Thermodynamics — JEE Main PYQs

5 moles of unknown gas is heated at constant volume from $10\ °\text{C}$ to $20\ °\text{C}$. The molar specific heat of this gas at constant pressure $c_p = 8 \text{ cal/mol.}°\text{C}$ and $R = 8.36 \text{ J/mol.}°\text{C}$. The change in the internal energy of the gas is ________ calorie.

An ideal gas undergoes a process maintaining relation between pressure ($P$) and volume ($V$) as $P = P_0\left(1 + \left(\frac{V_0}{V}\right)^2\right)^{-1}$, where $P_0$ and $V_0$ are constants. If two samples $A$ and $B$ (two moles each) with initial volumes $V_0$ and $3V_0$ respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, $T_B - T_A$ is __________.

($R$ = gas constant)

Consider the following statements:

A. Zeroth law of thermodynamics gives concept of temperature B. First law of thermodynamics gives concept of internal energy C. In isothermal expansion of ideal gas, $\Delta Q \neq \Delta W$ D. Product of intensive and extensive variables is extensive E. The ratio of any extensive variable to mass will be an extensive variable

Choose the correct combination of statements from the options given below:

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 =$ constant. The coefficient of volume expansion of the gas is __________.

A certain gas is isothermally compressed to $\left(\frac{1}{3}\right)^{\text{rd}}$ of its initial volume ($V_o = 3$ litre) by applying required pressure. If the bulk modulus of the gas is $3 \times 10^{5}\ \text{N/m}^2$, the magnitude of work done on the gas is _____ J.

A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same $P$, $V$, $T$. Heating is started from left side until pressure changes to $\frac{27P}{8}$. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is ________ litres. (for this ideal gas $\text{C}_\text{P}/\text{C}_\text{V} = 1.5$)

One mole of diatomic gas having rotational modes only is kept in a cylinder with a piston system. The cross-section area of the cylinder is $4\,\text{cm}^2$. The gas is heated slowly to raise the temperature by $1.2\,^{\circ}\text{C}$ during which the piston moves by $25\,\text{mm}$. The amount of heat supplied to the gas is ________ J. (Atmospheric pressure $= 100\,\text{kPa}$, $R = 8.3\,\text{J/mol.K}$) (Neglect mass of the piston)

Initial pressure and volume of a monoatomic ideal gas are $P$ and $V$. The change in internal energy of this gas in adiabatic expansion to volume $V_{final} = 27\,V$ is ________ J.

In an adiabatic process, which of the following statements is true ?

Identify the characteristics of an adiabatic process in a monoatomic gas.

(A) Internal energy is constant. (B) Work done in the process is equal to the change in internal energy. (C) The product of temperature and volume is a constant. (D) The product of pressure and volume is a constant. (E) The work done to change the temperature from $T_1$ to $T_2$ is proportional to $(T_2-T_1)$.

Choose the correct answer from the options given below :

During the melting of a slab of ice at 273 K at atmospheric pressure:

A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of 800 $\mathrm{cm^3}$ and temperature 27°C. The change in temperature when the gas is adiabatically compressed to 200 $\mathrm{cm^3}$ is:

(Take $\gamma = 1.5$ ; $\gamma$ is the ratio of specific heats at constant pressure and at constant volume)

[Figure: a closed vertical chamber; a piston (initially at height $L_0$ from the bottom) hangs from a spring attached to the top; after heating the piston rises to height $L_1$]

A piston of mass M is hung from a massless spring whose restoring force law goes as $F = -kx^3$, where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with 'n' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height $L_0$ to $L_1$, the total energy delivered by the filament is :(Assume spring to be in its natural length before heating)

An ideal gas exists in a state with pressure $P_0$, volume $V_0$. It is isothermally expanded to 4 times of its initial volume ($V_0$), then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is

Match List - I with List - II.

