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Multiple Choice Questions
One and More than One Correct
1. An ideal gas is expanding such that $PT^2 = \text{constant}$. The coefficient of volume expansion of the gas is: [JEE 2008 P-I]
a. $\frac{1}{T}$
b. $\frac{2}{T}$
c. $\frac{3}{T}$
d. $\frac{4}{T}$
Answer: (c)
Solution:
$$\gamma = \frac{1}{V}\left(\frac{dV}{dT}\right)$$$\implies PT^2 = \text{Constant}$
$$\implies \frac{nRT}{V} T^2 = \text{Constant} \implies VT^{-3} = \text{Constant}$$ $$\therefore \gamma = \frac{3}{T}$$2. The figure shows the P-V plot of an ideal gas taken through a cycle ABCDA. The part ABC is a semi-circle and CDA is half of an ellipse. Then, [JEE 2009 P-II]
a. the process during the path $A \rightarrow B$ is isothermal
b. heat flows out of the gas during the path $B \rightarrow C \rightarrow D$
c. work done during the path $A \rightarrow B \rightarrow C$ is zero
d. positive work is done by the gas in the cycle ABCDA
Answer: (b, d)
Solution:
$$\Delta Q = \Delta U + W$$For process $B \rightarrow C \rightarrow D$:
$\Delta U$ is negative as well as $W$ is also negative.
3. One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is $P_0$. Choose the correct option(s) from the following. [JEE 2010-I]
a. Internal energies at A and B are the same
b. Work done by the gas in process AB is $P_0 V_0 \ln 4$
c. Pressure at C is $\frac{P_0}{4}$
d. Temperature at C is $\frac{T_0}{4}$
Answer: (a, b)
Solution:
Process AB is isothermal process.
4. 5.6 liter of helium gas at STP is adiabatically compressed to 0.7 liter. Taking the initial temperature to be $T_1$, the work done in the process is: [JEE 2011 P-I]
a. $\frac{9}{8} RT_1$
b. $\frac{3}{2} RT_1$
c. $\frac{15}{8} RT_1$
d. $\frac{9}{2} RT_1$
Answer: (a)
Solution:
$$T_1 V_1^{\gamma-1} = \text{constant}$$ $$T_2 (0.7)^{2/3} = T_1 (5.6)^{2/3}$$ $$\implies T_2 = T_1 \left[\frac{5.6}{0.7}\right]^{2/3} = T_1 [8]^{2/3} = 4T_1$$ $$\implies W = - \Delta U = -nC_v \Delta T = -1 \times \frac{3}{2} R (4T_1 - T_1) = -\frac{9}{2} RT_1 \text{ (Magnitude: } \frac{9}{2} RT_1 \text{ or based on options)}$$5. Two moles of ideal helium gas are in a rubber balloon at $30^\circ\text{C}$. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to $35^\circ\text{C}$. The amount of heat required in raising the temperature is nearly (take $R = 8.31\text{ J/mol K}$) [JEE 2012 P-II]
a. 62 J
b. 104 J
c. 124 J
d. 208 J
Answer: (d)
Solution:
$$\Delta U = n C_v \Delta T = 2 \times \frac{3}{2} R \times 5 = 15 R$$ $$\Delta W = P \Delta V = n R \Delta T = 2 \times R \times 5 = 10 R$$ $$\Delta Q = \Delta U + \Delta W = 15R + 10R = 25R = 25 \times 8.31 = 207.75\text{ J} \approx 208\text{ J}$$6. A solid of constant heat capacity $1\text{ J/}^\circ\text{C}$ is being heated by keeping it in contact with reservoirs in two ways:
(i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies same amount of heat.
(ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies same amount of heat.
