Question

A heat engine with $0.20\mathrm{mol}$of a monatomic ideal gas initially fills a$2000{\mathrm{cm}}^3$cylinder at $600\mathrm{K}$. The gas goes through the following closed cycle:

Isothermal expansion to $4000{\mathrm{cm}}^3$.

Isochoric cooling to $300\mathrm{K}$.

Isothermal compression to$2000{\mathrm{cm}}^3$ .

Isochoric heating to $600\mathrm{K}$.

How much work does this engine do per cycle and what is its thermal efficiency?

Short answer

The work done by engine is$346\mathrm{J}$,Thermal efficiency is$24\%$.

Step-by-step solution

Calculation for work done of engine

Let us remember that the engine generates heat in the first and fourth processes while working on the first, and that the third process is working on the gas.

Work on the isothermal expansion will be completed.

$\mathrm{W}=\mathrm{nRTln}\left({\mathrm{V}}_2/{\mathrm{V}}_1\right)$

localid="1650290238937" $\mathrm{W}=0.2\mathrm{mol}\times 8.314\Omega \times 600\mathrm{Kln}\left(2\right)$

In the isothermal compression, the work done on the gas will be

$\mathrm{W}=0.2\mathrm{mol}\times 8.314\mathrm{\Omega }\times 300\mathrm{Kln}(0.5)$

$=-346\mathrm{J}$

The amount of heat used in isochoric heating will be

$\mathrm{Q}={\mathrm{nC}}_{\mathrm{v}}\mathrm{\Delta T}$

$=0.2\mathrm{mol}\times 12.5{\mathrm{m}}^3\times 300\mathrm{K}$

$=750\mathrm{J}$

The net work is,

$\mathrm{W}=692\mathrm{J}-346\mathrm{J}$

localid="1650290820029" $=346\mathrm{J}$

Calculation for thermal efficiency

The total amount of heat consumed was

${\mathrm{Q}}_{\mathrm{i}}=692\mathrm{J}+750\mathrm{J}$

$=1442\mathrm{J}$

The engine's efficiency is

$\mathrm{\eta }=\frac{\mathrm{W}}{{\mathrm{Q}}_{\mathrm{i}}}$

$=\frac{346\mathrm{J}}{1442\mathrm{J}}$

$=24\%$