List - I	List - II
(A) Isobaric	(I) $\Delta Q = \Delta W$
(B) Isochoric	(II) $\Delta Q = \Delta U$
(C) Adiabatic	(III) $\Delta Q =$ zero
(D) Isothermal	(IV) $\Delta Q = \Delta U + P\Delta V$

$\Delta Q$ = Heat supplied $\Delta W$ = Work done by the system $\Delta U$ = Change in internal energy P = Pressure of the system $\Delta V$ = Change in volume of the system

Choose the correct answer from the options given below :

An ideal gas has undergone through the cyclic process as shown in the figure. Work done by the gas in the entire cycle is _______ $\times\ 10^{-1}$ J.

(Take $\pi = 3.14$)

[Figure: P–V diagram with V (cm³) on the vertical axis marked 150, 250, 350 and P (kPa) on the horizontal axis marked 300, 400, 500; the cyclic process is a circle centred near (400 kPa, 250 cm³) spanning these ranges]

Match List - I with List - II.

List - I	List - II
(A) Isothermal	(I) $\Delta W$ (work done) $= 0$
(B) Adiabatic	(II) $\Delta Q$ (supplied heat) $= 0$
(C) Isobaric	(III) $\Delta U$ (change in internal energy) $\neq 0$
(D) Isochoric	(IV) $\Delta U = 0$

Choose the correct answer from the options given below :

A monoatomic gas having $\gamma = \frac{5}{3}$ is stored in a thermally insulated container and the gas is suddenly compressed to $\left(\frac{1}{8}\right)^{th}$ of its initial volume. The ratio of final pressure and initial pressure is:

($\gamma$ is the ratio of specific heats of the gas at constant pressure and at constant volume)

Given are statements for certain thermodynamic variables,

(A) Internal energy, volume (V) and mass (M) are extensive variables.

(B) Pressure (P), temperature (T) and density ($\rho$) are intensive variables.

(C) Volume (V), temperature (T) and density ($\rho$) are intensive variables.

(D) Mass (M), temperature (T) and internal energy are extensive variables.

Choose the correct answer from the options given below :

An ideal gas, initially at $0\,^{\circ}\mathrm{C}$ temperature, is suddenly compressed to one quarter of its volume. If the ratio of specific heat at constant pressure to the specific heat at constant volume is $3/2$, the change in temperature due to the thermodynamic process is _____ K.

[Figure: P-V diagram with points A $(2V_0, P_0)$, B $(3V_0, P_0)$, C $(3V_0, 2P_0)$ and D $(V_0, 2P_0)$; the path goes A → B (horizontal at $P_0$), B → C (vertical at $3V_0$), C → D (horizontal at $2P_0$)]

Using the given P-V diagram, the work done by an ideal gas along the path ABCD is :

Water of mass m gram is slowly heated to increase the temperature from $T_1$ to $T_2$. The change in entropy of the water, given specific heat of water is $1~\mathrm{J\,kg^{-1}\,K^{-1}}$, is :

An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly with temperature.

A. The work done by gas during the process is zero.

B. The heat added to gas is different from change in its internal energy.

C. The volume of the gas is increased.

D. The internal energy of the gas is increased.

E. The process is isochoric (constant volume process)

Choose the correct answer from the options given below:

The temperature of 1 mole of an ideal monoatomic gas is increased by $50°C$ at constant pressure. The total heat added and change in internal energy are $E_1$ and $E_2$, respectively. If $\frac{E_1}{E_2}=\frac{x}{9}$, then the value of x is _____.

The magnitude of heat exchanged by a system for the given cyclic process ABCA (as shown in figure) is (in SI unit) :

[Figure: P–V diagram with P (in kPa) on the vertical axis and V (in cc) on the horizontal axis; a circle spans 200–400 on both axes with A at the right (400 cc), B at the top (400 kPa), C at the left (200 cc) and D at the bottom (200 kPa); the cycle follows the upper semicircle through B and the horizontal line C→A through the centre, the lower semicircle through D being dotted]

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases.

Reason (R) : Free expansion of an ideal gas is an irreversible and adiabatic process.

