THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 108 Assignment # 1:
 
THERMAL PHYSICS
 
Wed. 05 Jan. 2005 - finish by Wed. 12 Jan.

1. "STAT - EC": Consider the following simplified model of a sort of stock market: A given stock ${\cal S}$ has a total of $N$ shares on the market for a fixed price $\varepsilon$. At a given time, $n$ of these shares are bought and the remaining $N-n$ are unwanted. Thus the net investment in ${\cal S}$ is $U = n\varepsilon$. [Here $\varepsilon$ and $U$ are measured in monetary units, say dollars; I have used the same notation as for energy for reasons that will soon become evident.] To keep things simple, we shall assume that the price $\varepsilon$ of a given stock does not change. Further, let's make the outrageous assumption that the stock market as a whole is a priori equally likely to be found in any one of the fully specified states accessible to it -- i.e. that a given amount of capital is equally likely to be distributed amongst the various stocks in any of the possible ways that give the same total.¹
   1. Invent a general definition for an economic analogue of temperature $\tau$ [measured in monetary units] that has the desired predictive power: that (given our starting assumptions) capital will tend to flow spontaneously from stocks with higher $\tau$ into others with lower $\tau$ and will stop flowing between two stocks only when they are in "economic equilibrium" - i.e. when they have the same "economic temperature" $\tau$.
   2. Now assume that the entire market is in "economic equilibrium" and is so much larger than any of its parts that we may treat it as a "capital reservoir" ${\cal R}$ at an "economic temperature" of $\tau$ = $100. Consider one share of one stock, valued at $\varepsilon_1$ = $200: What is the probability that it will be bought at any given time?
   3. Assuming that ${\cal R}$ is also huge compared to the entire offering of $N_1 = 1000$ shares of stock ${\cal S}_1$ valued at $\varepsilon_1$ = $200, what is the expected total investment $U_1$ in ${\cal S}_1$ when $\tau$ = $100?
   4. If the economic temperature drops to $\tau$ = $50, which stock will be likely to have the most capital $U$ invested in it, ${\cal S}_1$ with $N_1 = 1000$ shares at $\varepsilon_1$ = $200 per share or ${\cal S}_2$ with $N_2 = 1000$ shares at $\varepsilon_2$ = $100 per share?
2. MARS-EQUIVALENT ATMOSPHERIC PRESSURE: The composition of Mars' atmosphere is nominally 95.3% CO$_2$, 2.7% N$_2$, 1.6% Ar, 0.15% O$_2$ and 0.03% H$_2$O. Mean atmospheric pressure at the surface of Mars is 1-9 millibar, depending on altitude; the average is about 7 mb, compared to 1000 mb at sea level on Earth. At what altitude here on Earth would the atmospheric pressure be the same as that at the surface of Mars? (Assume an isothermal Earth atmosphere at 300 K. Are any other assumptions needed?)
3. ORTHO- vs. PARA-HYDROGEN: Molecular hydrogen, H$_2$, consisting of two protons bound together with two electrons, can form in either the "singlet" state called parahydrogen, in which the total spin (intrinsic angular momentum) of the molecule is zero, or in any one of three "triplet" states of orthohydrogen, in which the proton spins combine to make a total spin of $1 \hbar$ (the fundamental unit of angular momentum). For this problem, all you need to know is that the three triplet states are degenerate - i.e. they all have the same energy relative to the singlet state, namely $\varepsilon_3 = 2.375 \times 10^{-21}$ J. (The energy $\varepsilon_1$ of the singlet state can be taken to be zero, for reference.) Assume that the spin degrees of freedom of the H$_2$ molecules are unaffected by, but are in thermal equilibrium with, all their other degrees of freedom (like translational, rotational or vibrational). In this case, what fraction $f_3$ of H$_2$ molecules will be found (on average) in ortho states
   1. at room temperature (300 K)?
   2. at the boiling point of liquid nitrogen at atmospheric pressure (77 K)?
   3. at the freezing point of molecular hydrogen at atmospheric pressure (14 K)?