Question

The ideal gas in a Carnot engine extracts $1000\mathrm{J}$of heat energy during the isothermal expansion at $300°\mathrm{C}$. How much heat energy is exhausted during the isothermal compression at$50°\mathrm{C}$$?$

Short answer

Heat energy is exhausted during the isothermal compression is$560\mathrm{J}$.

Step-by-step solution

Formula for reduction of heat

The first law holds,

${\mathrm{Q}}_{\mathrm{h}}=\mathrm{W}+{\mathrm{Q}}_{\mathrm{c}}$

where ${\mathrm{Q}}_{\mathrm{h}}$ is the heat given to the gas from hot source , Q_c is the heat dumped to cold source and W is work output.

Efficiency,

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

$\mathrm{W}={\mathrm{\eta Q}}_{\mathrm{h}}$

This results is,

${\mathrm{Q}}_{\mathrm{c}}={\mathrm{Q}}_{\mathrm{h}}-\mathrm{W}={\mathrm{Q}}_{\mathrm{h}}(1-\mathrm{\eta })$

Calculation for reduction of heat

Efficiency is,

$\mathrm{\eta }=1-\frac{{\mathrm{T}}_{\mathrm{c}}}{{\mathrm{T}}_{\mathrm{h}}}$

Substituting this, we find the expression for the lost heat to be,

${\mathrm{Q}}_{\mathrm{c}}={\mathrm{Q}}_{\mathrm{h}}\left(1-\left(1-\frac{T_{\mathrm{c}}}{{\mathrm{T}}_{\mathrm{h}}}\right)\right)=\frac{{\mathrm{T}}_{\mathrm{c}}}{{\mathrm{T}}_{\mathrm{h}}}{\mathrm{Q}}_{\mathrm{h}}$

The lost heat will be

${\mathrm{Q}}_{\mathrm{c}}=\frac{50+273}{300+273}·1000=564\mathrm{J}$