Webpage title: Understanding the Energy Levels of Electrons in Atoms

[scientifico]

Hello, why do the electrons with higher quantic number N have more energy than the electrons with lower N if they need less energy to be removed from the atom?

Thank you!

 

[Ken G]

That's because the total energy of a bound state in a 1/r potential (or any potential that is zero at infinity) is generally taken to be negative, so what we mean by "higher" can be confusing. A high-n means the energy is more than a low-n, but that's because it is a smaller negative number at high n-- the magnitude of the energy is actually smaller for high n, because the total energy is negative (relative to a free electron). So to get the particle to go from bound to free, you at least need to get its energy to go from -X to 0, where "X" is a positive number that is smaller for high n. X is also the (positive) energy you need to give the electron to get it to ionize.

Another thing that confuses people is that the kinetic energy of an electron at high n is actually less than the kinetic energy of an electron in a ground state, for a 1/r potential. Here the ground state is a state of large (in magnitude) but negative energy, and this total comes from a very large positive kinetic energy combined with an even larger (in magnitude) negative potential energy. The sign of the energy is very important in all this, and is different from a particle in a simple harmonic potential and a particle in a box (which generally have positive total energy and greater kinetic energy for larger n).