Question

A gas is enclosed in a cylinder with a movable friction less piston. Its initial thermodynamic state at pressure $P_i={10}^{5}Pa$ and volume $V_i={10}^{-3}{m}^3$ changes to a final state at $P_f=\left(1/32\right)\times {10}^{5}Pa$ and $V_f=8\times {10}^{-3}{m}^3$ in an adiabatic quasi-static process, such that $P^3{V}^5$=constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $P_i$ followed by an isochoric (isovolumetric) process at volume $V_f$. The amount of heat supplied to the system in the two-step process is approximately:

• A. $112J$
• B. $294J$
• C. $588J$
• D. $813J$

Answer 1

The correct option is C $588J$

$P{V}^{5/3}=C$

Thus, $\gamma =\frac{5}{3}$

$\Delta Q_1=n{C}_P\Delta T=n\frac{\gamma R}{\gamma -1}\Delta T=\frac{\gamma P\Delta V}{\gamma -1}=\frac{5}{2}\times {10}^5(7\times {10}^{-3})=1750J$

$\Delta Q_2=n{C}_V\Delta T=n\frac{R}{\gamma -1}\Delta T=\frac{V\Delta P}{\gamma -1}=\frac{(\frac{1}{32}-1)\times {10}^5(8\times {10}^{-3})}{\frac{2}{3}}=-\frac{9300}{8}J$

$\Delta Q=\Delta Q_1+\Delta Q_2=1750-\frac{9300}{8}\approx 588J$