46 CHAPTER 1. BACKGROUND IDEAS
From this limited set of assumptions about theoretical entities called
molecules physicists can derive the equation of state for an ideal gas in terms
of the 4 quantifiable elements of volume, pressure, amount, and temperature.
Theequation of stateorideal gas lawis
PV =nRT.whereRis a measured constant, called the universal gas constant. This gives
a simple algebraic equation relating the 4 quantifiable elements describing a
gas. The equation of state or ideal gas law predicts very well the properties of
gases under the wide range of pressures, temperatures, masses and volumes
commonly experienced in everyday life. The ideal gas law predicts with
accuracy necessary for safety engineering the pressure and temperature in
car tires and commercial gas cylinders. This level of prediction works even
for gases we know do not satisfy the assumptions, such as air, which chemistry
tells us is composed of several kinds of molecules which have volume and do
not experience completely elastic collisions because of intermolecular forces.
We know the mathematical model is wrong, but it is still useful.
Nevertheless, scientists soon discovered that the assumptions of an ideal
gas predict that the difference in the constant-volume specific heat and the
constant-pressure specific heat of gases should be the same for all gases, a
prediction that scientists observe to be false. The simple ideal gas theory
works well for monatomic gases, such as helium, but does not predict so well
for more complex gases. This scientific observation now requires additional
assumptions, specifically about the shape of the molecules in the gas. The
derivation of the relationship for the observable in a gas is now more complex,
requiring more mathematical techniques.
Moreover, under extreme conditions of low temperatures or high pres-
sures, scientists observe new behaviors of gases. The gases condense into
liquids, pressure on the walls drops and the gases no longer behave according
to the relationship predicted by the ideal gas law. We cannot neglect these
deviations from ideal behavior in accurate scientific or engineering work. We
now have to admit that under these extreme circumstances we can no longer
ignore the size of the molecules, which do occupy some appreciable volume.
We also must admit that intermolecular forces must be considered. The two
effects just described can be incorporated into a modified equation of state
proposed by J.D. van der Waals in 1873. Van der Waals’ equation of state
is: (
P+
na
V^2)
(V−b) =RT