Page 64 - Modern physical chemistry

3.13 Distribution of Equivalent Particles in a Potential Field 53

pressure-volume-temperature measurements or from molecular parameters using statis-
tical mechanics.
A suitable analytic form for the compressibility factor for a given gas is

Z = ~~ = 1+A2(T)P + Aa(T)p2 + .... [3.71]
An alternate form for the gas is

_ PV _ B2(T) Ba(T)
z---l+--+--+ .... [3.72]
RT V V2
Formulas (3.71) and (3.72) are called virial equations of state; expressions A 2(T), A 3(T), ...
and B 2(T), Ba(T), .. , are called vinal coe.fficients.
When the series for different substances are compared at the same reduced temper-
ature and reduced pressure, the coefficients are similar, following the law of corresponding
states. The principal exceptions are hydrogen and helium at low temperatures, where
quantum mechanical effects become significant.
Around 1900, D. Berthelot modified the van der Waals equation to mimic the behav-
ior of gases at low pressures. Thus, he constructed the empirical equation

[3.73]

This yields the virial coefficients

1 [3.74]
A2 = 1:8 Pc1'r [1-~2 ).

A j = 0 for j = 3, 4, .... [3.75]
Formula (3.73) is known as the Berthelot equation of state; it is used for calculating devi-
ations from ideality at low pressures.

3. 13 Distribution of Equivalent Particles in a Potential Field

In the usual laboratory, all systems are in the gravitational field of the earth. At the
edge of a plasma, a considerable electric field exists. About an ion, a fairly long range
electric field may exist. The gravitational field acts on all particles; the electric field, on
charged particles. In any case, the spatial distribution of the particles is affected.
Consider a gaseous mixture at temperature T subjected to a potential gradient. For
simplicity, neglect the van der Waals interactions. The partial pressure Pi of the ith species
at a given level of potential is then given by equation (3.33). Letting n/V be concentra-
tion c i of the species, we have
[3.76]

and
dPi =RTdci' [3.77]
Let the potential of a molecule or ion of the ith species at a certain level be E i • The
force acting in the z direction on the particle is then - OE/dZ. Multiplying this by the