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JEE Mains | PYQ

Given below are two statements:
Statement I: The temperature of a gas is \( -73^{\circ}\text{C} \). When the gas is heated to \( 572^{\circ}\text{C} \), the root mean square speed of the molecules is doubled.
Statement II: The product of pressure and volume of an ideal gas will be equal to translational kinetic energy of the molecules.
In the light of the above statements, choose the correct answer from the options given below:

For three low density gases \( A, B, C \) pressure versus temperature graphs are plotted while keeping them at constant volume, as shown in the figure:

The temperature corresponding to the point ‘\( K \)’ is:

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)

A reversible engine has an efficiency of \( \frac{1}{4} \). If the temperature of the sink is reduced by \( 58^\circ \text{C} \), its efficiency becomes double. Calculate the temperature of the sink:

A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of \( 800 \ \text{cm}^3 \) and temperature \( 27^\circ \text{C} \). The change in temperature when the gas is adiabatically compressed to \( 200 \ \text{cm}^3 \) is: (Take \( \gamma = 1.5 \); \( \gamma \) is the ratio of specific heats at constant pressure and at constant volume)

An ideal gas undergoes a process maintaining relation between pressure \( (P) \) and Volume \( (V) \) as \( P=P_0\left(1+\left(\dfrac{V_0}{V}\right)^2\right)^{-1} \), 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 _______.
\( (R=\text{gas constant}) \)

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:

When a gas filled in a closed vessel is heated by raising the temperature by \( 1^\circ \text{C} \), its pressure increases by \( 0.4\% \). The initial temperature of the gas is ________________ \( \text{K} \).

A solid metallic cube having total surface are \( 24 \, \text{m}^2 \) is uniformly heated. If its temperature is increased by \( 10^\circ \text{C} \), calculate the increase in volume of the cube. (Given \( \alpha = 5.0 \times 10^{-4} \, ^\circ\text{C}^{-1} \).)

Due to cold weather a \( 1 \, \text{m} \) water pipe of cross-sectional area \( 1 \, \text{cm}^2 \) is filled with ice at \( -10^\circ \text{C} \). Resistive heating is used to melt the ice. Current of \( 0.5 \, \text{A} \) is passed through \( 4 \, \text{k}\Omega \) resistance. Assuming that all the heat produced is used for melting, what is the minimum time required?
(Given latent heat of fusion for water/ice \( 3.33 \times 10^{5} \, \text{J} \, \text{kg}^{-1} \), specific heat of ice \( = 2 \times 10^{3} \, \text{J} \, \text{kg}^{-1} \) and density of ice \( = 10^{3} \, \text{kg} / \text{m}^{3} \))

A Carnot engine has efficiency of \( 50\% \). If the temperature of sink is reduced by \( 40^\circ \mathrm{C} \), its efficiency increases by \( 30\% \). The temperature of the source will be:

The pressure \( P_{1} \) and density \( d_{1} \) of diatomic gas \( \left( \gamma = \dfrac{7}{5} \right) \) changes suddenly to \( P_{2} \; ( > P_{1} ) \) and \( d_{2} \) respectively during an adiabatic process. The temperature of the gas increases and becomes ________ times of its initial temperature. (given \( \dfrac{d_{2}}{d_{1}} = 32 \))

A monoatomic gas performs a work of \( \dfrac{Q}{4} \) where \( Q \) is the heat supplied to it. The molar heat capacity of the gas will be ________ \( R \) during this transformation. Where \( R \) is the gas constant.

The internal energy of air in \( 4 \, \text{m} \times 4 \, \text{m} \times 3 \, \text{m} \) sized room at \( 1 \) atmospheric pressure will be __________ \( \times 10^{6} \, \text{J} \). (Consider air as diatomic molecule)

What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?

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:

When \( 300 \, \text{J} \) of heat given to an ideal gas with \( C_{p} = \dfrac{7}{2} R \) its temperature raises from \( 20^\circ \text{C} \) to \( 50^\circ \text{C} \) keeping its volume constant. The mass of the gas is (approximately) __________ \( \text{g} \). \( ( R = 8.314/\text{mol.K} ) \).

For a perfect gas, two pressures \( P1 \) and \( P2 \) are shown in figure. The graph shows:

Which one is the correct option for the two different thermodynamic processes?

The pressure of a gas changes linearly with volume from \( A \) to \( B \) as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be:

The ratio of specific heats \( \left( \dfrac{C_p}{C_v} \right) \) in terms of degree of freedom \( (f) \) is given by :

The root mean square speed of molecules of nitrogen gas at \( 27^\circ \text{C} \) is approximately: (Given mass of a nitrogen molecule \( = 4.6 \times 10^{-26} \, \text{kg} \) and take Boltzmann constant \( k_{B} = 1.4 \times 10^{-23} \, \text{J K}^{-1} \))

In the reported figure, heat energy absorbed by a system in going through a cyclic process is \( \pi \, \text{J} \).

