Written Problem

The Carnot cycle is an idealization that involves four processes as shown $($see also Figure

$6-37$ in textbook$).$ As each of the processes is reversible, the entire cycle is also reversible.

$(1)(2)$: Isothermal expansion at $T=T_H$

$(2)(3)$ : Adiabatic expansion $($no heat transfer$)$

$(3)(4)$: Isothermal compression at $T=T_L$

$(4)(1)$: Adiabatic compression $($no heat transfer$)$

$($a$)$ Suppose the Carnot cycle is carried out in a closed system with an ideal gas as

working fluid. Show that

$Q_H=mR{T}_{H}ln(\frac{V_2}{V_1})$

where $m$ is the mass of the ideal gas. $($Recall that the internal energy of an ideal gas

depends only on the temperature T$.)$

You are now provided with more specifics: The ideal gas is $2g$ of air. The Carnot cycle

operates between $T_H=500K$ and $T_L=300K,$ with $V_2=2V_1.$

$($b$)$ What is the amount of heat $Q_L(kJ)$ that is transferred to the cold reservoir in every

cycle?

$($c$)$ Determine $W_{\text{n}\text{e}\text{t}}(kJ),$ the net amount of work that is produced per cycle.

Note that $W_{\text{n}\text{e}\text{t}}$ can be obtained by summing the ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}PdV$ contribution from each of the

four processes. This approach, however, is not recommended as there are much

easier ways to get to the answer.