Question

What is the difference between mass-average velocity and mole-average velocity during mass transfer in a moving medium? If one of these velocities is zero, will the other also necessarily be zero? Under what conditions will these two velocities be the same for a binary mixture?

Short answer

Question: Explain the difference between mass-average velocity and mole-average velocity during mass transfer in a moving medium and under what conditions they can be zero or equal for a binary mixture. Answer: Mass-average velocity is the average velocity of particles in a mixture, weighted by their mass fraction, while mole-average velocity is the average velocity weighted by their mole fraction. For both velocities to be zero, the mixture must be stagnant with no mass transfer. For these velocities to be equal in a binary mixture, the mass fractions and mole fractions must be proportional, which occurs when the molar masses of the two species are the same or when the product of the mole fraction and the molar mass is the same for both species.

Step-by-step solution

1. Definition of Mass-Average Velocity

Mass-average velocity, also known as "mass flux", is the average velocity of all particles (atoms, molecules, or ions) in a mixture, weighted by their mass. It can be calculated using the following equation: Mass-average velocity = \(\displaystyle\sum_{i=1}^{N} Y_i V_i\) where N = total number of species in the mixture, \(Y_i\) = mass fraction of species i, \(V_i\) = velocity of species i.

2. Definition of Mole-Average Velocity

Mole-average velocity, also known as "molar flux", is the average velocity of all particles in a mixture, weighted by their mole fraction. It can be computed using the following equation: Mole-average velocity = \(\displaystyle\sum_{i=1}^{N} X_i V_i\) where X = mole fraction of species i.

3. Comparison of Mass-average and Mole-average Velocities

The main difference between mass-average velocity and mole-average velocity is that the former is weighted by mass fraction, while the latter is based on the mole fraction. Therefore, the two velocities will have different values for any non-ideal (binary or multi-component) mixture, unless the mass fractions and mole fractions are proportional in the specific case.

4. Conditions for Zero Velocities

If one of the velocities (mass-average or mole-average) is zero, it does not necessarily imply that the other is also zero, as they are based on different weighting factors (mass fraction and mole fraction, respectively). However, in some cases (such as stagnant mixtures with no mass transfer), both velocities might be zero.

5. Conditions for Equal Velocities in a Binary Mixture

For the mass-average and mole-average velocities to be equal in a binary mixture, the mass fractions and mole fractions must be proportional. This occurs when the molar masses of the two species are the same, or when the product of the mole fraction and the molar mass is the same for both species: \(Y_1 = Y_2\) and \(X_1 = X_2\) or \(Y_1 M_1 = Y_2 M_2\) and \(X_1 M_1 = X_2 M_2\) In these cases, the mass-average and mole-average velocities would be equal.