A system of physical chemistry - Index of

2 8 A SYSTEM OF PHYSICAL CHEMISTRY

IS . But it has just been shown that Ar = fnu^/r. Hence the po-

tential energy, viz. is equal to (i/2);«?A But the term (i/2)w«^ is

the average kinetic energy **of** the particle.

Hence in a complete circular

vibration f/ic kinetic energy is just equalin magnitude to the potential energy .

W'e return now to the question **of** the diatomic molecule.

The linear vibration executed by one atom with respect to the other

involves one degree **of** freedom, i.e. the line **of** junction **of** the two atoms.

On the equipartition principle the kinetic energy involved per mole is

^RT. Since there is likewise an equal amount **of** potential energy, the

total energy due to vibration is iRT. Adding the amount due to the

(kinetic) energy **of** translation **of** the diatomic molecule as a whole, viz.

|RT, we obtain fRT as the total energy due to translation and vibration.

That is, the rise in this energy per i° is fR = 5'o cal. per mole

per degree, taking R = 2 cals. If now we take rotation **of** the molecule

as a whole into account we again have two degrees **of** freedom, to which

one must assign RT units **of** kinetic energy. The total energy **of** a

diatomic molecule, provided the law **of** equipartition is true, axxdi provided

ALL the degrees **of**freedom are effective, should be I^RT, and the

molecular heat therefore IR = 7

'o

approx.

Experiment shows, however, very different values. For hydrogen

at 0° C. the molecular heat Cy = 4*9 to 5*2 cal. per degree, and at

2000° C, Cv = 6'5 cal. {cf Nernst, Zeitsch. Elektrochem., 17, 272, 191 1).

For nitrogen at 0° d, = 4*84, and at 2000° d, = 67. For chlorine

at 0° C-,' = 5 "85, at 1200° Cy = 7*0. For oxygen at 0° d = 4'9) ^'^

2000° Cy = 6"7. These values are only approximately correct. It is

evident, however, that not only is there lack **of** agreement in the numerical

values between those observed and those calculated at lower tem-

peratures, but the fact that the molecular heat (f diatomic gases varies

considerably with the temperature is quite unaccounted for by the theory **of**

equipartition unless, indeed, the number **of** degrees **of** freedom is a

function **of** the temperature ; which is difficult to believe.

In the case **of** triatomic gas molecules the degrees **of** freedom in

respect **of** translation are three, the [kinetic] energy corresponding bemg

|RT. As regards vibration, there are possibly three vibrating pairs

each with one degree **of** freedom, corresponding to the quantity

3 X :^RT **of** kinetic energy. To this has to be added an equivalent

amount **of** potential energy, making 3RT as the total energy term in

respect **of** vibration. Hence translation and vibration apparently entail

fRT units **of** energy, and the increase in this for 1° rise in temperature

is fR = 9'o cal. per mole per degree. Logically we should likewise

add a terra for rotation **of** the molecule as a whole, which we have seen

amounts to #RT. The observed molecular heat Cy for CO2 at 18° is

709, and this becomes 10-47 ^'^ 2210° (Pier, I.e.). For water vapour

Cy at 50° C. = 5'96 (Nernst and Levy), and at 2327° C, C„ = 9'68

(Pier). Again the discordance between theory and experiment is very

apparent since theory predicts a constant molecular heat **of** either 9 or

12 cals. per mole per degree.