Energy of the Electrons

The electron's total energy in the nth orbit is the sum of its kinetic energy and its (negative) electrostatic potential energy, $E_n=\text{K.E.}+\text{P.E.}=\frac{1}{2}m_ev_n^2-\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r_n}$. Substituting the expressions for $v_n$ and $r_n$ derived in the previous section and simplifying gives the compact result $E_n=-\frac{m_eZ^2e^4}{8\epsilon_0^2h^2n^2}$, or, after substituting the values of all the physical constants, the far more useful practical form $E_n=-\frac{13.6\,Z^2}{n^2}$ eV.

The NEGATIVE sign is essential and physically meaningful: it signals that the electron is bound to the nucleus, meaning energy must be SUPPLIED to it to bring its total energy up to zero (the point at which the electron becomes free of the atom entirely). As n increases, $E_n$ becomes LESS negative -- the energy increases toward (but never reaches) zero as $n\to\infty$. The n=1 orbit, with the most negative (lowest) energy, is called the ground state, the atom's naturally lowest-energy and hence most stable configuration; any orbit with n > 1 is called an excited state, and an electron occupying the nth orbit is said to be in the nth energy state.

For hydrogen (Z=1) specifically, the ground-state energy is exactly $-13.6$ eV. The energy needed to move an electron from the ground state up to some excited state n is called the excitation energy of that state, equal to $E_n-E_1$; for example, the minimum excitation energy, to the very first excited state (n=2), is $-3.4-(-13.6)=10.2$ eV. Carrying this logic to its natural conclusion, the energy needed to remove the ground-state electron completely from the atom (raising its energy all the way to zero) is called the ionization energy -- exactly 13.6 eV for hydrogen -- and this is numerically identical to the (positive) binding energy of the ground-state electron: the energy that would be RELEASED if a free proton and a free electron, starting infinitely far apart, were brought together to form a hydrogen atom in its ground state. …