Core Syllabus Module

Thermodynamics

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Target Advanced

LEVELJEE Advanced

The specific heat capacity of a metal at low temperature ( $T$ ) is given as $C_{p} (kJ K^{- 1} kg^{- 1}) = 32 (\frac{T}{400})^{3}$ . A 100 g vessel of this metal is to be cooled from 20 K to 4 K by a special refrigerator operating at room temperature (27°C). The amount of work required to cool the vessel is

(A)

equal to 0.002 kJ

(B)

greater than 0.148 kJ

(C)

between 0.148 kJ and 0.028 kJ

The $p$ - $T$ diagram for an ideal gas is shown in the figure, where $A C$ is an adiabatic process, find the corresponding $p$ - $V$ diagram

The rectangular box shown in figure has a partition which can slide without friction along the length of the box. Initially each of the two chambers of the box has one mole of a monoatomic ideal gas ( $γ = 5/3$ ) at a pressure $p_{0}$ , volume $V_{0}$ and temperature $T_{0}$ . The chamber on the left is slowly heated by an electric heater. The walls of the box and the partition are thermally insulated. Heat loss through the lead wires of the heater is negligible. The gas in the left chamber expands pushing the partition until the final pressure in both chambers becomes $243 p_{0} /32$ . Determine (a) the final temperature of the gas in each chamber and (b) the work done by the gas in the right chamber.

JEE Advanced 2014

LEVELJEE Advanced

Comprehension Passage

In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with 2 moles of an ideal monoatomic gas at 700 K and the upper compartment is filled with 2 moles of an ideal diatomic gas at 400 K. The heat capacities per mole of an ideal monoatomic gas are

C_{V} = \frac{3}{2} R, C_{p} = \frac{5}{2} R

, and those for an ideal diatomic gas are

C_{V} = \frac{5}{2} R, C_{p} = \frac{7}{2} R

Question 1:

Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be

Comprehension Passage

A fixed thermally conducting cylinder has a radius

R

and height

L_{0}

. The cylinder is open at its bottom and has a small hole at its top. A piston of mass

M

is held at a distance

L

from the top surface, as shown in the figure. The atmospheric pressure is

p_{0}

Question 1:

The piston is now pulled out slowly and held at a distance $2 L$ from the top. The pressure in the cylinder between its top and the piston will then be

(A)

p_{0}

(B)

\frac{p _{0}}{2}

(C)

\frac{p _{0}}{2} + \frac{M g}{π R ^{2}}

(D)

\frac{p _{0}}{2} - \frac{M g}{π R ^{2}}

Question 2:

While the piston is at a distance $2 L$ from the top, the hole at the top is sealed. The piston is then released to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

(A)

(\frac{2 p _{0} π R ^{2}}{π R ^{2} p _{0} + M g}) (2 L)

(B)

(\frac{p _{0} π R ^{2} - M g}{π R ^{2} p _{0}}) (2 L)

(C)

(\frac{p _{0} π R ^{2} + M g}{π R ^{2} p _{0}}) (2 L)

(D)

(\frac{p _{0} π R ^{2}}{π R ^{2} p _{0} - M g}) (2 L)

Question 3:

The piston is taken completely out of the cylinder. The hole at the top is sealed. A water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $ρ$ . In equilibrium, the height $H$ of the water column in the cylinder satisfies

(A)

ρ g (L_{0} - H)^{2} + p_{0} (L_{0} - H) + L_{0} p_{0} = 0

(B)

ρ g (L_{0} - H)^{2} - p_{0} (L_{0} - H) - L_{0} p_{0} = 0

(C)

ρ g (L_{0} - H)^{2} + p_{0} (L_{0} - H) - L_{0} p_{0} = 0

(D)

ρ g (L_{0} - H)^{2} - p_{0} (L_{0} - H) + L_{0} p_{0} = 0

JEE Advanced 2017

LEVELJEE Advanced

A flat plane is moving normal to its plane through a gas under the action of a constant force $F$ . The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?