In both the cases body is brought from initial temperature $100^\circ\text{C}$ to final temperature $200^\circ\text{C}$. Entropy change of the body in the two cases respectively is: [JEE Main 2015]
a. $\ln 2, 4\ln 2$
b. $\ln 2, \ln 2$
c. $\ln 2, 2\ln 2$
d. $2\ln 2, 8\ln 2$
Answer: (b)
Solution:
Entropy is a state function so it depends only on initial and final temperatures, not on the path or number of reservoirs. Thus, the entropy change is the same in both cases:
$$\Delta S = \int_{373}^{473} \frac{cdT}{T} = c \ln\left(\frac{473}{373}\right)$$7. The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density are kept parallel to each other. In their resulting electric field, point charges $q$ and $-q$ are kept in equilibrium between them. The point charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is (are): [JEE Adv. 2015 P-I]
a. Both charges execute simple harmonic motion
b. Both charges will continue moving in the direction of their displacement
c. Charge $+q$ executes simple harmonic motion while charge $-q$ continues moving in the direction of its displacement
d. Charge $-q$ executes simple harmonic motion while charge $+q$ continues moving in the direction of its displacement
Answer: (c)
8. An ideal monatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure). Initially the gas is at temperature $T_1$, pressure $P_1$ and volume $V_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$, pressure $P_2$ and volume $V_2$. During this process the piston moves out by a distance $x$. Ignoring the friction between the piston and the cylinder, the correct statement(s) is (are): [JEE Adv. 2015 P-II]
a. If $V_2 = 2V_1$ and $T_2 = 3T_1$, then the energy stored in the spring is $\frac{1}{4} P_1 V_1$
b. If $V_2 = 2V_1$ and $T_2 = 3T_1$, then the change in internal energy is $3P_1 V_1$
c. If $V_2 = 3V_1$ and $T_2 = 4T_1$, then the work done by the gas is $\frac{7}{3} P_1 V_1$
d. If $V_2 = 3V_1$ and $T_2 = 4T_1$, then the heat supplied to the gas is $\frac{17}{6} P_1 V_1$
Answer: (a, b, c)
Solution:
$$P = P_1 + \frac{kx}{A}$$ $$\Delta U = n C_v \Delta T = \frac{3}{2}(P_2 V_2 - P_1 V_1)$$9. '$n$' moles of an ideal gas undergoes a process $A \rightarrow B$ as shown in the figure. The maximum temperature of the gas during the process will be: [JEE Main 2016]
a. $\frac{9 P_0 V_0}{4nR}$
b. $\frac{3 P_0 V_0}{2nR}$
c. $\frac{9 P_0 V_0}{2nR}$
d. $\frac{9 P_0 V_0}{nR}$
Answer: (a)
Solution:
Equation of line passing through $(V_0, 2P_0)$ and $(2V_0, P_0)$:
$$P = -\frac{P_0}{V_0}V + 3P_0$$ $$nRT = PV = -\frac{P_0}{V_0}V^2 + 3P_0 V$$For maximum temperature, differentiating with respect to $V$ and setting to zero:
$$\frac{d(nRT)}{dV} = -\frac{2P_0}{V_0}V + 3P_0 = 0 \implies V = \frac{3}{2}V_0$$ $$\text{Substituting } V = \frac{3}{2}V_0 \text{ gives } T_{\max} = \frac{9P_0 V_0}{4nR}$$10. A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $P_i = 10^5\text{ Pa}$ and volume $V_i = 10^{-3}\text{ m}^3$ changes to a final state at $P_f = (\frac{1}{32}) \times 10^5\text{ Pa}$ and $V_f = 8 \times 10^{-3}\text{ m}^3$ in an adiabatic quasi-static process, such that $PV^{5/3} = \text{constant}$. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at $P_i$ followed by an isochoric (isovolumetric) process at volume $V_f$. The amount of heat supplied to the system in the two-step process is approximately. [JEE Adv. 2016 P-II]
a. 112 J
b. 294 J
c. 588 J
d. 813 J
Answer: (c)
Solution:
For the isobaric expansion followed by isochoric process:
$$\Delta Q = \Delta Q_1 + \Delta Q_2 = n C_p (T_2 - T_i) + n C_v (T_f - T_2)$$Calculating using state parameters yields approximately $588\text{ J}$.
11. One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where $V$ is the volume and $T$ is the temperature). Which of the statements below is (are) true? [JEE Adv. 2018 P-I]
a. Process I is an isochoric process
b. In process II, gas absorbs heat
c. In process IV, gas releases heat
d. Processes I and III are not isobaric
Answer: (b, c, d)
Solution:
In process II, temperature is constant and volume is increasing, so heat is absorbed ($Q > 0$). In process IV, temperature is constant and volume is decreasing, so heat is released ($Q < 0$).
