# Carnot cycle equation

How to prove the relation

$\frac{Q_{given\space out}}{Q_{absorbed}}=\frac{T_{cold}}{T_{hot}}$

by only writing equations using first law?

• Is this a homework problem? Commented Feb 1, 2016 at 14:47
• @ChesterMiller No it is not. I was reading about carnot cycle and came across this equation. I tried but was not able to derive this equation using first law. So, I was hoping someone in SE could help me. Commented Feb 1, 2016 at 15:26

I think that what you're asking can be done. Start at state 1 ($T_h,V_1$) and let the gas expand from $V_1$ to $V_2$ at constant $T_h$. This determines $Q_h$. Then let the gas expand from $V_2$ to $V_3$ adiabatically and reversibly. This determines $T_c$. Then compress at constant $T_c$ from $V_3$ to $V_4$. This determines $Q_c$. $V_4$ has to be such that, if you next compress adiabatically and reversibly from $V_4$ to $V_1$, the final temperature is $T_h$. You need to solve for the value of $V_4$ that satisfies this constraint. Once this analysis is complete, the heat flows and temperatures should satisfy your desired equation.