Fig. 25.—The Bohr model of the hydrogen atom in
the simplified form (with circles instead of ellipses).
If the electron passes from an outer orbit to an inner one; for example, if it goes from number 4 to number 2, or from number 2 to number 1, the electric force which attracts it to the nucleus will do work just as the force of gravity does work when a stone falls to the ground. A part of this work is used to increase the kinetic energy of the electron, making its velocity in the inner orbit greater than in the outer, but the rest of the work is transformed into radiation energy which is emitted from the atom in the form of monochromatic light. In consequence of the second postulate the frequency of the emitted radiation is proportional to the energy loss. When the electron has reached the innermost orbit (the one denoted by 1 in the figure), it cannot get any nearer the nucleus and hence cannot emit any more radiation unless it first is impelled to pass from its inner orbit to an outer orbit again by the absorption of external energy sufficient to bring about this change. Once in the outer orbit again, it is in a state to produce radiation by falling in a second time. The innermost orbit represents thus the electron’s equilibrium state, and corresponds to the normal state of the atom.
If we try to illustrate the matter with an analogy from the theory of sound, we can do so by comparing the atom not with a stringed instrument, but with a hypothetical musical instrument of a wholly different kind. Let us imagine that we have placed one over another and concentrically a series of circular discs of progressively smaller radii, and let us suppose that a small sphere can move around any one of these without friction and without emitting sound. In such a motion the system may be said to be in a “stationary state.” Sooner or later the sphere may fall from the first disc on to one lower down and continue to roll around on the second, having emitted a sound, let us assume, by its fall. By passing thus from one stationary state to another it loses a quantity of energy equal to the work which would be necessary to raise it again to the disc previously occupied, and to bring it back to the original state of motion. We can assume that the energy which is lost in the fall reappears in a sound wave emitted by the instrument, and that the pitch of the sound emitted is proportional to the energy sent out. If, moreover, we imagine that the lowermost disc is grooved in such a way that the sphere cannot fall farther, then this fanciful instrument can provide a very rough analogy with the Bohr atom. We must beware, however, of stretching the analogy farther than is here indicated.
It must be specially emphasized here that the frequency of the sound emitted in the above example has no connection with the frequency of revolution of the sphere. In the Bohr atom, likewise, the frequency of revolution ω of the electron in its stationary orbit has no direct connection with the frequency of the radiation emitted when the electron passes from this orbit to another. This is a very surprising break with all previous views on radiation, a break whose revolutionary character should not be under-estimated. But, however unreasonable it might seem to give up the direct connection between the revolutional frequency and the radiation frequency, it was absolutely necessary if the Rutherford atomic model was to be preserved. And as we shall now see, the new point of view of the Bohr theory leads naturally to an interpretation of the Balmer-Ritz formula, which had previously not been connected with any other physical theory.
The quantity of energy E, which the atom gives up when the electron passes from an outer to an inner orbit, or which, conversely, is taken in when the electron passes from an inner to an outer orbit, may, as has been indicated, be regarded as the difference between the energy contents of the atom in the two stationary states. This difference may be expressed in the following way. Let us imagine that we eject the electron from a given orbit (e.g. No. 2 in the diagram) so that it is sent to “infinity,” or, in other words, is sent so far away from the nucleus that the attraction of the latter becomes negligible. To bring about this removal of the electron from the atom demands a certain amount of energy, which we can call the ionizing work corresponding to the stationary orbit in question. We may here designate it as A₂. To eject the electron from the orbit No. 4 will demand a smaller amount of ionizing work, A₄. The difference A₂ - A₄ is accordingly the work which must be done to transfer the electron from the orbit No. 2 to the orbit No. 4. This is, however, exactly equal to the quantity E of energy which will be emitted as light when the electron passes from orbit No. 4 to orbit No. 2. If we call the frequency of this light ν, then from the relations E = hν and E = A₂ - A₄, we have
hν = A₂ - A₄
If, now, in place of this specific example using the stationary orbits 2 and 4 we take any two orbits designated by the numbers n″ (for the inner) and n′ (for the outer), we can write for the frequency of the radiation emitted for a transition between these arbitrary states
| hν = Aₙ″ - Aₙ′ or ν = | Aₙ″ | - | Aₙ′ |
| h | h |
We have now reached the point where we ought to bring in the Balmer-Ritz formula for the distribution of the lines in the hydrogen spectrum. This formula may be written ([see p. 59])
| ν = | K | - | K |
| n″ ² | n′ ² |