Question

At a given temperature and pressure, do you think the mass diffusivity of air in water vapor will be equal to the mass diffusivity of water vapor in air? Explain.

Short answer

Answer: No, the mass diffusivity of air in water vapor will not be equal to the mass diffusivity of water vapor in air, even when the temperature and pressure are constant, due to the difference in molecular weights of the two substances.

Step-by-step solution

Understanding Mass Diffusivity

Mass diffusivity (D) is a measure of how quickly one substance can diffuse through another. It depends on factors such as temperature, pressure, and the properties of the substances involved. In this exercise, we are dealing with air and water vapor, so we will focus on those two substances.

Factors Affecting Mass Diffusivity

Some factors that affect mass diffusivity are temperature, pressure, and molecular weight of the substances involved. In this case, we have two scenarios: 1. The mass diffusivity of air in water vapor. 2. The mass diffusivity of water vapor in air. Even though temperature and pressure are given to be constant, the molecular weights of air and water vapor are different.

Molecular Weight and Mass Diffusivity

The molecular weight of a substance, denoted as M, plays a role in determining the mass diffusivity. The larger the difference in molecular weights between the two substances, the lower the mass diffusivity. In this case, the molecular weights of air and water vapor are not equal, with air having a molecular weight of approximately 29 g/mol and water vapor having a molecular weight of approximately 18 g/mol.

Conclusion

Since the molecular weights of air and water vapor are not equal, the mass diffusivity of air in water vapor will not be equal to the mass diffusivity of water vapor in air, even when the temperature and pressure are constant. This is because the molecular weights of the substances have a direct impact on the mass diffusivity, regardless of the temperature and pressure.

Key concepts

Molecular Weight

Molecular weight is an important factor when considering how substances interact with each other, particularly in diffusion processes. It is essentially the mass of one molecule of a substance, usually measured in grams per mole (g/mol). The molecular weight can influence how quickly a molecule will move, or diffuse, when mixed with another substance.
In our context, the difference in molecular weights between air and water vapor significantly affects their mass diffusivity - the rate at which molecules spread from areas of high concentration to areas of lower concentration.

• The molecular weight of air is approximately 29 g/mol.
• The molecular weight of water vapor is roughly 18 g/mol.

These differing values mean that the heavier molecules of air will diffuse through water vapor at a different rate compared to the lighter water vapor diffusing through air.

Diffusion in Gases

Diffusion in gases is a process where gas molecules spread out and mix with another gas. This mixing continues until a uniform distribution is achieved. The speed and efficiency of this process depend largely on factors like the molecular weight of the gases involved. In general, lighter gas molecules will diffuse more quickly than heavier ones.
In our case of air and water vapor, heterogeneity in molecular weight means each will diffuse at different rates. This is why, under identical external conditions, the mass diffusivity of air in water vapor cannot match that of water vapor in air.
The principle is similar to pouring milk into coffee; both liquids will eventually mix, but the process happens faster depending on certain properties, like temperature and density. For gases, it's molecular weight that plays a big role in how quickly they mix.

Temperature and Pressure Effects

While this exercise specifies that temperature and pressure are constant, it's crucial to understand how these factors normally affect diffusion.

• Temperature: Higher temperatures generally increase diffusivity because molecules move faster, enhancing their ability to spread out quickly.
• Pressure: At higher pressure, more molecules are present in a given space, which can influence how frequently molecules collide and thereby affect diffusion.

However, even at constant temperature and pressure, the intrinsic properties of the molecules, such as their molecular weight, remain a dominant factor in determining mass diffusivity. Hence, the mass diffusivity difference between air and water vapor persists, irrespective of these constant external conditions.