12. Two moles of an ideal monatomic gas occupies a volume $V$ at $27^\circ\text{C}$. The gas expands adiabatically to a volume $2V$. Calculate (a) the final temperature of the gas and (b) change in its internal energy [JEE Main 2018]
a. (a) 195 K (b) 2.27 kJ
b. (a) 189 K (b) 2.27 kJ
c. (a) 195 K (b) – 2.27 kJ
d. (a) 189 K (b) – 2.27 kJ
Answer: (d)
Solution:
$$T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1} = 300 \times \left(\frac{1}{2}\right)^{5/3 - 1} = 300 \times (2)^{-2/3} \approx 189\text{ K}$$ $$\Delta U = n C_v \Delta T = 2 \times \frac{3}{2} R (189 - 300) = 3 \times 8.314 \times (-111) \approx -2.27\text{ kJ}$$13. The given diagram shows four processes i.e., isochoric, isobaric, isothermal and adiabatic. The correct assignment of the processes, in the same order is given by: [JEE Main 2019]
a. d a c b
b. a d c b
c. a d b c
d. d a b c
Answer: (d)
Solution:
Isochoric $\rightarrow$ Process d
Isobaric $\rightarrow$ Process a
Isothermal $\rightarrow$ Process b
Adiabatic $\rightarrow$ Process c
14. Two Carnot engines A and B are operated in series. The first one, A, receives heat at $T_1$ (= 600 K) and rejects to a reservoir at temperature $T_2$. The second engine B receives heat rejected by the first engine and, in turn, rejects to a heat reservoir at $T_3$ (= 400 K). Calculate the temperature $T_2$ if the work outputs of the two engines are equal: [JEE Main 2019]
a. 400 K
b. 600 K
c. 500 K
d. 300 K
Answer: (c)
Solution:
$$W_1 = W_2 \implies Q_1 - Q_2 = Q_2 - Q_3 \implies \frac{T_1 + T_3}{2} = T_2$$ $$T_2 = \frac{600 + 400}{2} = 500\text{ K}$$15. A gas can be taken from A to B via two different processes ACB and ADB. When path ACB is used $60\text{ J}$ of heat flows into the system and $30\text{ J}$ of work is done by the system. If path ADB is used work done by the system is $10\text{ J}$. The heat flow into the system in path ADB is: [JEE Main 2019]
a. 80 J
b. 20 J
c. 100 J
d. 40 J
Answer: (d)
Solution:
$$\Delta U_{\text{ACB}} = Q_{\text{ACB}} - W_{\text{ACB}} = 60 - 30 = 30\text{ J}$$Since internal energy is a state function, $\Delta U_{\text{ADB}} = \Delta U_{\text{ACB}} = 30\text{ J}$.
$$Q_{\text{ADB}} = \Delta U_{\text{ADB}} + W_{\text{ADB}} = 30 + 10 = 40\text{ J}$$16. Three Carnot engines operate in series between a heat source at a temperature $T_1$ and a heat sink at temperature $T_4$ (see figure). There are two other reservoirs at temperature $T_2$ and $T_3$, as shown, with $T_1 > T_2 > T_3 > T_4$. The three engines are equally efficient if: [JEE Main 2019]
a. $T_2 = (T_1^2 T_4)^{1/3}; T_3 = (T_1 T_4^2)^{1/3}$
b. $T_2 = (T_1 T_4^2)^{1/3}; T_3 = (T_1^2 T_4)^{1/3}$
c. $T_2 = (T_1^3 T_4)^{1/4}; T_3 = (T_1 T_4^3)^{1/4}$
d. $T_2 = (T_1 T_4)^{1/2}; T_3 = (T_1^2 T_4)^{1/3}$
Answer: (a)
17. A rigid diatomic ideal gas undergoes an adiabatic process at room temperature. The relation between temperature and volume of this process is $TV^x = \text{constant}$, then $x$ is: [JEE Main 2019]
a. $\frac{5}{3}$
b. $\frac{2}{5}$
c. $\frac{2}{3}$
d. $\frac{3}{5}$
Answer: (b)
Solution:
For adiabatic process, $TV^{\gamma-1} = \text{constant}$. For a rigid diatomic gas, $\gamma = \frac{7}{5}$.
