6. [14 points total] Methane on Titan
Consider a methane (CH4) lake on Titan, a large moon of Saturn. Suppose the surface temperature by this lake is T = 95 K. We will explore the vapor pressure of methane near the surface of the lake and the ability of methane to escape Titan's atmosphere into space.
a. [6 points] The Clausius-Clapeyron equation is given in Schroeder as:
dP = (L / R) * (dT / T^2)
where L is the latent heat of vaporization, R is the gas constant, and dP and dT represent small changes in pressure and temperature, respectively.
Assuming we can treat methane as an ideal gas, show that the vapor pressure as a function of temperature can be written as:
P(T) = P(T_ref) * exp((L / R) * (1 / T_ref - 1 / T))
where T_ref is some reference temperature for which the vapor pressure is known.
b. [2 points] The atmosphere immediately above this methane lake is saturated with methane vapor in equilibrium with the liquid methane on the lake's surface. At the triple point of methane, T_rp = 90.67 K and P_rp = 0.117 bar with 1 bar = 10^5 Pa. The latent heat of vaporization for methane is L = 8.19 kJ/mol. What is the vapor pressure of methane above the surface of this lake?
c. [4 points] The escape velocity from the surface of Titan is 2639 m/s. The Maxwell-Boltzmann speed distribution can be written in terms of a scaled speed z = u / (sqrt(2RT/M)) (where u_max is the peak of the Maxwell-Boltzmann speed distribution) as:
D(x) = 4 * pi * (M / (2 * pi * R * T))^1.5 * x^2 * exp(-M * x^2 / (2 * R * T))
Write an integral whose answer will give the probability of a methane molecule near Titan's surface having a speed between x and x + dx. Evaluate the integral. You must, however, include correct numerical values for the limits of the integral.
d. [2 points] Below is a graph of D(x).
Would you expect the probability of a methane molecule near Titan's surface to have a speed in excess of the escape speed to be significant (approaching 1% or an appreciable fraction of a percent)? Ideally, your explanation should refer to the graph above.
