Question

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

Short answer

Answer: As the temperature increases, the mass diffusivity of a gas mixture increases, which means gas molecules will diffuse faster at higher temperatures. On the other hand, as the pressure increases, the mass diffusivity of a gas mixture decreases, due to increased number of collisions between gas molecules.

Step-by-step solution

Understand mass diffusivity

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.

Formula for mass diffusivity

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.

Effect of temperature on mass diffusivity

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.

Effect of pressure on mass diffusivity

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.

Key concepts

Binary Gas Mixture

A binary gas mixture consists of two different types of gas molecules, usually referred to as Gas A and Gas B. In such a mixture, the gases intermingle and diffuse into each other, a process driven by the random movement of molecules. This diffusion allows the gases to spread throughout the available space, gradually creating a uniform composition. The rate at which this diffusion occurs can be quantified using mass diffusivity, a crucial parameter in understanding gas behavior. The diffusion process in a binary gas mixture is influenced by several factors, including the molecular weights and sizes of the gases involved. Lighter and smaller molecules usually diffuse faster compared to heavier and larger ones. This property is central to applications such as gas separation, environmental monitoring, and chemical engineering.

Chapman-Enskog Theory

The Chapman-Enskog theory is a fundamental framework for analyzing gas diffusion, providing insights into how different factors affect the mass diffusivity of gases. Developed by David Enskog and Sydney Chapman in the early 20th century, this theory offers a mathematical approach to model the distribution of molecular velocities in gases, underpinned by statistical mechanics. According to this theory, mass diffusivity (D) for a binary gas mixture is given by:\[ D = \frac{3}{16} \frac{(k_B T)^{3/2}}{(\pi \mu)^{1/2} \sigma^2 \Omega_D} \]Here, various parameters like temperature (T), Boltzmann’s constant (k_B), molecular properties, and collision interactions (\Omega_D) are taken into account to predict how diffusion changes. The Chapman-Enskog formulation is widely used to predict diffusion behaviors in scientific research and practical applications in industries like manufacturing and refining.

Temperature Effect on Diffusion

The effect of temperature on the diffusion of gases is significant and largely straightforward. As temperature increases, the energy and speed of gaseous molecules also increase, enhancing their ability to mix and move through each other. This is reflected mathematically in the Chapman-Enskog theory formula, where mass diffusivity ( D ) is proportional to T^{3/2} . Therefore, as temperature rises, diffusion speeds up, leading to faster mixing and spreading of the gases within a binary mixture. This characteristic is particularly important in processes where temperature control is essential, such as gas reactions and chemical synthesis. In practical applications, understanding this temperature dependence helps optimize processes like combustion in engines or efficiency in chemical reactors, ensuring favorable conditions for desired reactions.

Pressure Effect on Diffusion

Pressure plays a less direct role in gas diffusion compared to temperature. While it does not appear explicitly in the Chapman-Enskog equation for mass diffusivity, pressure affects the concentration and spacing of gas molecules. According to the ideal gas law, pressure is directly proportional to the concentration of gas molecules, given by \( \frac{n}{V} = \frac{P}{RT} \).When pressure increases, the density of the gas mixture also increases, resulting in more frequent collisions among gas molecules. Although this might seem like it would enhance diffusion, the reverse is true. More frequent collisions reduce the average distance molecules travel before hitting another molecule, thus decreasing overall diffusivity.Understanding pressure's impact on diffusion is critical for processes where maintaining low mass diffusivity is necessary, such as ensuring the containment of gases in pressurized vessels or preventing rapid spreading in gas leaks.