Is it true that an evaporating molecule has the same kinetic energy as a molecule in a pot of boiling water?
Is it true that an evaporating molecule has the same kinetic energy as a molecule in a pot of boiling water?


Well for one the kinetic energy for a given temperature is an average kinetic energy of a distribution of energis. So for the question:


  "Is it true that an evaporating molecule has the same kinetic energy as a molecule in a pot of boiling water?"


The answer is maybe or there probably are molecules that have the same energy in each phase, but I think the question you want to ask is:


  Is it true that all molecules in steam have the same average kinetic energy as all the molecules in a pot of boiling water?


From a thermodynamic perspective, at the boiling temperature the liquid and vapor phases are in equilibrium and have the same temperature. Since temperature is the measure of the average kinetic energy, if temperature is equal average kinetic energy is equal.

So simplistically speaking yes they are the same. You can stop with reading here or veer with me into super technical semantics and minutia—your choice.


  Reasoning for thinking this: boiling is when the vapor pressure = the pressure of the atmosphere.


Now it is commonly taught as you state: "boiling is when the vapor pressure = the pressure of the atmosphere". The thing is though that this is a restatement of "at the boiling temperature the liquid and vapor phases are in equilibrium and have the same temperature" which is similar but not equivalent. You said:


  It does so because it gained kinetic energy ${x}$, and ${x}$ was great enough to counteract the pressure of the air surrounding it.


Boiling occurs at the bottom of the pot which has a higher hydrostatic pressure due to the height of the water $(P = \rho gh)$. So the molecule must not just over come the pressure of the atmosphere but also the pressure of the water height. This increased pressure mean boiling occurs at a higher temperature than water at the top of the pot (see: water saturation pressure). As the steam bubble rises, the pressure decreases ($h$ decreases) and the bubble expands the steam will cool due to the work done by expansion (see: internal energy) lowering the equilibrium vapor pressure and condense at the interface of the bubble. The thermal transport is not instantaneous though so the interface inside the bubble will absorb heat from the latent heat of condensation, and in turn be hotter than the surrounding water thus the bubble will be hotter in the center than at the interface with water. Since the vapor is (very) slightly hotter, the average kinetic energy will be (very) slightly hotter, thus (very) technically it would not be correct that all the molecules in steam have the same average kinetic energy as all the molecules in a pot of boiling water