A Carnot engine operates between $\mathbf{235}\mathbf{°}\mathbf{C}$and$\mathbf{115}\mathbf{°}\mathbf{C}$, absorbing$\mathbf{6}\mathbf{.}\mathbf{30}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{J}$per cycle at the higher temperature. (a) What is the efficiency of the engine? (b) How much work per cycle is this engine capable of performing?

The engine operates at temperatures,${\mathrm{T}}_{\mathrm{H}}=235°\mathrm{C}=508\mathrm{K}$ and${\mathrm{T}}_{\mathrm{L}}=115°\mathrm{C}=388\mathrm{K}$

The energy absorbed at the higher temperature,${\mathrm{Q}}_{\mathrm{H}}=6.30\times {10}^4\mathrm{J}$

We can find the efficiency of the Carnot engine using the corresponding formula. Then, multiply this by the amount of heat absorbed per cycle at the higher temperature, and we can get work done per cycle by the Carnot engine.

Formula:

The efficiency of the Carnot engine,

$\mathrm{\epsilon }=\frac{{\mathrm{T}}_{\mathrm{H}}-{\mathrm{T}}_{\mathrm{L}}}{{\mathrm{T}}_{\mathrm{H}}}$ (1)

The work done per cycle of the engine,

$\mathrm{W}={\mathrm{\epsilon Q}}_{\mathrm{H}}$ (2)

Using equation (1) and the given values, we can get the efficiency of the engine as given:

$$\begin{gathered} \mathrm{\epsilon }=\left(\frac{508\mathrm{K}-388\mathrm{K}}{508\mathrm{K}}\right)\times 100\% \\ =\left(0.236\right)\times 100\% \\ =23.6\% \end{gathered}$$

Therefore, the efficiency of the Carnot engine is $23.6\%$.

Work done per cycle by Carnot engine using equation (2) and given values can be given as:

$$\begin{gathered} \mathrm{W}=\left(0.236\right)\left(6.30\times {10}^4\mathrm{J}\right) \\ =1.49\times {10}^4\mathrm{J} \end{gathered}$$

Therefore, work done per cycle by Carnot engine is$1.49\times {10}^4\mathrm{J}$ .