The following question was in my mid-sem test:
If $W_1$ and $W_2$ be the work done in an isothermal process and adiabatic process between same initial state and final state respectively, then
(a) $W_1$=$W_2$
(b) $W_1$>$W_2$
(c) $W_1$<$W_2$
(d) $W_1\le W_2$
I was thinking that the statement in the problem is wrong. Let the initial state be $(P_1,V_1)$ and final state be $(P_2,V_2)$. Let the isothermal curve passes through the initial state and final state. Also there is a relation between the slope of adiabatic curve and the isothermal curve.
slope of adiabatic curve = $\gamma \frac{dP}{dV}$ where $\frac{dP}{dV}$ is the slope of the isothermal curve.
Now if the adiabatic curve passes through the initial state $(P_1,V_1)$ then it won't pass through the final state.
Am I right or wrong? Is the statement in the problem means something else?
Best Answer
If both processes are reversible, then, as you said, they can't pass through the same initial and final states. If they are irreversible, I'm not so sure.
In any event, the intent of the problem seems to have been to get you to use the first law of thermodynamics. Both processes have the same $\Delta U$ because the initial and final states are the same. So, $W_I-W_A=Q_I-Q_A$. But, $Q_A=0$. Therefore, $$W_I=W_A+Q_I$$$Q_I$ can be either positive of negative. So, it seems to me, none of the answers are right.