Question

Your classmate says that if all the molecules in a particular liquid had the same speed, and some were able to evaporate, the remaining liquid would not undergo cooling. Do you agree or disagree and what is your explanation to support your answer.

Answer 1

Answer: in an ideal system, your classmate is right

Explanation:

Here I suppose that we are in a closed environment.

If all the particles had the same speed, and some of them are able to evaporate, then you have coexistence between liquid state and gas state, In this coexistence state, there is a line of pressure as a function of temperature where the coexistence is possible.

As some of the liquid evaporates, the volume of the gas is bigger than the one of the liquid, so the pressure in the system will increment a little bit, and the temperature will answer to it until the equilibrium is reached.

In equilibrium, the temperature will not change if we don't do anything to the system (assuming adiabatic walls).

here your classmate may be right

A thing you may know is that the temperature is an intensive quantity, which means that it does not depend on the number of particles in the mixture, so the fact that some of the particles "leave" the liquid does not imply that the liquid changes their temperature. And there is also useful to notice that the change of phase is at a constant temperature.

Answer 2

The molecules which evaporate presumably take heat away from the liquid. So, I'd disagree with the classmate. Whether the amount of cooling would differ from the usual case wherein the molecules have different speeds is another question. 
I guess the argument goes something along the lines of that the faster moving and therefore most kinetically energetic molecues evaporate and take away most heat. But if there's no faster moving molecules, 'cos they all have the same speed well, then presumably they'd all take away the same amount of heat. So, maybe the cooling would be less. No cooling though ??? Hmmmm dunno .... i think not ....