Homework Assignment # 7

Due Monday Nov 11, 2019

Do Problems 8-1, 8-3, 8-4, 8-8, 8-10, and 8-11 as well as the following:

  1. (For yourself, you don't need to hand these in) Solve all the problems you missed on Exam. 2.
  2. For a monatomic gas, the only contribution to the total energy arises from translational motion. We know that $$q_{trans}=\left (\frac{2\pi m k T}{h^2}\right )^{3/2} V$$
    1. Show that the entropy of a monatomic gas can be written as a sum of a term which is independent of mass plus an additional term which depends just on the mass of the gas $$S = D(T,V)+f(m)$$
    2. Derive the expression for $f(m)$.
    3. The molar entropies of the noble gases are, under standard conditions
      Element mass /amu S01 bar (J/K mol)
      He 4 126.15
      Ne 20 146.33
      Ar 40 154.80
      Kr 84 164.09
      where 1 amu = 1.66$10^{-27}$ kg

      To check your answer to part (b), for each gas $S-f(m)$ will be a constant, equal to $D(T,V)$. Use the data for He to derive the value of $D$ at standard temperature and pressure for He.

    4. Then, using their different masses, determine the entropies for the other noble gasses, and compare with the values given in the table. Show that this applies.
  3. We know that $\langle E \rangle = kT^2 (\partial \ln Q/ \partial T)_V$ and (see Section VI of the supplemental material, we will also show this in class) $ P=kT (\partial \ln Q/\partial V)_T$ . Use the equivalence of the mixed 2nd derivatives $$\frac{\partial^2 f(x,y)}{\partial x \partial y}= \frac{\partial^2 f(x,y)}{\partial y \partial x}$$ to show that the right hand side of Eq. (8.22) equals the left hand side even when we use these statistical-mechanical definitions of $U$ and $P$.
  4. Consider a two-level system. The lower level is doubly-degenerate, with energy $E_1 = -\varepsilon$ and $g_1=2$. The upper level is also non-degenerate ($g_2 = 1$) with energy $E_2 =+\varepsilon$
    1. Write an expression for $q(T)$ in terms of $\varepsilon$ and $x$, where $x=\exp(-\varepsilon/kT)$
    2. Determine an expression for $\langle E \rangle$ as a function of $\varepsilon$ and $T$. The limits at $T\to 0$ and $T\to\infty$ should agree with your physical intuition. We have discussed before the $T\to 0$ and $T\to \infty$ limits for similar two-level systems.
    3. On physical grounds, what is the entropy at $T= 0$ and $T=\infty$.
    4. Remembering that $\ln x = -\beta \varepsilon$, determine an expression for the entropy in the following form $$S(T) = \frac{\varepsilon}{T}\frac{P(x)}{Q(x)} + R(x)$$ where $P(x)$, $Q(x)$, and $R(x)$ are polynomials in the variable $x$
    5. From this expression, determine the $T\to 0$ and $T\to \infty$ values of $S(T)$. Your answer should agree with the answer to part (c) above.
    6. At $T=0$, $x=0$. Determine a power series expansion for $S(T)$ valid at low $T$ (small $x$), retaining all terms up through ${\cal O}(x^2)$