* Multiple Correct Options

(A)

At a later time the external force

F

balances the resistive force

(B)

The plate will continue to move with constant non-zero acceleration, at all times

(C)

The resistive force experienced by the plate is proportional to

v

(D)

The pressure difference between the leading and trailing faces of the plate is proportional to

uv

JEE Advanced 2022

LEVELJEE Advanced

List-I describes thermodynamic processes in four different systems. List-II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

List-I

(P)

1 0^{- 3} kg

of water at

10 0^{\circ} C

is converted to steam at the same temperature, at a pressure of

1 0^{5} Pa

. The volume of the system changes from

1 0^{- 6} m^{3}

1 0^{- 3} m^{3}

in the process. Latent heat of water

= 2250 kJ/kg

(Q)

0.2

moles of a rigid diatomic ideal gas with volume

V

at temperature

500 K

undergoes an isobaric expansion to volume

3 V

. Assume

R = 8.0 J mol^{- 1} K^{- 1}

(R)

One mole of a monatomic ideal gas is compressed adiabatically from volume

V = \frac{1}{3} m^{3}

and pressure

2 kPa

to volume

\frac{V}{8}

(S)

Three moles of a diatomic ideal gas whose molecules can vibrate, is given

9 kJ

of heat and undergoes isobaric expansion.

List-II

(1)

2 kJ

(2)

7 kJ

(3)

4 kJ

(4)

5 kJ

(5)

3 kJ

JEE Advanced 2020

LEVELJEE Advanced

A thermally isolated cylindrical closed vessel of height 8 m is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass 8.3 kg. Thus the partition is held initially at a distance of 4 m from the top, as shown in the schematic figure below. Each of the two parts of the vessel contains 0.1 mole of an ideal gas at temperature 300 K. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached, the distance of the partition from the top (in m) will be ______. (take the acceleration due to gravity = 10 ms^{-2} and the universal gas constant = 8.3 J mol^{-1} K^{-1}).

JEE Advanced 2022

LEVELJEE Advanced

An ideal gas of density $ρ = 0.2 kg m^{- 3}$ enters a chimney of height $h$ at the rate of $α = 0.8 kg s^{- 1}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_{1} = 0.1 m^{2}$ and the upper end is $A_{2} = 0.4 m^{2}$ . The pressure and the temperature of the gas at the lower end are $600 Pa$ and $300 K$ , respectively, while its temperature at the upper end is $150 K$ . The chimney is heat insulated so that the gas undergoes adiabatic expansion. [Take: $g = 10 ms^{- 2}$ and the ratio of specific heats of the gas $γ = 2$ . Ignore atmospheric pressure.] Which of the following statement(s) is(are) correct?

* Multiple Correct Options

(A)

The pressure of the gas at the upper end of the chimney is

300 Pa

(B)

The velocity of the gas at the lower end of the chimney is

40 ms^{- 1}

and at the upper end is

20 ms^{- 1}

(C)

The height of the chimney is

590 m

(D)

The density of the gas at the upper end is

0.05 kg m^{- 3}

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume,

C_{V} = 2 R

. Here, R is the gas constant. Initially, each side has a volume

V_{0}

and temperature

T_{0}

. The left side has an electric heater, which is turned on at very low power to transfer heat Q to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to

V_{0} /2

. Consequently, the gas temperatures on the left and the right sides become

T_{L}

and

T_{R}

, respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question 1:

The value of $\frac{T _{R}}{T _{0}}$ is

(A)

2

(B)

3

(C)

2

(D)

3

Question 2:

The value of $\frac{Q}{R T _{0}}$ is

(A)

4 (22 + 1)

(B)

4 (22 - 1)

(C)

(52 + 1)

(D)

(52 - 1)

JEE Advanced 2019

LEVELJEE Advanced

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V-T) diagram. The correct statement(s) is/are: [R is the gas constant]

* Multiple Correct Options

(A)

Work done in this thermodynamic cycle (

1 \to 2 \to 3 \to 4 \to 1

) is

∣ W ∣ = \frac{1}{2} R T_{0}

(B)

The ratio of heat transfer during processes

1 \to 2

and

2 \to 3

\frac{Q _{1 \to 2}}{Q _{2 \to 3}} = \frac{5}{3}

(C)

The above thermodynamic cycle exhibits only isochoric and adiabatic processes.

