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.
EQUIPARTITION OF KINETIC ENERGY 27 attraction can only hold when the particle is not far removed from the centre, i.e. when r is small. If A is this attractive force per unit dis- tance from the centre, then on the assumption just made, the force acting inwards at a distance r is A/*. We have now to calculate this force in terms of the motion of the particle. Consider the particle traversing the circular path shown in the figure (Fig. 4). Suppose that the particle is at the point a, travelling with a velocity u in the direction ax. After a short interval of time 8/ it is at b travel- ling with a velocity u in the direction ib. The arc at = u8f, and so the radian measure of the angle aob or cTd is 86= u^ijr. If Tc = Td=u, then the velocity has changed from one represented in magnitude and direc- tion by Tc to one represented by Td. By the triangle of velocities, the change in velocity is represented in magnitude and direction by the line cd. The direction of cl is the same as that of to ; and its magnitude is 2u sin 86/2. In the limit when 8/ is infinitesimally small sin Sdj-, == 8^/2, or the change of velocity is u80, i.e., u^f/r. Fig. 4. Hence the acceleration inward (viz. the velocity inward divided by 8/ is equal to u'^Jr. But force = mass x acceleration. Hence the force acting inwards and preventing the particle from flying oiff is mu^/n This must be identical with Ar. That is— mu^/r = Ar. Further, the potential energy of the particle at a (namely, the work which must be done upon the particle to bring it from the position of rest, the centre of the circle, to the point a on the circumference) is the product of the force acting into the distance traversed. The lorce varies at every stage of the radius, so that it is necessary to integrate the work expression lor each increment dr in order to obtain an expression for the potential energy of the particle at the point a. That is the potential energy of the particle when it is on the circumference is Ardr which