A system of physical chemistry - Index of

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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

oft 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.

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