can an electron gain enough energy to escape the atom?
Also, are the number of particles in a mole a constant?
I’m confused 
Yes, electrons can gain enough energy to escape an atom. Many years ago I worked with folks who excited atoms with a single electron above the “filled” shells to very high energy levels. These so=called “Rydberg” atomes had a reputation for being “floppy” and delicate – it was easy for them to pick up thermal energy and self-ionize.
The number of atoms in a mole IS constant.
Electrons escaping from an atom? Sure, happens all the time. For example, an x-ray photon striking the atom can transfer enough energy to the atom forthe atom to emit an electron. An atom with one (or more) electorn short is charged positively - it’s a positive ion. Hence the term “ionizing radiation” which applies to X-rays, gamma rays, etc.
Also, if you dissolve some type of salt in water, the atoms come apart and become ions - some have an extra electron, some are missing an electron.
A “mole” is a dimensionless unit - i.e. it’s just a number, like “gross” or “dozen.” So the number of atoms in a mole is by definition constant. Just like the number of roses in a dozen is always constant (12).
thanks, I was confused… but no longer
An electron CAN escape an atom… if it has an accomplice on the outside willing to sneak in a key in a cake, or crash through the wall at an appropriate time in a truck…
The salient question here is: how long can it remain at large before the Atomic Guard catches up to it?