/ 
IONISATION, DIFFUSION, ROTATION 
261 
If the volume is enclosed by actual walls / remains constant on the 
average, so that S i m7 2 = _ ( X x + yY + zZ) (182-2), 
but we must include in ( X , Y, Z) the forces on the molecules when they 
are reflected from the walls, viz. the external pressure. If the walls are 
fictitious, we replace each escaping molecule by a molecule entering at the 
same spot so that I is kept constant; the transfer of momentum involved 
in these replacements is still represented by the pressure at the boundary. 
The pressures on the 6 faces of a centimetre cube give a contribution 
— 3p to E (xX + yY + zZ). The remaining part of the expression is easily 
shown to amount to YYrR 1: where R x * is the force (positive if repulsive) 
between two molecules at a distance r and the summation extends over 
each pair of molecules (counted once). Hence (182-2) becomes 
p = fZfmF 2 + ^YYrR 1 (182-3). 
Consider molecules which repel one another, and let </> be the potential 
of the field of force so that nuf) is the potential energy of a molecule. If 
Oq is the number of molecules per cu. cm. in regions where </> is zero, the 
number at any other point is (46-1) 
cr = J°° [~~j a 0 e-(*mV+m*)IRT 4t tVUV (182-41) 
= o. 0 e-m*/fir (182*42). 
From (182-41) we also see that the mean value of |wF 2 is independent of 
cf) and equal to f RT. Hence by (182-3) 
p = u'RT + lYYrRy (182-43), 
where a is the actual number of molecules per cu. cm. 
Suppose that the molecules are rigid and of diameter d so that the 
centres of two molecules cannot approach within distance d. We may 
regard them as kept apart by a repulsive force of enormous intensity at 
distance d, with </> increasing from 0 to oo in an infinitesimal range at the 
value r = d. 
Let us calculate the contribution of one rigid molecule to the virial. 
The average number of other molecules within a distance r to r + dr is 
aAnr 2 dr; and for these R x r = (— md(f>ldr)r. Hence, using (182-42), the 
contribution to YYR 1 r is 
— 47rr 2 dr a 0 e~ m ' l> / RT mr~ (182-44), 
so that the first summation of gives 
— 47Ta 0 m ( r 3 e~ m<l> l RT dcf> 
.0 
= 4:7TCF 0 d 3 RT , 
since d(f> only differs from zero at r = d. 
* The suffix is used to distinguish it from Boltzmann’s constant.