A system of physical chemistry - Index of

26 A SYSTEM OF PHYSICAL CHEMISTRY

molecule shall be exactly 3/2^T, where k is the gas constant per single

molecule. We know from Maxwell's distribution law that the kinetic

eneriiy varies in general from molecule to molecule. What the equipartition

principle does mean is that on the average the kinetic energy

**of** a molecule is 3/2/^T.

Let us now consider the special case **of** a monatomic gas. As we have

already seen, it can possess no vibrational energy ; whether it possesses

rotational energy is at this stage a doubtful point. Its energy, due to

translational motion, for i gram-mole at T°abs. is fRT, when R = I'gSs

cals. Now the specific heat C„ at constant volume is simply the increase

in the total energy per degree. If we are dealing with i gram-mole or

gram atom as unit **of** mass, the heat term will be the so-called molecular

or atomic heat (Z^. It is clear from definition that the increase in the

^

(kinetic) energy **of** translation per 1° rise in temperature = ^(|RT)

= fR = 2-98 cals. per mole. Recent experiments **of** Pier in Nernst's

Laboratory (Zeitsch. Elektrochem., 15, 546, 1909; 16, 897, 1910) have

shown that the molecular heat **of** argon is 2 '98 cals. per mole, and fur-

ther that this is independent **of** temperature. This agreement suggests

that one should neglect the rotational energy **of** the monatomic mole-

cule (vibration **of** atoms inside the molecule is naturally impossible

since the gas is monatomic). In fact the monatomic molecule seems

to function as a massive point. Agreement **of** this order between cal-

culated and experimental values is, however, not found in other cases.

Thus, taking the case **of** a diatomic gas, the number **of** degrees **of** free-

dom in virtue **of** translation is again 3. The number **of** degrees **of**

freedom in virtue **of** vibration we have considered as i ; that is 4 degrees

in virtue **of** translation and vibration. The corresponding kinetic energy

**of** such a molecule will be 4 x -^RT = 4-0 T cal./mole, if the law

**of** equipartition be assumed. Of course this does not represent all the

energ}' due to translation and vibration. In vibrations we have potential

energy as well as kinetic which must be taken account **of**. It can

be shown by a simple calculation that the potential energy **of** a particle

undergoing what we might call " circular vibration " is just equal to the

kinetic energy **of** the vibration. The calculation is as follows :—

Consider a particle whose mass is m travelling in a circle round a

centre **of** gravity with velocity u. Suppose r is the radius **of** the circular

path. A motion **of** this kind involves the action **of** two opposing forces,

one tending to draw the particle towards the centre, its position **of** rest,

the other due to the motion **of** the particle tending to make it

fly

**of**t at

a tangent. The two forces must just balance in order to make the

circular movement permanent. Let us suppose that the diameter **of**

the circle, i.e. the amplitude **of** the vibration, to be so small that the

force tending to draw the particle back to the centre is proportional to

the distance **of** the particle from the centre. This very simple law **of**

'

Energy **of** translation is necessarily entirely kinetic, and it is to the distribution

**of** kint tic energy amongst degrees **of** freedom that the law **of** equipartition is properly

to be applied.