1 66 A SYSTEM OF PHYSICAL CHEMISTRY lie between x^ and .V2 is given by: 4/ Jtt area of P1M1M2P2, where OMj = Xi and OM2 = x^. By methods of analysis or quadrature of a carefully drawn graph numerical values can be obtained. The following table from Meyer's Kinetic Theory of Gases (translated by Baynes) illustrates the point :— Of 1000 molecules of oxygen at 0° C, whose r-m-s-speed is known (by methods to be explained presently) to be 461*2 metres per second— 13 to 81 „ 166 ,, 214 .. 202 ,, 151 >. 91 » 76 „ 14 molecules have speeds below 82 from 167 215 203 152 92 77 „ 400 500 „ 600 above 700 100 metres per second. 100 to 200 200 ,, 300 300 ,, 400 500 600 700
Hence we derive— APPENDIX I 167 . 2 - - 3 R ^ -wE = RT, or E = - - . T. ' 3 2 nv But nv is the number of molecules in the ; gram-molecule so is R/^Z' the gas constant per molecule, usually denoted by k. Thus we obtain, as the connection between the temperature of the gas and its mean molecular motion— E = |/^T (8) We have so far dealt with a gas composed of similar molecules, but if we consider a gas mixture containing molecules of different masses, little alteration is required in the formulae. It can be shown that among each group of molecules of one type, the relative distribution in speed and direction is similar to that already outlined for a simple gas, and is unaffected by the presence of other types of molecules. This is a statistical generalisation of the well-known law of Dalton concerning gas mixtures. Thus, each group of molecules has a certain r-m-s-speed, but the value of this varies from group to group ; however, a very simple relation connects them. If we denote types of molecules by suffixes, I, 2, 3, etc., then— ;«i^i^ = m.2^>^ = m-^i^ — etc. where /«i, m.2, m-^, etc., are the masses of each type of molecule. In terms of average kinetic energies per group of molecules this is— Ej = E2 = E3 = etc. Referring to (8) we see that this dynamical conclusion is the statistical statement of the equality of temperature which obtains throughout the gas mixture when a steady state has been attained. On account of this relation it becomes more convenient to state the distribution law in terms of limits of kinetic energy. This is easily done. Thus, revert- ing to the case of a simple gas for a moment, we introduce the following changes into (4) :— Write and we obtain after a few steps — E = \mr E = \mc^ dE = mc . dc 4-N^^-^3 E^ . . - 3=/-^E as the number of molecules with kinetic energies between E and E + ^E at an instant. We may avail ourselves of (8) and obtain instead of this expression the following one :— f^ . EK g - ^/^^ . ^E . . . (9) as the number of molecules limited in the manner mentioned.