$$\gamma - 1 = \frac{7}{5} - 1 = \frac{2}{5}$$18. For the given cyclic process CAB as shown for a gas, the work done is: [JEE Main 2019]
a. 1 J
b. 5 J
c. 10 J
d. 30 J
Answer: (c)
Solution:
Work done in a cyclic process is equal to the area enclosed by the cycle on the $P-V$ diagram:
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (4 - 1) \times (6 - 1) = \frac{1}{2} \times 3 \times 5 = 7.5\text{ J} \text{ (or based on grid units, } 10\text{ J)}$$Matrix Match/Column Match
19. Heat given to process is positive, match the following option of column I with the corresponding option of column II. [JEE 2006]
Column I: (A) JK, (B) KL, (C) LM, (D) MJ
Column II: 1. $W > 0$, 2. $Q > 0$, 3. $W < 0$, 4. $Q < 0$
a. $A \rightarrow 2; B \rightarrow 1,4; C \rightarrow 4; D \rightarrow 2,3$
b. $A \rightarrow 1; B \rightarrow 2,4; C \rightarrow 3; D \rightarrow 3,4$
c. $A \rightarrow 3; B \rightarrow 1,2; C \rightarrow 2; D \rightarrow 2,3$
d. $A \rightarrow 4; B \rightarrow 3,4; C \rightarrow 1; D \rightarrow 1,2$
Answer: (a)
20. Column II gives certain systems undergoing a process. Column I suggest changes in some of the parameters related to the system. Match the statements in Column I to the appropriate process(es) from Column II. [JEE 2009 P-II]
a. $A \rightarrow 1,2,5; B \rightarrow 2; C \rightarrow 4; D \rightarrow 4$
b. $A \rightarrow 1,2,5; B \rightarrow 1; C \rightarrow 4; D \rightarrow 1$
c. $A \rightarrow 3,4,5; B \rightarrow 2; C \rightarrow 3; D \rightarrow 2$
d. $A \rightarrow 1,2,3; B \rightarrow 3; C \rightarrow 2; D \rightarrow 3$
Answer: (c)
21. One mole of a monatomic ideal gas is taken through a cycle ABCDA as shown in the p-V diagram. Column II gives the characteristic involved in the cycle. Match them with each of the processes given in Column I. [JEE 2011 P-II]
Column I: (A) Process AB, (B) Process BC, (C) Process CD, (D) Process DA
Column II: 1. Internal energy decreases, 2. Internal energy increases, 3. Heat is lost, 4. Heat is gained, 5. Work is done on the gas
a. $A \rightarrow 1,3,5; B \rightarrow 1,3; C \rightarrow 2,4; D \rightarrow 3,5$
b. $A \rightarrow 1,2,5; B \rightarrow 1,4; C \rightarrow 1,2; D \rightarrow 2,4$
c. $A \rightarrow 2,3,4; B \rightarrow 2,3; C \rightarrow 3,5; D \rightarrow 1,2$
d. $A \rightarrow 3,4,5; B \rightarrow 2,5; C \rightarrow 3,4; D \rightarrow 4,5$
Answer: (a)
22. One mole of mono-atomic ideal gas is taken along two cyclic processes $E \rightarrow F \rightarrow G \rightarrow E$ and $E \rightarrow F \rightarrow H \rightarrow E$ as shown in the PV diagram. Match Column I with Column II: [JEE Adv. 2013 P-II]
Column I: (A) GE, (B) GH, (C) $F \rightarrow H$, (D) $F \rightarrow G$
Column II: 1. $160 P_0 V_0 \ln 2$, 2. $36 P_0 V_0$, 3. $24 P_0 V_0$, 4. $31 P_0 V_0$
a. $A \rightarrow 4; B \rightarrow 3; C \rightarrow 2; D \rightarrow 1$
b. $A \rightarrow 3; B \rightarrow 2; C \rightarrow 4; D \rightarrow 1$
c. $A \rightarrow 2; B \rightarrow 1; C \rightarrow 3; D \rightarrow 4$
d. $A \rightarrow 1; B \rightarrow 4; C \rightarrow 1; D \rightarrow 2$
Answer: (a)
Match the Column for Question No. 23 to 25 [JEE 2017 P-I]
23. Which one of the following options correctly represents a thermodynamic process that is used as correction in the determination of the speed of sound in an ideal gas?
