# Thermodynamics, maximum work

1. Apr 22, 2012

### fluidistic

1. The problem statement, all variables and given/known data
Each of three identical bodies has an equation of state U=NCT, with NC=2 cal/K. Their initial temperatures are 200 K, 250 K and 540 K. What is the maximum amount of work that can be extracted in a process in which these three bodies are brought to a final common temperature?

2. Relevant equations
Not sure.

3. The attempt at a solution
I know that the efficiency isn't constant during the entire process because the temperature of the bodies change.
I've made a sketch with the following characteristics: An amount of heat $dQ_1$ leaves the body whose initial temperature is 540 K (let's call it body 1 with temperature $T_1$). This amount of heat enters a machine that produces an amount of work $dW_1$ and transmit an amount of heat $dQ_2$ to the body (let's call it body 2 with temperature $T_2$) that is initially at a temperature of 250 K. A similar process occur between the body 2 and 3.
I think that the relation between $dW_1$, $dQ_1$ and $dQ_2$ is the following in case of maximum efficiency/maximum work produced:
$dW_1=-dQ_1 \left ( 1-\frac{T_2}{T_1} \right ) =-CdT_1\left ( 1-\frac{T_2}{T_1} \right )$.
Also $dQ_2=dQ_1-dW_1 =CdT_1+CdT_1 \left ( 1-\frac{T_2}{T_1} \right )$ and $cdT_2=dQ_2-dQ_3=CdT_1 \left ( 2-\frac{T_2}{T_1} \right ) \Rightarrow dT_2=dT_1 \left ( 2-\frac{T_2}{T_1} \right )$.
Now I'm not sure what I've done is relevant nor do I know how to proceed further. I must get $dW_1$ and $dW_2$ (amount of work produced by a machine operating between body 2 and 3).