Molecules

Among the discrete set of states of a given selection of atoms there need not necessarily but there may be a lowest level, implying a close approach of the nuclei to each other.

Atoms I I am adopting the version which is usually given in popular treatment and which suffices for our present purpose. But I have the bad conscience of one who perpetuates a convenient error. The true story is much more complicated, inasn1uch as it includes the occasional indeterminateness with regard to the state the system is in such a state form a molecule.

The point to stress here is, that the molecule will of necessity have a certain stability;

The configuration cannot change, unless at least the energy difference, necessary to ’lift’ it to the next higher level, is supplied from outside.

Hence this level difference, which is a well-defined quantity, determines quantitatively the degree of stability of the molecule. I t will be observed how intimately this fact is linked with the very basis of quantum theory, viz. with the discreteness of the level scheme.

This order of ideas of the Heitler-London theory:

• has been thoroughly checked by chemical facts; and
• has proved explained the basic fact of chemical valency and many details about the structure of molecules, their binding-energies, their stabilities at different temperatures, and so on.

Their Stability Dependent On Temperature

How is the stability of a molecule at different temperatures?

Take our system of atoms at first to be actually in its state of lowest energy. The physicist would call it a molecule at the absolute zero of temperature.

To lift it to the next higher state or level a definite supply of energy is required. The simplest way of trying to supply it is to ‘heat up’ your molecule.

You bring it into an environment of higher temperature (‘heat bath’), thus allowing other systems (atoms, molecules) to ilnpinge upon it. Considering the entire irregularity of heat motion, there is no sharp temperature limit at which the ’lift’ will be brought about with certainty and immediately. Rather, at any temperature (different from absolute zero) there is a certain smaller or greater chance for the lift to occur, the chance increasing of course with the temperature of the heat bath. The best way to express this chance is to indicate the average time you will have to wait until the lift takes place, the ’time of expectation’.

From an investigation, due to M. Polanyi and E. Wigner,1 the ’time of expectation’ largely depends on the ratio of two energies, one being just the energy difference itself that is required to effect the lift (let us write W for it), the other one characterizing the intensity of the heat motion at the temper- ature in question (let us write T for the absolute tempt:rature and kTfor the characteristic energy).2

It stands to reason that the chance for effecting the lift is smaller, and hence that the time of expectation is longer, the higher the lift itself compared with the average heat energy, that is to say, the greater the ratio W:kT. What is amazing is how enormously the time of expectation depends on comparatively small changes of the ratio W:kT. To give an example (following Delbriick): for W 30 times kT the time of expectation might be as short as los., but would rise to 16 months when W is 50 times kT, and to 30,000 years when W is 60 times kT!

Mathematical Interlude

It might be as well to point out in mathematical language - for those readers to whom it appeals - the reason for this enormous sensitivity to changes in the level step or temperature, and to add a few physical remarks of a similar kind. The reason is that the time of expectation, call it t, depends on the ratio W/kTby an exponential function, thus

t == re W/ kT

r is a certain small constant of the order of 10- 13 or 10-14 S .

This particular exponential function is not an accidental feature.

It recurs again and again in the statistical theory of heat, forming, as it were, its backbone. I t is a measure of the improbability of an energy amount as large as W gathering accidentally in some particular part of the system, and it is this improbability which increases so enormously when a considerable multiple of the ‘average energy’ kTis required. I ZeitschriftjUr Physik, Chemie (A), Haber-Band (1928 ), p. 439. 2k is a numerically known constant, called Boltzmann’s constant; ~kT is the average kinetic energy of a gas atom at temperature T.

A W == 30kT is extremely rare.

That it does not yet lead to an enormously long time of expectation (only lo s. in our exam- ple) is, of course, due to the smallness of the factor T. This factor has a physical meaning. I t is of the order of the period of the vibrations which take place in the system all the time. You could, very broadly, describe this factor as meaning that the chance of accumulating the required amount W, though very small, recurs again and again ‘at every vibration’, that is to say, about 10 13 or 10 14 times during every second.