Problem 1: Short questions (15 points each = 45 points)

a)

$$P=\left(\genfrac{}{}{0ex}{}{4}{2}\right){(1∕4)}^2{(3∕4)}^2=27∕128$$

b)

$$\Phi =\frac{1}{4}n⟨v⟩\sim \frac{1}{\sqrt{m}}$$

$${\Phi }_{\mathit{𝑁𝑒}}∕{\Phi }_{\mathit{𝐴𝑟}}=\sqrt{m_{\mathit{𝐴𝑟}}∕m_{\mathit{𝑁𝑒}}}=\sqrt{2}$$

c)

$$\lambda =\frac{1}{\sqrt{2}\mathit{𝑛𝜎}}$$

$n$ stays constant upon changing $T$ $\Rightarrow$ $\lambda$ unchanged

Problem 2: Molecules on surface (60 points)

a), b), c)

$$p_{\mathit{𝑢𝑝}}=\frac{e^{-\mathit{𝛽𝜖}}}{2+e^{-\mathit{𝛽𝜖}}} ,p_x=\frac{1}{2+e^{-\mathit{𝛽𝜖}}} ,p_y=\frac{1}{2+e^{-\mathit{𝛽𝜖}}}$$

d)

$T\to 0,\mathit{ 𝛽}\to \mathit{\infty }\Rightarrow p_{\mathit{𝑢𝑝}}\to 0$

e)

$T\to \infty ,\mathit{ 𝛽}\to 0 \Rightarrow p_{\mathit{𝑢𝑝}}\to 1∕3$

a)

$$⟨E⟩=\frac{𝜖e^{-\mathit{𝛽𝜖}}}{2+e^{-\mathit{𝛽𝜖}}}$$

b)

$T\to 0,\mathit{ 𝛽}\to \mathit{\infty }\Rightarrow ⟨E⟩\to 0$

c)

$T\to \infty ,\mathit{ 𝛽}\to 0 \Rightarrow ⟨E⟩\to 𝜖∕3$

Problem 3: Two-dimensional ideal gas (45 points)

a)

$$g(v_x,v_y)=\frac{m}{2\pi k_{B}T}{e}^{-\mathit{𝛽𝑚}{\overrightarrow{v}}^2∕2}$$

b)

$$⟨E⟩=\frac{m}{2}(⟨v_x^2⟩+⟨v_y^2⟩)=k_{B}T$$

c)

$$\begin{array}{rcll}f(E)& =& \int \; <!--\; nolimits\; -->d{v}_{x}d{v}_{y}g(v_x,v_y)\delta (E-m(v_x^2+v_y^2)∕2)& \\ & =& \frac{m}{2\pi k_{B}T}\int \; <!--\; nolimits\; -->\mathit{𝑣𝑑𝑣𝑑𝜙}{e}^{-\frac{1}{2}m{v}^2∕k_{B}T}\delta (E-m{v}^2∕2)& \\ & =& \frac{m}{k_{B}T}{\int \; <!--\; nolimits\; -->}_0^{\infty }\frac{\mathit{𝑑𝑥}}{m}{e}^{-\mathit{𝛽𝑥}}\delta (E-x)& \\ & =& \frac{1}{k_{B}T}{e}^{-E∕k_{B}T}& \end{array}$$

_________________________________________________________________________________________