Railway Tanker

A long, cylindrical tank is placed on a carriage that can slide without friction on rails (see Figure 1.1). The mass of the empty tanker is M = 180 kg.  Initially, the tank is filled with an ideal gas of mass m = 120 kg at a pressure P₀ = 150 atm at an ambient temperature T₀ = 300K. Then one end of the tank is heated to 335 K while the other end is kept fixed at 300 K. Find the pressure in the tank and the new position of the center of mass of the tanker when the system reaches equilibrium.

Figure 1.1

SOLUTION

The new equilibrium pressure of the gas will be the same throughout the tanker, whereas the temperature across its length will vary: higher at the heated wall, and cooler at other end. Expanding the temperature T along the length of the tanker in a Taylor series and keeping the first two terms (since the temperature difference between the walls is small compared to T₀), we have

(1)

We may write the ideal gas law as a function of position in the tanker:

(2)

Where n (x) is the gas concentration. Rearranging, we have

(3)

The total number N of molecules in the cylinder is given by

(4)

where A is the cross-sectional area of the tanker. Alternatively, we can integrate (1.3) exactly and expand the resulting logarithm, which yields the same result. The total number of molecules originally in the tank is

(5)

Since the total number of molecules in the gas before and after heating is the same, (no phase transitions), we may equate (4) and (5), yielding

(6)

The center of mass (inertia) X₀ of the gas found with the same accuracy is given by

(7)

As we have assumed that the tanker slides on frictionless rails, the center of mass of the system will not move but the center of the tanker will move by an amount ∆X such that

(8)

Substituting (7) into (8) and rearranging give

(9)