we need to analyze its effect indirectly. The collision diameter, \sigma, which is independent of pressure, can be considered constant. However, pressure affects the concentration (n/V) of the gas mixture, where n is the number of moles of the gas and V is the volume. As per the ideal gas equation, PV = nRT \Rightarrow \frac{n}{V}= \frac{P}{RT} As the pressure increases, the concentration increases which means that the number of collisions between gas molecules will also increase. An increased number of collisions will reduce the average distance traveled by the molecules between successive collisions, leading to a decrease in mass diffusivity. In summary, (a) As the temperature increases, the mass diffusivity of a gas mixture increases. (b) As the pressure increases, the mass diffusivity of a gas mixture decreases.