1. The ideal gas law, PV = nRT, provides a simple description of the relationship between

pressure, P, volume, V, and temperature, T, where R is the ideal gas constant, and n is

the number of moles of gas. The ideal gas law is a linear approximation that is accurate

for gases at low pressure and moderate temperature. In other conditions it does not

provide a useful description. In such cases, we use a more complex description of this

relationship that takes into account the volume of the gas molecules themselves, as well

as intermolecular forces. These considerations lead to the van der Waals equation: 0.112 where a and b are constants that depend on the particular gas.

The ideal gas law is accurate when the number of moles, n, is small compared to the

volume, V, so that the volume taken up by the gas molecules can be neglected, and interactions betwem gas particles are negligible. We will show how the ideal gas law is

reached from the van der Waals equation under this condition. (a) Set a: = 3. Rewrite the van der Waals equation in terms of a", and isolate P. You

should obtain: RT

3

P = — 2

(1:) 1 _ in; cm (b) Determine the derivative of P[a) with respect to r, treating T and R as constants. (c) Find the linear approximation for 13(3) centered at a; = [1. This corresponds to ideal

conditions: when there are few particles in a relatively large volume, so that g as 0). (d) Use the linear approximation to recover the ideal gas law.