Extraterrestrial Kinetic Theory of Gases Michael T. Marron University of Wisconsin-Parkside, Kenosha, WI 53141
asking students to reproduce derivations done in class. In its present form, i t is appropriate for a take-home exam. For an in-class exam, the functions for part a could be provided. One can readily imagine variations involving other simple functional forms for either application level. Question On the planet Taru, Boltzmann's Law has been declared counterrevolutionary by the Politburo. Revisionist members of the Central Committee have persuaded Politburo members to adopt a new law for energy distribution to he named after Paracelsus, a famous Taruvian worker-chemist. The new law will he put into effect immediately, and all things will be made to conform. Consistent with the Taruvian sense of fair play, energy is to be distributed linearly among all things with most having none and with a maximum value of E, = I12 mc2,,; nothing may have an energy greater than Em,. The maximum speed, c,,, was set hy decree to the most probable value = (2kTl under the old, discredited Boltzmann regime, e,, m)'I2. This pleased the proletariat very much. A plot of a Paracelsian energy probability density function is shown in Figure 3. (a) Derive expressions for both the energy and speed probability density functions for Paracelsian gases. Be sure to normalize your functions. (b) Compute the most probable and the average value of the molecular speed for a Paracelsian gas. (c) What is the average molecular energy of a Paracelsian gas? (d) How many molecules in a mole of Paracelsian gas will have speeds less than %em,,? (e) When Boltzmann's Law was in effect the ideal gas law was P V = nRT; what is the new ideal gas law for a Paracelsian gas?
Figure 3. Plot of a "Paracelsian" energy probability density function.
tained by changing variables using the substitution E = mc2 and being careful to account for the difference in volume elements so that h(E)dE = g(c)dc. This gives g(c) = 4c(l - ( c / c ~ , , ) ~ ) / c ~ ~ ~ . (h) The most probable speed is computed by finding the maximum value of g(c) using dg
(z)~=