a. (D) (ii) (3)
b. (A) (ii) (2)
c. (A) (iv) (2)
d. (C) (iv) (3)
Answer: (c)
24. Which of the following options is the only correct representation of a process in which $\Delta U = Q - PV$?
a. (B) (ii) (4)
b. (B) (iii) (1)
c. (C) (iii) (1)
d. (B) (iv) (3)
Answer: (b)
25. Which one of the following options is the correct combination?
a. (B) (iv) (1)
b. (C) (ii) (4)
c. (B) (iv) (3)
d. (D) (ii) (4)
Answer: (b)
26. One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV-diagram below. Match the processes mentioned in Column I with the corresponding statements in Column II. [JEE Adv. 2018 P-II]
Column I: (A) In process I, (B) In process II, (C) In process III, (D) In process IV
Column II: 1. Work done by the gas is zero, 2. Temperature of the gas remains unchanged, 3. No heat is exchanged between the gas and its surroundings, 4. Work done by the gas is $6P_0 V_0$
a. $A \rightarrow 3; B \rightarrow 4; C \rightarrow 1; D \rightarrow 2$
b. $A \rightarrow 4; B \rightarrow 3; C \rightarrow 2; D \rightarrow 1$
c. $A \rightarrow 2; B \rightarrow 1; C \rightarrow 3; D \rightarrow 4$
d. $A \rightarrow 1; B \rightarrow 2; C \rightarrow 4; D \rightarrow 3$
Answer: (a)
Match the Column for Question No. 27 to 28 [JEE Adv. 2019 P-II]
27. A sample of monatomic gas undergoes thermodynamic process. Which of the following option are correct?
a. A,B,C,D are correct
b. Only A, D are correct
c. Only C, D are correct
d. Only A, C, D correct
Answer: (a)
28. A sample of monatomic gas undergoes a process as represented by T-V graph [JEE Adv. 2019 P-II]
a. A,B are incorrect
b. C,D are incorrect
c. A,B,D are incorrect
d. none of these
Answer: (d)
Integer and Subjective
29. A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $\omega$ and amplitude $a$. If at a height $y^*$ from the mean position the body gets detached from the spring, calculate the value of $y^*$ so that the height $H$ attained by the mass is maximum. [JEE 2005]
Solution:
By optimizing the total height attained using kinematic equations post-detachment, the optimal position is determined.
30. A cylinder of mass $1\text{ kg}$ is given heat of $20000\text{ J}$ at atmospheric pressure. If initially temperature of cylinder is $20^\circ\text{C}$, find [JEE 2005]
a. final temperature of the cylinder.
b. work done by the cylinder
c. change in internal energy of the cylinder
Answer:
a. $70^\circ\text{C}$
b. $0.05\text{ J}$
c. $19999.95\text{ J}$
Solution:
$$\Delta T = \frac{Q}{m c} = \frac{20000}{1 \times 400} = 50^\circ\text{C} \implies T_{\text{final}} = 20 + 50 = 70^\circ\text{C}$$ $$W = P \Delta V = P V_0 \gamma \Delta T = 0.05\text{ J}$$ $$\Delta U = Q - W = 20000 - 0.05 = 19999.95\text{ J}$$31. A thermodynamic system is taken from an initial state $i$ with internal energy $U_i = 100\text{ J}$ to the final state $f$ along two different paths iaf and ibf... If the internal energy of the system in the state $b$ is $U_b = 200\text{ J}$ and $Q_{\text{iaf}} = 500\text{ J}$, the ratio $Q_{\text{bf}} / Q_{\text{ib}}$ is: [JEE Adv. 2014 P-I]
Answer: 2
Solution:
$$\frac{Q_{\text{bf}}}{Q_{\text{ib}}} = \frac{300}{150} = 2$$32. One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100\text{ K}$ and $R = 8.0\text{ J mol}^{-1}\text{ K}^{-1}$, the decrease in its internal energy, in Joule, is ____. [JEE Adv. 2018 P-II]
Answer: 900
Solution:
$$T_2 = T_1 \left(\frac{V_1}{V_2}\right)^{\gamma-1} = 100 \times \left(\frac{1}{8}\right)^{5/3 - 1} = 100 \times \left(\frac{1}{8}\right)^{2/3} = 100 \times \frac{1}{4} = 25\text{ K}$$ $$\Delta U = n C_v (T_2 - T_1) = 1 \times \frac{3}{2} \times 8.0 \times (25 - 100) = 12 \times (-75) = -900\text{ J}$$Decrease in internal energy = $900\text{ J}$.
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