(D)

The ratio of heat transfer during processes

1 \to 2

and

3 \to 4

\frac{Q _{1 \to 2}}{Q _{3 \to 4}} = \frac{1}{2}

JEE Main 2019

LEVELJEE Advanced

The number density of molecules of a gas depends on their distance $r$ from the origin as, $n (r) = n_{0} e^{- α r^{4}}$ . Then, the total number of molecules is proportional to

(A)

n_{0} α^{- 3/4}

(B)

n_{0} α^{1/2}

(C)

n_{0} α^{1/4}

(D)

n_{0} α^{- 3}

JEE Main 2019

LEVELJEE Advanced

A $25 \times 1 0^{- 3} m^{3}$ volume cylinder is filled with $1 mole$ of $O_{2}$ gas at room temperature ( $300 K$ ). The molecular diameter of $O_{2}$ and its root mean square speed are found to be $0.3 nm$ and $200 m/s$ , respectively. What is the average collision rate (per second) for an $O_{2}$ molecule?

(A)

\sim 1 0^{10}

(B)

\sim 1 0^{12}

(C)

\sim 1 0^{11}

(D)

\sim 1 0^{13}

JEE Main 2015

LEVELJEE Advanced

Consider a spherical shell of radius $R$ at temperature $T$ . The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u = \frac{U}{V} \propto T^{4}$ and pressure $p = \frac{1}{3} (\frac{U}{V})$ . If the shell now undergoes an adiabatic expansion, the relation between $T$ and $R$ is

(A)

T \propto e^{- R}

(B)

T \propto e^{- 3 R}

(C)

T \propto \frac{1}{R}

(D)

T \propto \frac{1}{R ^{3}}

LEVELJEE Advanced

Three moles of an ideal gas ( $C_{p} = \frac{7}{2} R$ ) at pressure, $p_{A}$ and temperature $T_{A}$ is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally gas is compressed at constant volume to its original pressure $p_{A}$ . (a) Sketch $p - V$ and $p - T$ diagrams for the complete process. (b) Calculate the net work done by the gas, and net heat supplied to the gas during the complete process.

JEE Advanced 1999

LEVELJEE Advanced

Two moles of an ideal monoatomic gas initially at pressure $p_{1}$ and volume $V_{1}$ undergo an adiabatic compression until its volume is $V_{2}$ . Then the gas is given heat $Q$ at constant volume $V_{2}$ . (a) Sketch the complete process on a $p - V$ diagram. (b) Find the total work done by the gas, the total change in internal energy and the final temperature of the gas. (Give your answer in terms of $p_{1}, V_{1}, V_{2}, Q$ and $R$ )

JEE Advanced 2017

LEVELJEE Advanced

Comprehension Passage

An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding

p

V

diagrams in column 3 of the table. Consider only the path from state 1 to state 2.

