The Bohr Model of Hydrogenic atoms
This model was created by Nils Bohr to explain the form of the emission spectrum of atomic hydrogen, which consists of series of discrete lines. It had already been noted that the series of lines could all be fitted to an expression of the form:
where n1 and n2 are both positive integers, and n2 = n1 + 1, n1 + 2, n1 + 3 etc.
The constant RH is called the Rydberg constant for the hydrogen atom, and has the numerical value 109677 cm-1. Bohr’s model was an attempt to explain this result.
The model, originally based on the hydrogen atom, was based upon three assumptions:
- Electrons move in fixed, circular orbits around the nucleus, obeying classical mechanics.
- The angular momentum of the electron in an atom is restricted (quantised) into integer multiples of h/2π.
- Radiation is emitted by the electron only when it changes orbits.
For each orbit to be stable, the Coulombic attractive force between the positive nucleus and the negative electron must balance the centrifugal force that tends to fling the electron away from the nucleus. Thus, from classical mechanics, we may write:
where Z is the nuclear charge of the atom, numerically equivalent to the atomic number of the element under consideration, and v is the instantaneous linear momentum of the electron.
From classical mechanics, we know that the angular momentum, ω, of the electron, is given by the expression v = rω.
The above equation rearranges to give:
The total energy of the electron in each orbit is given by the sum of the electron’s kinetic and potential energies, which can be written out explicitly:
The expression has been manipulated using the relationship derived above.
We now turn to trying to remove r from the expression. The assumption made by Bohr that the angular momentum is quantised into integer multiples of h/2π allows us to write:
where n is any positive integer.
N.B. It is implicit in this equation that we are considering the electron as moving around a stationary nucleus, allowing us to immediately write down its angular momentum as mevr from classical mechanics.
In practice, this is quite a good approximation, as the nucleus is so much more massive than the electron, but since the nucleus is not infinitely massive, it remains an approximation. This is taken into account in a manner described later on.
We may manipulate this expression to give us:
The second equality follows from the equality of the first and third terms, as derived above. Manipulation of the second equality gives us an expression for r:
which we can substitute into our expression for the total energy of an orbit to give:
From the assumption that the emission of energy corresponds to transition from one orbit to another, we may write the energies of the emission spectrum as the difference of two terms like the one above (i.e. the emission energy is the difference of two orbit energies):
This expression may be simply manipulated into the form of the Rydberg formula, and when the values of the fundamental constants are substituted in (using Z = 1, for the nuclear charge of a hydrogen atom is one) then the agreement with the Rydberg formula is almost exact.
It is not exact due to the approximation introduced by assuming that the nucleus is infinitely massive, and is thus a fixed point around which the electron orbits. In reality, the electron and nucleus both orbit around their combined centre of mass, which will be close to but not at the centre of the nucleus. This effect may be completely compensated for by replacing the mass of the electron with the reduced mass, μ, defined as:
where mn is the nuclear mass. Note that since mn >> me , the denominator is approximately the same as mn, and hence the reduced mass is approximately equal to the mass of the electron.
Using the correct reduced mass and the appropriate value of Z, the above expression may be used to generate the frequencies of the transitions in any hydrogenic atom or ion. (A hydrogenic species is one which has only one electron, for example He+ , Li2+.) Note that because the reduced mass is involved, the Rydberg constant is slightly different for each element.
Though Bohr’s model is very successful for hydrogenic atoms, it is fundamentally flawed in several ways.
- It cannot be extended to atoms or ions with more than one electron.
- It cannot offer any explanation for the different intensities of the various transitions.
- It does not explain why many lines are found, at high resolution, to actually be composed of closely associated groups of lines. (Such groups are known as multiplets.)
- Most damagingly, there is no justification for the assumptions made as the basis of the derivation. For example, there is no obvious reason for the angular momenta to be quantised into units of h/2π , this appears to form part of the theory solely as a convenience to make the theory work!
We shall see that the quantum mechanical approach to explaining the orbits of electrons is much more successful. Though many of the conclusions of quantum mechanics relate closely to the results of the Bohr theory for a hydrogenic atom, the assumptions Bohr was forced to make arise naturally from the quantum mechanical treatment of the problem.
As a brief example, we may provide a reasonable justification for the quantisation of angular momentum in terms of the de Broglie relation. We write out the relation as follows, and substitute for the linear momentum:
Now, one complete orbit has the same length as the circumference of the circle that the orbit traces out, so is given by 2πr. This distance must be equal to an integral number of wavelengths. (If it were not, the wave traced out by the particle on its first orbit would be out of phase with those traced out on all subsequent orbits.)
Over an infinite number of orbits, the out of phase waves would cancel each other out to zero amplitude, which implies that the particle cannot exist under such circumstances. Only if the waves traced out on all orbits overlap exactly, i.e. if the orbit is an integral number of wavelengths, is the situation a satisfactory one under which the particle can exist.
We can express this requirement mathematically as:
where n is an integer. Rearrangement of this readily yields:
which immediately indicates quantisation of angular momentum in units of h/2π.