Consider a sample containing 5.00 moles of a monatomic ideal gas that is taken from state $\mathrm{A}$ to state $\mathrm{B}$ by the following two pathways:
Pathway one:
$$
\begin{array}{l}{P_{\mathrm{A}}=3.00 \mathrm{atm}} \\ {V_{\mathrm{A}}=15.0 \mathrm{L}}\end{array} \stackrel{1}{\longrightarrow} \begin{array}{l}{P_{\mathrm{D}}= 3.00 \mathrm{atm}} \\ {V_{\mathrm{D}}=55.0 \mathrm{L}}\end{array}\\
\quad\quad\quad\quad\quad\quad\stackrel{2}{\longrightarrow} \begin{array}{l}{P_{\mathrm{D}}=6.00 \mathrm{atm}} \\ {V_{\mathrm{D}}=20.0 \mathrm{L}}\end{array}
$$
Pathway two:
$$
\begin{array}{l}{P_{\mathrm{A}}=3.00 \mathrm{atm}} \\ {V_{\mathrm{A}}=15.0 \mathrm{L}}\end{array} \stackrel{3}{\longrightarrow} \begin{array}{l}{P_{\mathrm{D}}=6.00 \mathrm{atm}} \\ {V_{\mathrm{D}}=15.0 \mathrm{L}}\end{array}\\
\quad\quad\quad\quad\quad\quad\stackrel{4}{\longrightarrow} \begin{array}{l}{P_{\mathrm{D}}=6.00 \mathrm{atm}} \\ {V_{\mathrm{D}}=20.0 \mathrm{L}}\end{array}
$$
For each step, assume that the external pressure is constant and equals the final pressure of the gas for that step. Calculate $q, w, \Delta E,$ and $\Delta H$ for each step in $\mathrm{kJ}$ , and calculate overall values for each pathway. Explain how the overall values for the two pathways illustrate that $\Delta E$ and $\Delta H$ are state functions, whereas $q$ and $w$ are path functions. Hint: In a more rigorous study of thermochemistry, it can be shown that for an ideal gas:
$$
\begin{aligned} \Delta E &=\mathrm{nC}_{\mathrm{v}} \Delta T \text { and } \\ \Delta H &=\mathrm{nC}_{\mathrm{p}} \Delta T \end{aligned}
$$
where $\mathrm{C}_{\mathrm{v}}$ is the molar heat capacity at constant volume and $\mathrm{C}_{\mathrm{p}}$ is the molar heat capacity at constant pressure. In addition, for a monotomic ideal gas, $\mathrm{C}_{\mathrm{v}}=\frac{3}{2} \mathrm{R}$ and $\mathrm{C}_{\mathrm{p}}=\frac{5}{2} \mathrm{R}$ .