91Ó°ÊÓ

Carnot engine temperatures,

The temperature of the heat source $=T_H=182°C$

The temperature of the heat sink $=T_C=0°C$

Mass $\left(m\right)=10Kg$

Distance $\left(h\right)=10m$

Energy extracted from the hot reservoir$=Q_H=25J$

Conversion of temperature into kelvin

We know that the temperature conversion is given by

$T_{Kelvin}=T_{celsius}+273K$

Therefore,

Conversion of Carnot engine temperature

$$\begin{gathered} T_C=0°C \\ {T}_C=273+0=273K \\ {T}_H=182°C \\ {T}_H=273+182=455K \end{gathered}$$

Temperature of the heat source $=T_H={182}^{0}C$

Temperature of the heat sink $=T_C=0^{0}C$

We know that the thermal efficiency of a heat engine is the ratio of the work done by the engine to the heat drawn out of the hot reservoir.

Here, $T_H$is the temperature of hot reservoir that is source and $T_C$is the temperature of cold reservoir in the above equation, the efficiency is given by

$$\begin{gathered} \mathrm{η}=1-\frac{T_C}{T_H} \\ \mathrm{η}=1-\frac{273}{455} \\ \mathrm{η}=0.4 \end{gathered}$$

Energy extracted from the hot reservoir $=Q_H=25J$

mass$m=10kg$

distance $h=10m$

Work done is defined as the energy extracted from the hot reservoir times the efficiency of the Carnot engine.

$∴W_{out}=\mathrm{η}{Q}_H$

Substituting the values of efficiency and heat absorbed in the above equation, net work done per cycle is calculated as

$$\begin{gathered} W_{Out}=0.4\left(25J\right) \\ {W}_{Out}=10J \end{gathered}$$

Now energy used to lift the mass through a distance is given by $mgh$

Therefore, the energy used to lift the mass is calculated as

$$\begin{gathered} mgh=10×9.81×10 \\ mgh=980J \end{gathered}$$

Now , the number of cycles is given by

$n=\frac{energytoliftmass}{W_{out}}$

Substituting the values of energy to lift the mass and energy spent per cycle in the above equation,

$n=\frac{980}{10}=98$

Conclusion: The number of cycles is $98$.