Heat Engines

Q. A heat engine operates between a cold reservoir at temperature $T_2=400\text{K}$ and a hot reservoir at temperature $T_1.$ It takes $300\text{J}$ of heat from the hot reservoir and delivers $240\text{J}$ of heat to the cold reservoir in a cycle. The minimum temperature of the hot reservoir has to be $\text{K}$

Q. For which combination of temperatures the efficiency of Carnot's engine is highest?

1. $80\text{K},60\text{K}$
2. $100\text{K},80\text{K}$
3. $60\text{K},40\text{K}$
4. $40\text{K},20\text{K}$

Q. An ideal gas undergoes a cyclic process $abcda$ which is shown by a pressure - density curve. Identify the option with incorrect statement.

1. Work done by the gas in process $bc$ is zero
2. Work done by the gas in process $cd$ is negative
3. Internal energy of the gas at state $a$ is greater than that at state $c$
4. Net work done by the gas in the cycle is negative

Q. An ideal gas is taken through a cyclic thermodynamic process through four steps. The amounts of heat involved in these steps are $Q_1=5960\text{J},Q=-5585\text{J},Q_3=-2980\text{J}$ and $Q_4=3645\text{J}$ respectively. The corresponding quantities of work done by gas for each steps are $W_1=2200\text{J},W_2=-825\text{J},W_3=-1100\text{J}$ and $W_4=765\text{J}$ respectively. The efficiency of the cycle is

1. $10.82\%$
2. $20.85\%$
3. $60\%$
4. $79.18\%$

Q. An engine is designed to operate between $480\text{K}$ and $300\text{K}$. Assuming that the engine actually produces $1.2\text{kJ}$ of mechanical energy per $\text{kcal}$ of heat absorbed. Find the ratio of actual efficiency of engine to the theoretical maximum efficiency.
(Take $1\text{cal}=4.2\text{J}$)

1. $\frac{4}{3}$
2. $\frac{8}{28}$
3. $\frac{16}{21}$
   
4. None of the these

Q. A gas undergoes a cyclic process $a\to b\to c\to a$ which is as shown in the $PV$ diagram. The process $a\to b$ is isothermal , $b\to c$ is adiabatic and $c\to a$ is a straight line on the $P-V$ diagram. Work done in process $ab$ and $bc$ is $5\text{J}$ and $4\text{J}$ respectively. Calculate the efficiency of the cycle, if the area enclosed by the diagram $\text{abca}$ in the figure is $3\text{J}$.

1. $0.4$
2. $0.6$
3. $0.75$
4. $0.8$

Q. An ideal gas is taken through a cyclic thermodynamic process through four steps. The amounts of heat involved in these steps are $Q_1=5960\text{J}$, $Q_2=-5585\text{J}$, $Q_3=-2980\text{J}$ and $Q_4=3645\text{J}$ respectively. The corresponding quantities of work involved are $W_1=2200\text{J}$, $W_2=-825\text{J},W_3=-1100\text{J}$ and $W_4$ respectively . Find the efficiency of the cycle.

1. $10.82\%$
2. $17.44\%$
3. $28.53\%$
4. $30.86\%$

Q. One mole of a monoatomic gas of molar mass $M$ undergoes a cyclic process as shown in the figure. Here, $\rho$ is the density and $P$ is the pressure of the gas. Find the efficiency of the cycle.

1. $\frac{3}{5}(1-\ln 2)$
2. $\frac{2}{5}(1-\ln 2)$
3. $\frac{3}{5}\ln 2$
4. $\frac{2\ln 2}{5}$

Q. Calculate the thermal efficiency of a Carnot's heat engine working between ice point and steam point.

1. $26.81\%$
2. $33.33\%$
3. $12.97\%$
4. $83.19\%$

Q. An ideal gas undergoes a cyclic process $abcda$ which is shown by a pressure - density curve. Identify the option with incorrect statement.

