Question

You are designing a Carnot engine that has $2\mathrm{mol}$of ${\mathrm{CO}}_2$as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of $527°\mathrm{C}$and a maximum pressure of 5.00 atm. With a heat input of 400 J per cycle, you want 300 J of useful work. For how many cycles must this engine run to melt completely a $10.0\mathrm{kg}$block of ice originally at $0.0°\mathrm{C}$ , using only the heat rejected by the engine?

Short answer

The number of cycles must this engine run to melt completely a $10.0\mathrm{kg}$ block of ice originally at $0.0°\mathrm{C}$is $334\times {10}^2$cycle.

Step-by-step solution

  Latent heat of fusion of ice

The amount of heat required to melt the ice without raising its temperature is called latent heat of fusion of ice.

It can be given by $\mathrm{Q}=\mathrm{mL}$.

Here, $\mathrm{Q}$is the amount heat gain by ice, is the mass of ice and $\mathrm{L}$is latent heat of fusion of ice .

Heat rejected by Carnot engine

The total amount of heat gain by Carnot engine is equal to heat spends on useful work and rejected heat.

${\mathrm{Q}}_{\mathrm{G}}=\mathrm{W}+{\mathrm{Q}}_{\mathrm{R}}$

The conservation of heat energy

The heat gain by one body is equals to the heat lost by another body.

${\mathrm{Q}}_{\mathrm{Lost}}={\mathrm{Q}}_{\mathrm{gain}}$

Calculation of heat rejected by Carnot engine

Using relation

${\mathrm{Q}}_{\mathrm{G}}=\mathrm{W}+{\mathrm{Q}}_{\mathrm{R}}$

Now, putting the values of constants in above relation

$$\begin{gathered} {\mathrm{Q}}_{\mathrm{G}}=\mathrm{W}+{\mathrm{Q}}_{\mathrm{R}} \\ 400=300+{\mathrm{Q}}_{\mathrm{R}} \\ {\mathrm{Q}}_{\mathrm{R}}=100\mathrm{J} \end{gathered}$$

Here, ${\mathrm{Q}}_{\mathrm{R}}$is the rejected heat by engine So, conventionally it must be negative. ${\mathrm{Q}}_{\mathrm{R}}=-100\mathrm{J}$

Calculation of the number of cycles must the engine run to melt completely a  10.0kg block of ice originally at  0.0°C

Using the conservation of heat energy

Heat rejected by engine number of cycles =Heat gain by ice

$$\begin{gathered} {\mathrm{nQ}}_{\mathrm{R}}=-\mathrm{mL} \\ \mathrm{n}=\frac{-\mathrm{mL}}{{\mathrm{Q}}_{\mathrm{R}}} \end{gathered}$$

Now, putting the values of constants

$$\begin{gathered} \mathrm{n}=\frac{-10\times 334\times {10}^3}{-100} \\ \mathrm{n}=334\times {10}^2 \end{gathered}$$

So, role="math" localid="1664278813409" $\mathrm{n}=334\times {10}^2\mathrm{cycles}$

Thus, the number of cycles must this engine run to melt completely a $10.0\mathrm{kg}$ block of ice originally at $0.0°\mathrm{C}$is $334\times {10}^2$cycle.