Electrons in Atoms: The Bohr Model of Hydrogen

James Richard Fromm

Chemists are primarily concerned with the electrons in atoms and with the interactions of the electrons in one atom with those of other atoms. Our concern with the nucleus is generally restricted to its control of the electrons. In this and in several following sections, we explore the electronic structure of the isolated atom.

The experiments of Rutherford in 1911 had clearly shown that the subatomic constituents of an atom were not uniformly distributed within it. The small central nucleus contained the comparatively massive protons and neutrons while the much less massive electrons occupied the remainder of the atomic volume. Spectroscopic measurements of the emission and absorption of light, discussed in a previous section, made it clear that each different type of atom had a different, and complex, internal energy structure. A quantitative understanding of the internal structure of the simplest atom, hydrogen, was first achieved by the Danish physicist Neils Bohr in 1913. The Bohr model of the hydrogen atom was a major step toward modern quantum mechanics.

Bohr Model of Atoms

The Bohr model of an atom postulated a structure in which a single electron moved in a circular orbit around the central nucleus, much as a satellite orbits the earth or the planets orbit the sun. Satellites may circle the earth at any distance outside the atmosphere; their orbits may have any radius. As a satellite travels in its orbit there is a balance between the inward force of gravitation, mm'g/r2, and the outward or centrifugal force of the moving satellite, mv2/r. In these equations m is the mass of the satellite, m' is the mass of the central earth, r is the radius of the orbit, and v is the velocity of the satellite in its orbit. The forces are equal in a stable orbit; mm'g/r2 = mv2/r gives by rearrangement v2r = m'g. Since the mass of the earth m'and the gravitational constant g are both constants, a satellite can assume an orbital radius if and only if it possesses the appropriate velocity. Acceleration of the satellite must therefore increase both its velocity and its orbital radius so it moves outward, while deceleration will move it inwards. Astronauts use this procedure, decelerating their capsules or shuttles by firing rockets to move inward until they reenter atmosphere.

The Bohr model of an atom explains the interaction of light with matter as being interaction between one quantum of light and one electron in one atom. The emission of light is a loss of energy, and a loss of energy would occur as an electron decays from a higher (outer) orbit to a lower (inner) orbit of lower energy. Light is absorbed when an electron in an atom is excited to a higher-energy orbit from a lower-energy orbit. In both cases, DE = E(f) = hf. (In this equation, the symbol D is used to indicate difference; the more common symbol for difference is the Greek upper-case delta.)

Bohr's model postulated that electrons in atoms, unlike satellites in Earth orbit, could take up only certain orbits. As a consequence electrons in atoms could engage in orbit changes which had only certain values of changes in E and thus of hf. Mathematically, this is most easily expressed by stating that the angular momentum of the electron is quantized. In equation form this is mvr = nh/2(pi), where m is the mass of the electron, v is its velocity, r is the radius of its orbit, h is the Planck constant, and n is a quantum number which can have only the integer values: 1, 2, 3, ...

Consider an atom which consists of a single electron circling a nucleus. The outward centrifugal force of the moving electron, mv2/r, is balanced by the force of electrostatic attraction. The force of electrostatic attraction between two charged objects is given by the Coulomb law:

F = Q1Q2/4(pi)er2

In the Coulomb law, Q1 and Q2 are the charges on each of two bodies, r is the radial distance separating them, and e (more commonly symbolized by the lower-case Greek letter epsilon) is the permittivity of the medium separating them. For an atom whose nucleus consists of z protons circled by a single electron, the Coulomb law giving the attractive force between them is

F = -ze2/4(pi)e0r2

The equation assumes this form because both the negative electron and the positive proton have one elementary charge e and the permittivity is that of vacuum or free (empty) space. The permittivity of vacuum, e0, has the value 8.8541878176... x 10-12 farads/meter, F/m. The equivalent SI units of C2/J m are more useful units for e and e0 for our purposes. The force of electrostatic attraction, like that of gravitational attraction, decreases as the square of the distance separating the charges.

The two forces are equal and opposite, so their sum is zero and ze2/4(pi)e0r2 = mv2/r. The radius of the orbit can have any value, because the velocity of the electron can have any value. The radius of any orbit is given by

r = ze2/4(pi)e0mv2

The key assumption of the Bohr model is that the circumference of the orbit must be an integer multiple of the de Broglie wavelength of the electron. Mathematically, this is 2(pi)r = nh/mv. This quantization assumption was originally made by Bohr in the form of quantization of momentum, as mentioned above, because the work of Bohr came before that of de Broglie, but the two equations are equivalent. Quantization of any of the physical quantities describing the electron (orbit radius, circumference, angular momentum, velocity, energy) requires that the rest are quantized as well. The quantized orbital velocity of an electron is then v = nh/2(pi)mr. Substitution of the quantized electron velocity in the equation for radius r gives the radius of a Bohr orbit as

r = n2h2e0/(pi)mze2

and the velocity of the electron in that orbit as

v = ze2/2nhe0

We now turn our attention to the energy associated with electrons moving in the Bohr orbits of atoms. The kinetic energy of any moving object is mv2/2, and the potential energy of the separated electronic and nuclear charges is -ze2/4(pi)e0r, so the total energy of an electron is the sum of these. Substitution of the previous equations for the quantized radius and velocity gives the energy of an electron in a Bohr orbit as:

E = -mz2e4/8n2h2e02

For an electron free of the atom, at an effectively "infinite" distance from it, r is infinite, n is infinite, and so the energy E is zero. The energy of an electron bound to the atom in any orbit, relative to that of a free electron, is given by the above equation. For a mole of atoms, the number of them with which chemists usually work, the value must be multiplied by the Avogadro number NA:

E = -mz2e4NA/8n2h2e02

Application of the Bohr Model of Atoms to Hydrogen

We now apply the Bohr model of atoms to the simplest of atoms, that of hydrogen. The innermost Bohr orbit of hydrogen has the lowest quantum number, n = 1, and has a radius r of 0.0529 nm. The next outer orbit, for which n = 2, has a radius r = (0.0529)(22) = 0.212 nm. The radius of any Bohr orbit of hydrogen can be calculated in this way. These calculated Bohr radii are comparable in magnitude to the radii of atoms and ions as measured in various ways.

Combination of the values of the fundamental constants gives -1311.8/n2 kJ/mole for the energies of the electrons in the Bohr orbits of hydrogen. Continuation of these energy calculations for the successive integers n gives the values of the Table below.

Table: Energies and Sizes of the Hydrogen Bohr Orbits
n E, kJ/mole Factor r, pm
1 -1311.8 1/1+2 53
2 -327.95 1/2+2 212
3 -145.76 1/3+2 476
4 -81.99 1/4+2 846
5 -52.47 1/5+2 1322
6 -36.44 1/6+2 1904
7 -26.77 1/7+2 ---
--- --- --- ---
inf. 0.0 0.0 inf.

The Bohr model of the hydrogen atom is a model, but it is not usually called a theory, since Bohr did not attempt to explain why the properties of an electron in hydrogen is quantized, or, to put it another way, why certain orbits are allowed for electrons in hydrogen while others are not. Although Bohr did not explain why quantization existed, the Bohr model is a model which has considerable explanatory power. Bohr realized that the structure of energy levels for electrons in hydrogen, taken together with the Planck equation E(f)= hf, provided an explanation for the observed structure of frequencies of light emitted and absorbed by the hydrogen atom. The energy difference between two allowed Bohr orbits in hydrogen is identical to the energy of one quantum of light emitted or absorbed by an atom of hydrogen.

Copyright 1997 James R. Fromm