Kinetic theory for the molecular condensation flux

[Punchlinegirl]

Homework Statement

Start with the kinetic theory formula for the molecular condensation flux, $$\Phi_cond$$ = 1/4 n_v*c_v, where n_v is the number density of vapor molecules just above a liquid water surface and c_v is the mean speed of the vapor molecules. Derive an expression for calculating the evaporation flux, $$\Phi_evap$$ in terms of the saturation vapor pressure of water e_s(T) at any given temperature T. The vapor may be treated as an ideal gas.

Homework Equations

see above

The Attempt at a Solution

I know that $$\Phi_cond$$=1/4 n_v *c_v = e/ (2$$\pi$$m_v*k_b*T)^1/2.
And that at equilibrium the evaporation flux and condensation flux are equal, but I know you can't say there equal here since it's for any time T. However, I have no idea how to get started. I would think you'd need to do a differential equation such as dn/dt is equal to the flux, but I'm not even sure how to get this set up. If someone could help, it would be greatly appreciated!

 

[Jhero]

Thank you for your question. To derive an expression for the evaporation flux, \Phi_evap, in terms of the saturation vapor pressure of water, e_s(T), we can use the same approach as for the condensation flux.

First, we need to define the number density of liquid water molecules, n_l, just below the liquid water surface. This can be related to the number density of vapor molecules, n_v, by the ideal gas law: n_l = n_v * e_s(T) / (k_b * T), where k_b is the Boltzmann constant.

Next, we can use the same formula for the condensation flux, \Phi_cond = 1/4 * n_v * c_v, but substitute n_v with n_l from the previous equation. This gives us \Phi_cond = 1/4 * n_l * c_v * e_s(T) / (k_b * T).

Since at equilibrium, the evaporation flux and condensation flux are equal, we can equate \Phi_cond and \Phi_evap and solve for \Phi_evap. This gives us \Phi_evap = 1/4 * n_l * c_v * e_s(T) / (k_b * T).

Therefore, the expression for calculating the evaporation flux in terms of the saturation vapor pressure of water, e_s(T), is \Phi_evap = 1/4 * n_l * c_v * e_s(T) / (k_b * T), where n_l is the number density of liquid water molecules just below the liquid water surface, c_v is the mean speed of the vapor molecules, and T is the temperature.

I hope this helps. Let me know if you have any further questions.