In the light of the above statements, choose the correct answer from the options given below :

A Carnot engine (E) is working between two temperatures 473K and 273K. In a new system two engines - engine $E_1$ works between 473K to 373K and engine $E_2$ works between 373K to 273K. If $\eta_{12}$, $\eta_1$ and $\eta_2$ are the efficiencies of the engines E, $E_1$ and $E_2$, respectively, then

A sample of gas at temperature $T$ is adiabatically expanded to double its volume. Adiabatic constant for the gas is $\gamma = 3/2$. The work done by the gas in the process is:

($\mu = 1$ mole)

The heat absorbed by a system in going through the given cyclic process is :

[Figure: a circular cyclic process on a P–V diagram; vertical axis labelled P (in cc) with dashed lines at 60 and 340, horizontal axis labelled V (in kPa) with dashed lines at 60 and 340, the circle traversed clockwise]

During an adiabatic process, if the pressure of a gas is found to be proportional to the cube of its absolute temperature, then the ratio of $\frac{C_P}{C_V}$ for the gas is :

The specific heat at constant pressure of a real gas obeying $PV^2 = RT$ equation is:

A total of 48 J heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by $2°\mathrm{C}$. The work done by the gas is : Given, $R = 8.3\ \mathrm{J\,K^{-1}\,mol^{-1}}$.

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio $\frac{V_a}{V_d}$ and the ratio $\frac{V_b}{V_c}$ is:

[Figure: P-V diagram with two adiabatic curves a-d and b-c intersecting two isothermal curves; volumes marked $V_a$, $V_d$, $V_b$, $V_c$ on the V-axis]

A diatomic gas $(\gamma = 1.4)$ does $100\ \mathrm{J}$ of work in an isobaric expansion. The heat given to the gas is :

A sample of 1 mole gas at temperature T is adiabatically expanded to double its volume. If adiabatic constant for the gas is $\gamma = \dfrac{3}{2}$, then the work done by the gas in the process is :

The volume of an ideal gas ($\gamma = 1.5$) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is :

A real gas within a closed chamber at $27°C$ undergoes the cyclic process as shown in figure. The gas obeys $PV^3 = RT$ equation for the path $A$ to $B$. The net work done in the complete cycle is (assuming $R = 8$ J/mol K) :

[Figure: P–V diagram with $P$ in $\mathrm{N/m^2}$ and $V$ in $\mathrm{m^3}$; curve from A (2, 20) to B (4, 10), horizontal line from B back to C (2, 10), and vertical line from C up to A]

The pressure and volume of an ideal gas are related as $PV^{\frac{3}{2}} = K$ (Constant). The work done when the gas is taken from state A $(P_1, V_1, T_1)$ to state B $(P_2, V_2, T_2)$ is :

A diatomic gas ($\gamma = 1.4$) does 200 J of work when it is expanded isobarically. The heat given to the gas in the process is :

0.08 kg air is heated at constant volume through $5^{\circ}\mathrm{C}$. The specific heat of air at constant volume is $0.17\ \mathrm{kcal/kg^{\circ}C}$ and $J = 4.18$ joule/cal. The change in its internal energy is approximately.

A thermodynamic system is taken from an original state A to an intermediate state B by a linear process as shown in the figure. It's volume is then reduced to the original value from B to C by an isobaric process. The total work done by the gas from A to B and B to C would be :

[Figure: P–V diagram; pressure axis in Dyne/cm² marked 8000 (state A) and 4000 (states B, C); volume axis in m³ marked 3 and 7; straight line from A (3, 8000) to B (7, 4000), then isobaric line from B back to C (3, 4000)]

Two thermodynamical processes are shown in the figure. The molar heat capacity for process A and B are $\mathrm{C_A}$ and $\mathrm{C_B}$. The molar heat capacity at constant pressure and constant volume are represented by $\mathrm{C_P}$ and $\mathrm{C_V}$, respectively. Choose the correct statement.

[Figure: log P vs log V graph; process A is a straight line through origin at angle $\tan^{-1}\gamma$, process B is a straight line at $45°$]

Choose the correct statement for processes $A$ & $B$ shown in figure.

[Figure: P–V diagram with two downward-sloping curves; curve A is lower and less steep, curve B is higher and steeper, both with arrows directed toward increasing V]

The given figure represents two isobaric processes for the same mass of an ideal gas, then

[Figure: V–T graph with two straight lines from origin O, line $P_2$ steeper (above) line $P_1$]