Page 90 - Physical chemistry understanding our chemical world

QUANTIFYING THE INTERACTIONS AND THEIR INFLUENCE 57

p = 3.65887 bar − 0.05895 bar
p = 3.59992 bar

The pressure calculated with the ideal-gas equation (Equation (1.13)) is 3.63 bar, so
the value we calculate with the van der Waals equation (Equation (2.2)) is 1 per cent
smaller. The experimental value is 3.11 bar, so the result with the van der Waals equation
is superior.

The lower pressure causes coalescence of gas particles, which decreases their kinetic
energy. Accordingly, the impact between the aggregate particle and the container’s
walls is less violent, which lowers the observed pressure.

The virial equation

An alternative approach to quantifying the interactions and deviations from the ideal-
gas equation is to write Equation (1.13) in terms of ‘virial coefficients’:

V 2

p = RT (1 + B p + C p + ...) (2.3)
n
where the V ÷ n term is often rewritten as V m and called the molar volume.
Equation (2.3) is clearly similar to the ideal-gas equation,
Equation (1.13), except that we introduce additional terms, each The word ‘virial’ comes
expressed as powers of pressure. We call the constants, B , C etc., from the Latin for force

‘virial coefficients’, and we determine them experimentally. We or powerful.

call B the second virial coefficient, C the third, and so on.
Equation (2.3) becomes the ideal-gas equation if both B and C are tiny. In fact,

these successive terms are often regarded as effectively ‘fine-tuning’ the values of p or
V m .The C coefficient is often so small that we can ignore it; and D is so minuscule

that it is extremely unlikely that we will ever include a fourth virial coefficient in any
calculation. Unfortunately, we must exercise care, because B constants are themselves

a function of temperature.
Worked Example 2.2 What is the molar volume V m of oxygen gas
Care: the odd-looking
O
at 273 K and p ? Ignore the third and subsequent virial terms, and
−1

take B =−4.626 × 10 −2 bar . units of B’ require us
to cite the gas con-
stant R in SI units with
From Equation (2.3)
prefixes.
V RT

V m = = × (1 + B p)
n p