What happens to the energy of an electron as it goes farther from the nucleus

Electrons closer to the nucleus will have a lower, meaning more negative, potential energy. Electrons further away from the nucleus will have smaller valued negative potential energy (absolute value is smaller), meaning electrons farther from the nucleus have greater energy. It might be counter-intuitive, but it makes sense when you realise that it takes more energy to remove an inner-shell electron and less energy to remove an outer-shell electron.

Mathematically,

$$E = - \frac{Z^2 R}{n^2}$$

where $R$ is the Rydberg constant, $Z$ is the atomic number and $n$ is the principal quantum number, where the greater the value of $n$ (the further away the electron is), the greater the value for $E$ and vice-versa. The negative sign is very significant.

If we want to also compare how kinetic energy varies in relation to the potential energy, we can use the virial theorem

$$\langle{T}\rangle= -\frac{1}{2}\langle V \rangle$$

which states that the average values of the electron's kinetic energy is half that of the potential energy. The negative sign this time implying that the kinetic energy of an electron decreases as distance from the nucleus increases. This is consistent.

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