What is a heat engine?

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(N/A) heat engine is a device that converts heat energy into mechanical work. It operates by taking heat from a high-temperature reservoir (source),performing work on the surroundings,and rejecting the remaining heat to a low-temperature reservoir (sink). The efficiency of a heat engine is defined as the ratio of the work done to the heat absorbed from the source,given by $\eta = \frac{W}{Q_H} = 1 - \frac{Q_L}{Q_H}$,where $W$ is the work done,$Q_H$ is the heat absorbed from the source,and $Q_L$ is the heat rejected to the sink.

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Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures $T_{1}$ and $T_{2}$. The temperature of the hot reservoir of the first engine is $T_{1}$ and the temperature of the cold reservoir of the second engine is $T_{2}$. $T$ is the temperature of the sink of the first engine,which is also the source for the second engine. How is $T$ related to $T_{1}$ and $T_{2}$,if both engines perform equal amounts of work?

$A$ Carnot engine $C_1$ operates between temperatures $T_1$ and $T_2$ $(T_1 > T_2)$. $A$ second Carnot engine $C_2$ uses all the heat rejected by the engine $C_1$ and operates between temperatures $T_2$ and $T_3$ (where $T_2 > T_3$). The efficiency of this combined ($C_1$ and $C_2$ together) engine is

$A$ diatomic ideal gas is used as the working substance in a Carnot engine. During the adiabatic expansion process,if the volume of the gas increases from $V$ to $32V$,what is the efficiency of the engine?

Does a Carnot engine working under an isothermal state perform any useful work?

Three designs are proposed for an engine which is to operate between $500 \ K$ and $300 \ K$. Design $A$ claims to produce $150 \ J$ of work per $1000 \ J$ of heat input, design $B$ $450 \ J$ of work per $1000 \ J$, and design $C$ $300 \ J$ of work per $1000 \ J$. Which of the designs would you choose?

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