Page 247 - Instant notes

The structure of the hydrogen atom 233

central nucleus of positive charge Ze (Z is the atomic number) is described by the
Coulomb potential:

where ε 0 is the vacuum permittivity. For hydrogen, Z=1. The minus sign indicates
attraction between the opposite charges of the electron and the nucleus. V is zero when
the electron and nucleus are infinitely separated and decreases as the particles approach.
The Schrödinger equation for a single particle moving in this potential energy can be
solved exactly. The imposition of appropriate boundary conditions (that the
wavefunctions approach zero at large distance) restricts the system to certain allowed
wavefunctions and their associated energy values. The allowed quantized energy values
are given by the expression:

where µ=m em n/(m ε+m n) is the reduced mass of the electron m e and nucleus m n. The
energy level formula applies to any one-electron atom (called hydrogenic atoms), e.g.
+
2+
2+
H, He , Li , Be , etc.
The difference between any pair of energy levels in a hydrogenic atom is:

and the values of the physical constants give exact agreement (using appropriate units)
with the Rydberg constant derived experimentally from the frequencies of the lines in the
2
4
2
hydrogen emission spectrum (Z=1), µe /8ε 0h =hcR H.
The distribution of energy levels for the hydrogen atom:

is shown in Fig. 2. The quantum number n is called the principal quantum number.
The energies are all negative with respect to the zero of energy at n=∞ which corresponds
to the nucleus and electron at infinite separation. The energy of the ground state (the state
with the lowest allowed value of the quantum number, n=1) is:
E 1=−hcR H

and is an energy hcR H more stable than the infinitely separated electron and nucleus. The
energy required to promote an electron from the ground state (n= 1) to infinite distance
from the nucleus (n=∞) is called the ionization energy, I. For hydrogen,
−1
I=hcR H=2.179×10 −18 J, which corresponds to 1312 kJ mol or 13.59 eV.