1. Work done by the gas in process $bc$ is zero
2. Work done by the gas in process $cd$ is negative
3. Internal energy of the gas at state $a$ is greater than that at state $c$
4. Net work done by the gas in the cycle is negative

Q. A heat engine is involved with exchange of heat of $1915J,-40J,+125J\text{and}-QJ$, during one cycle achieving an efficiency of $50.0$%. The value of $Q$ is

1. $980\text{J}$
2. $400\text{J}$
3. $640\text{J}$
4. $40\text{J}$

Q. An ideal gas acting as a working substance for a heat engine. The heat interaction involved during the process are $Q_1=9605\text{J}$ and $Q_2=-8565\text{J}$ respectively. Then efficiency of the heat engine is

1. $10.82\%$
2. $79.18\%$
3. $60\%$
4. $21\%$

Q. An ideal engine has an efficiency of $\frac{1}{12}$. When the temperature of sink is reduced by ${70}^{\circ }\text{C}$, its efficiency gets doubled. The temperature of the source is

1. ${567}^{\circ }\text{C}$
2. ${467}^{\circ }\text{C}$
3. ${767}^{\circ }\text{C}$
4. ${367}^{\circ }\text{C}$

Q. Find the efficiency of the thermodynamic cycle shown in figure for an ideal diatomic gas.

1. $\frac{1}{4}$
2. $\frac{1}{9}$
3. $\frac{1}{8}$
4. $\frac{1}{18}$

Q. In a Carnot heat engine, the temperature of source and sink are $500\text{K}$ and $375\text{K}$. If the engine consumes $25\times {10}^5\text{J}$ per cycle, then work done by engine per cycle is

1. $6.25\times {10}^5\text{J}$
2. $100\times {10}^5\text{J}$
3. $2.5\times {10}^5\text{J}$
4. $0.625\times {10}^5\text{J}$

Q. A heat engine receives $50\text{kcal}$ of heat from the source per cycle, and operates with an efficiency of $20\%$. Find the heat rejected to the sink per cycle.

1. $50\text{kcal}$
2. $10\text{kcal}$
3. $42\text{kcal}$
4. $40\text{kcal}$

Q. An ideal gas is taken through a cyclic thermodynamic process through four steps. The amounts of heat involved in these steps are $Q_1=5960\text{J}$, $Q_2=-5585\text{J}$, $Q_3=-2980\text{J}$ and $Q_4=3645\text{J}$ respectively. The corresponding quantities of work involved are $W_1=2200\text{J}$, $W_2=-825\text{J},W_3=-1100\text{J}$ and $W_4$ respectively . Find the efficiency of the cycle.

1. $10.82\%$
2. $17.44\%$
3. $28.53\%$
4. $30.86\%$

Q. A carnot engine, whose efficiency is 40% takes in heat from a source maintained at a temperature of $500\text{K}$. It is desired to have an engine of efficiency 60%. Then, the intake temperature for the same exhaust (sink) must be

1. efficiency of carnot engine cannot be made larger than 50%
2. $1200\text{K}$
3. $750\text{K}$
4. $600\text{K}$

Q. An ideal reversible heat engine converts one sixth of the heat input into work. If the temperature of the sink is reduced by ${62}^{\circ }C$, its efficiency is doubled. Find the temperature of the source and sink respectively (In $\text{Kelvin}$).

1. $325,375$
2. $372,310$
3. $372,248$
4. $472,310$

Q. An ideal gas is taken through a cycle $1231$ (see figure) and the efficiency of the cycle was found to be $25\%$. When the same gas goes through the cycle $1341$, the efficiency is $10\%$. Find the efficiency of the cycle $12341$.