W

denotes the corresponding work done on the system. The equations and plots in the table have standards notations and used in thermodynamic processes. Here

γ

is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is

n

. $\begin{array}{lll} \textbf{Column 1} & \textbf{Column 2} & \textbf{Column 3} \\ \text{(I) } W_{1 \rightarrow 2} = \frac{1}{\gamma - 1}(p_2V_2 - p_1V_1) & \text{(i) Isothermal} & \text{(P) Graph P} \\ \text{(II) } W_{1 \rightarrow 2} = -pV_2 + pV_1 & \text{(ii) Isochoric} & \text{(Q) Graph Q} \\ \text{(III) } W_{1 \rightarrow 2} = 0 & \text{(iii) Isobaric} & \text{(R) Graph R} \\ \text{(IV) } W_{1 \rightarrow 2} = -nRT \ln\left(\frac{V_2}{V_1}\right) & \text{(iv) Adiabatic} & \text{(S) Graph S} \end{array}$

Question 1:

22. Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

23. Which of the following options is the only correct representation of a process in which $Δ U = Δ Q - p Δ V$ ?

24. Which one of the following options is the correct combination?

One mole of a diatomic ideal gas ( $γ = 1.4$ ) is taken through a cyclic process starting from point $A$ . The process $A \to B$ is an adiabatic compression. $B \to C$ is isobaric expansion, $C \to D$ an adiabatic expansion and $D \to A$ is isochoric. The volume ratio are $V_{A} / V_{B} = 16$ and $V_{C} / V_{B} = 2$ and the temperature at $A$ is $T_{A} = 300 K$ . Calculate the temperature of the gas at the points $B$ and $D$ and find the efficiency of the cycle.

LEVELJEE Advanced

Two moles of helium gas ( $γ = 5/3$ ) are initially at temperature $2 7^{\circ} C$ and occupy a volume of $20 L$ . The gas is first expanded at constant pressure until the volume is doubled. Then it undergoes an adiabatic change until the temperature returns to its initial value. (a) Sketch the process on a $p$ - $V$ diagram. (b) What are the final volume and pressure of the gas? (c) What is the work done by the gas?

LEVELJEE Advanced

One gram mole of oxygen at 27°C and one atmospheric pressure is enclosed in a vessel. (a) Assuming the molecules to be moving with $v_{rms}$ , find the number of collisions per second which the molecules make with one square metre area of the vessel wall. (b) The vessel is next thermally insulated and moved with a constant speed $v_{0}$ . It is then suddenly stopped. The process results in a rise of the temperature of the gas by 1°C. Calculate the speed $v_{0}$ .

Micro-Topic Strategy Matrix

Kinetic Theory of Gases

Thermodynamic Processes

First Law of Thermodynamics

Second Law of Thermodynamics

Heat Engines and Refrigerators

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The p-T diagram for an ideal gas is shown in the figure, where AC is an adiabatic process, find the corresponding p-V diagram

Comprehension Passage

Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

Now consider the partition to be free to move without friction so that the pressure of gases in both compartments is the same. Then total work done by the gases till the time they achieve equilibrium will be

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The piston is now pulled out slowly and held at a distance 2L from the top. The pressure in the cylinder between its top and the piston will then be

List-I describes thermodynamic processes in four different systems. List-II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

List-I

List-II

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of T0​TR​​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of RT0​Q​ is

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V-T) diagram. The correct statement(s) is/are: [R is the gas constant]

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number density of molecules of a gas depends on their distance r from the origin as, n(r)=n0​e−αr4. Then, the total number of molecules is proportional to

Comprehension Passage

22. Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 23. Which of the following options is the only correct representation of a process in which ΔU=ΔQ−pΔV?

24. Which one of the following options is the correct combination?

The $p$ - $T$ diagram for an ideal gas is shown in the figure, where $A C$ is an adiabatic process, find the corresponding $p$ - $V$ diagram

The piston is now pulled out slowly and held at a distance $2 L$ from the top. The pressure in the cylinder between its top and the piston will then be

The value of $\frac{T _{R}}{T _{0}}$ is

The value of $\frac{Q}{R T _{0}}$ is

The number density of molecules of a gas depends on their distance $r$ from the origin as, $n (r) = n_{0} e^{- α r^{4}}$ . Then, the total number of molecules is proportional to

23. Which of the following options is the only correct representation of a process in which $Δ U = Δ Q - p Δ V$ ?