Thermodynamics, Statistical Physics, and Quantum Mechanics

89

4.6 Teacup Engine (Princeton, Moscow Phys-Tech)

If the cup were vacuumtight, the number of molecules leaving the surface

would be the same as the number of molecules returning to the surface.

The mass flow rate of the molecules hitting the surface (see Problem 4.14)

is

where is the vapor density corresponding to the saturation, is the

average velocity of the molecules, and A is the surface area of the ice. The

mass flow rate of the molecules actually returning to the surface is

where is the stickingcoefficient(the probabilitythat themoleculehitting

the surface will stick to it). Let us assume for now that (we will

see later that this is not true, but that actually gives us the lower limit

of the distance). If the cup is open we can assume that the number of

molecules leaving the surface is the same as in the closed cup, but there

are few returning molecules. We then find that the time for complete

evaporation of the ice is

where we take g as the mass of the ice, and

from Problem 4.13. Substituting (S.4.6.4) into (S.4.6.3), we obtain

Once again using the ideal gas law, we may obtain

Substituting(S.4.6.6) into(S.4.6.5) yields

THERMODYNAMICS AND STATISTICAL PHYSICS