1. $18.6\%$
2. $23.2\%$
3. $32.5\%$
4. $44\%$

Q. One mole of a diatomic ideal gas $(\gamma =1.4)$ is taken through a cyclic process starting from point $A$. The process $A\to B$ is an adiabatic compression, $B\to C$ is an isobaric expansion, $C\to D$ is an adibatic expansion and $D\to A$ is an isochoric process. The volume ratios are $\frac{V_A}{V_B}=16,\frac{V_C}{V_B}=2$ and temperature $T_B=909\text{K},T_C=1818\text{K}$. Find the efficiency of the cycle, when work done $W_{AB}=-1522.5R\text{J},W_{BC}=909R\text{J},W_{CD}=2567.5R\text{J}$

1. $32.84\%$
2. $61.40\%$
3. $28.56\%$
4. $35.61\%$

Q. Two Carnot engines ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ are operated in succession. The first one, $A$ receives heat from a source at $T_1=800\text{K}$ and rejects to a sink at $T_2\text{K}$ . The second engine $B$ receives heat rejected by the first engine and rejects it to the another sink at $T_3=300\text{K}$. If the work outputs of the two engines are equal, then the value of $T_2$ is

1. $100\text{K}$
2. $300\text{K}$
3. $550\text{K}$
4. $700\text{K}$

Q. A Carnot engine operates at an efficiency of $40\%$ with the sink at ${27}^{\circ }C$. Find the amount by which the temperature of the source must be increased in order to increase the efficiency by $10\%$.

1. $600\text{K}$
2. $100\text{K}$
3. $500\text{K}$
4. $300\text{K}$

Q. In which process in a Carnot's heat engine, heat $\left(Q_1\right)$ is absorbed from a heat source?

1. Reversible isothermal expansion
2. Reversible isothermal compression
3. Reversible adiabatic expansion
4. Reversible adiabatic compression

Q. A real heat engine has an efficiency of $40\%$. The work output of the engine is $100\text{kJ}$ per cycle. How much energy is extracted from the high temperature reservoir per cycle ?

1. $40\text{kJ}$
2. $250\text{kJ}$
3. $400\text{kJ}$
4. Can be calculated only if the engine is a Carnot engine.

Q. A Carnot heat engine works between the temperatures ${427}^{\circ }\text{C}$ and ${27}^{\circ }\text{C}$. What amount of heat should it consume per second to deliver mechanical work at the rate of $1.0\text{kW}$?

1. $418\text{cal/s}$
2. $41.8\text{cal/s}$
3. $4.18\text{cal/s}$
4. $0.418\text{cal/s}$

Q. Two Carnot engines ${}^{\prime }{A}^{\prime }$ and ${}^{\prime }{B}^{\prime }$ are operated in succession. The first one, $A$ receives heat from a source at $T_1=800\text{K}$ and rejects to a sink at $T_2\text{K}$ . The second engine $B$ receives heat rejected by the first engine and rejects it to the another sink at $T_3=300\text{K}$. If the work outputs of the two engines are equal, then the value of $T_2$ is

1. $100\text{K}$
2. $300\text{K}$
3. $550\text{K}$
4. $700\text{K}$

Q. An ideal engine has an efficiency of $\frac{1}{12}$. When the temperature of sink is reduced by ${70}^{\circ }\text{C}$, its efficiency gets doubled. The temperature of the source is

1. ${567}^{\circ }\text{C}$
2. ${467}^{\circ }\text{C}$
3. ${767}^{\circ }\text{C}$
4. ${367}^{\circ }\text{C}$

Q. A fixed mass of gas undergoes a change represented by $ABCDA$ as shown in figure. In some of the changes, work is done on the gas and in others, work is done by the gas. In which pair of the changes, work is done on the gas?

1. $AB$ and $CD$
2. $AB$ and $BC$
3. $BC$ and $CD$
4. $CD$ and $DA$

Q. A thermodynamic cycle is comprised of four processes $1\to 2$ , $2\to 3$ , $3\to 4$ and $4\to 1$ . Heat & work interactions of these processes are given as

Process	Heat transfer (J)	Work done (J)
1-2	0	150 (by the gas)
2-3	100 (from the gas)	0
3-4	0	50 (on the gas)
4-1	200 (to the gas)	0

The thermal efficiency of the cycle is -

1. $20\%$
2. $30\%$
3. $40\%$
4. $50\%$