Find the work and heat for two different paths.

One mole of an ideal monatomic gas $($molar $Cv=\frac{3}{2}R,$ molar $Cp=\frac{5}{2}R)$ is contained in a cylinder with a piston. The gas begins $($state $1)$ with volume, $V_1=10$ liter, pressure, $P_1=2\mathrm{.}47$atm, and temperature $T_1=300K.$ Two processes $("$paths$")$ are carried out, each of which produces the same final values $($state $2)$: $V_3=15$ liter, $P_3=3\mathrm{.}28$atm, and $T_3=600K.$

Path A: heat at constant $V$ until $P_2=4\mathrm{.}92$ atm and $T_2=600K,$ then expand at constant $T$ to the final values above.

Path B: expand at constant $T$ until $V_2=15L$ and $P_2=1\mathrm{.}64$ atm, then heat at constant $V$ to the final values above.

A useful fact: for an ideal gas, both energy and enthalpy depend only on temperature, and don't change if the temperature doesn't change.

Using the expressions for heat and work from class, find the heat change, $q_A,$ and work change, $w_A,$ for path $A,$ and the heat change, $q_B,$ and work change $w_B,$ for path $B.$ Also find the total energy change, $U,$ for each path.