how many degrees of freedom has the system composed of a certain amount of water and a certain amount of air?

Question

Neglect the rare gases and the carbon dioxide contained in air

Answer 1

It's not completely clear what you are asking, but I'll assume you are asking how many "thermodynamic degrees of freedom" the system in question has.

The Gibbs phase rule says that the number of degrees of freedom, f, of a closed system at equilibrium is given by:

f = c - p + 2,

where c is the minimum number of chemical components needed to describe the system, and p is number of phases in the system.

With the information given in the question, you can't actually give a single answer; it depends on the amount of water present in the system. It also depends on how many components you choose to include from air. I'll assume, because you are told to neglect the "rare gases", that we're to consider that dry air is made up of just nitrogen and oxygen.

If there is more than enough water present in the system to saturate the air with respect to water vapor, then there will be a liquid phase present, as well as a gas phase. There are therefore 2 phases and 3 components (H2O, N2, and O2). The system then has

3 - 2 + 2 = 3 degrees of freedom

If, on the other hand, there is not enough water present to saturate the air with water vapor (i.e., the relative humidity is < 100%), then there will only be a gas phase present, and the system has:

3 - 1 + 2 = 4 degrees of freedom.