Consider a van der Waals gas contained in the apparatus described in Problem 3.4-1 (i.e., in the "constant volume gas thermometer").
a) Assuming it to be known in advance that the gas obeys a van der Waals equation of state, show that knowledge of two reference temperatures enables one to evaluate the van der Waals constants a and b.
b) Knowing the constants a and b, show that the apparatus can then be used as a thermometer, to measure any other temperature.
c) Show that knowledge of three reference temperatures enables one to determine whether a gas satisfies the van der Waals equation of state, and if it does, enables one to measure any other temperature.
Problem 3.4-1
A "constant volume ideal gas thermometer" is constructed as shown (schematically) in Fig. 3.3. The bulb containing the gas is constructed of a material with a negligibly small coefficient of thermal expansion. The point A is a reference point marked on the stem of the bulb. The bulb is connected by a flexible tube to a reservoir of liquid mercury, open to the atmosphere. The mercury reservoir is raised or lowered until the mercury miniscus coincides with the reference point A. The height h of the mercury column is then read.
a) Show that the pressure of the gas is the sum of the external (atmospheric) pressure plus the height h of the mercury column multiplied by the weight per unit volume of mercury (as measured at the temperature of interest).
b) Using the equation of state of the ideal gas, explain how the temperature of the gas is then evaluated.
c) Describe a "constant pressure ideal gas thermometer" (in which a changing volume is directly measured at constant pressure).
