Question

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

Short answer

Answer: (a) As the temperature increases, the mass diffusivity of a gas mixture also increases. (b) As the pressure increases, the mass diffusivity of a gas mixture generally decreases.

Step-by-step solution

Understanding the concept of mass diffusivity

Mass diffusivity (also known as diffusion coefficient) characterizes the rate at which one substance (usually gas) dissolves or spreads into another substance. It depends on factors like temperature, pressure, and the properties of the substances in a mixture.

Formula for mass diffusivity

The mass diffusivity for a gas mixture can be estimated using Chapman-Enskog theory, which results in the following formula for binary mixture (with species A and B): D_AB = \dfrac{3\sqrt{\frac{8k_BT}{\pi \mu_AB}}}{(\sigma_{AB}^2 \Omega_D^{AB})} where D_AB is the mass diffusivity of species A in species B, k_B is Boltzmann’s constant, T is the temperature, \mu_AB is the reduced mass of species A and B, \sigma_{AB} is the effective collision diameter, and \Omega_D^{AB} is the dimensionless collision integral related to diffusion, which depends on T*=\dfrac{k_BT}{\sqrt{\epsilon_{AB}}} (T* is the reduced temperature and \epsilon_{AB} is the energy corresponding to the depth of the potential well).

Dependence on Temperature

From the above formula, we can see that mass diffusivity is directly proportional to the square root of the temperature, all other factors remaining constant: D_AB ∝ √T This means that if the temperature increases, the mass diffusivity of the gas mixture will also increase. This is because the molecules have more kinetic energy at higher temperatures, which causes them to move faster and therefore diffuse more rapidly.

Dependence on Pressure

The pressure dependence of mass diffusivity is not explicit in the Chapman-Enskog formula, but it can be related to the density of the gas mixture, which is inversely proportional to the molar volume (V_m): P = \dfrac{n_RT}{V_m} where P is pressure, n is the amount of moles of gas, and R is the gas constant. Keeping the number of moles and temperature constant, we can write the relation as: V_m ∝ \dfrac{1}{P} Higher pressure implies higher density, i.e., more molecules per unit volume. Thus, higher pressures might reduce the mass diffusivity due to more frequent collisions and less room for the molecules of species A to penetrate the mixture of species B. In summary: \((a)\) As the temperature increases, the mass diffusivity of a gas mixture also increases. \((b)\) As the pressure increases, the mass diffusivity of a gas mixture generally decreases.