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:

Let \( \gamma_{1} \) be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and \( \gamma_{2} \) be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio, \( \frac{\gamma_{1}}{\gamma_{2}} \) is:

A Carnot engine takes \( 5000 \ \text{kcal} \) of heat from a reservoir at \( 727^\circ \text{C} \) and gives heat to a sink at \( 127^\circ \text{C} \). The work done by engine is:

Starting with the same initial conditions, an ideal gas expands from volume \( V_1 \) to \( V_2 \) in three different ways. The work done by the gas is \( W_1 \) if the process is purely isothermal, \( W_2 \) if the process is purely adiabatic and \( W_3 \) if the process is purely isobaric. Then, choose the correct option.

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

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

Energy of \( 10 \) non rigid diatomic molecules at temperature \( T \) is:

Two particles \( A \) and \( B \) of equal masses are suspended from two massless springs of spring constants \( K_{1} \) and \( K_{2} \) respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of \( A \) and \( B \) is:

\( 10 \) mole of oxygen is heated at constant volume from \( 30^\circ\text{C} \) to \( 40^\circ\text{C} \). The change in the internal energy of the gas is __________ \( \text{cal} \). (The molecular specific heat of oxygen at constant pressure, \( C_p = 7 \, \text{cal} / \text{mol}^\circ\text{C} \) and \( R = 2 \, \text{cal}./\text{mol}.^\circ\text{C} \))

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.
Reason (R): In isothermal process, \( PV = \) constant, while in adiabatic process \( PV^{\gamma} = \) constant. Here \( \gamma \) is the ratio of specific heats, \( P \) is the pressure and \( V \) is the volume of the ideal gas.
In the light of the above statements, choose the correct answer from the options given below:

The efficiency of a Carnot's engine, working between steam point and ice point, will be:

The root mean square velocity of molecules of gas is:

The relation between root mean square speed \( v_{\mathrm{rms}} \) and most probable speed \( v_p \) for molar mass \( M \) of oxygen gas molecule at the temperature of \( 300 \, \text{K} \) will be:

Two metallic blocks \( M_1 \) and \( M_2 \) of same area of cross-section are connected to each other (as shown in figure). If the thermal conductivity of \( M_2 \) is \( K \) then the thermal conductivity of \( M_1 \) will be: [Assume steady state heat conduction]

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of \( \Delta Q : \Delta U : \Delta W \) is ______.

Match List I with List - II.

\( \Delta Q = \text{Heat supplied} \)
\( \Delta W = \text{Work done by the system} \)
\( \Delta U = \text{Change in internal energy} \)
\( P = \text{Pressure of the system} \)
\( \Delta V = \text{Change in volume of the system} \)
Choose the correct answer from the options given below :

One mole of an ideal gas at \( 27^\circ \text{C} \) is taken from A to B as shown in the given PV indicator diagram. The work done by the system will be \( \_\_\_\_\_\_\_\_ \times 10^{-1} \, \text{J} \).
[Given: \( R = 8.3 \, \text{J mol}\,\text{K} \), \( \ln 2 = 0.6931 \)]

A thermodynamic system is taken from an original state \( D \) to an intermediate state \( E \) by the linear process shown in the figure. Its volume is then reduced to the original volume from \( E \) to \( F \) by an isobaric process. The total work done by the gas from \( D \) to \( E \) to \( F \) will be

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

The temperature of a gas is \( -78^\circ \text{C} \) and the average translational kinetic energy of its molecules is \( K \). The temperature at which the average translational kinetic energy of the molecules of the same gas becomes \( 2K \) is:

The pressure \( (P) \) and temperature \( (T) \) relationship of an ideal gas obeys the equation \( PT^{2} = \text{constant} \). The volume expansion coefficient of the gas will be:

The root mean square speed of molecules of a given mass of a gas at \( 27^\circ \text{C} \) and \( 1 \) atmosphere pressure is \( 200 \, \text{m s}^{-1} \). The root mean square speed of molecules of the gas at \( 127^\circ \text{C} \) and \( 2 \) atmosphere pressure is \( \dfrac{x}{\sqrt{3}} \, \text{m s}^{-1} \). The value of \( x \) will be ______.

The number of molecules in one litre of an ideal gas at \( 300 \, \text{K} \) and \( 2 \, \text{atmosphere} \) pressure with mean kinetic energy \( 2 \times 10^{-9} \, \text{J} \) per molecule is:

A poly-atomic molecule \( (C_{V} = 3R, \, C_{P} = 4R, \) where \( R \) is gas constant) goes from phase space point \( A \, (P_{A} = 10^{5} \, \text{Pa}, \, V_{A} = 4 \times 10^{-6} \, \text{m}^{3}) \) to point \( B \, (P_{B} = 5 \times 10^{4} \, \text{Pa}, \, V_{B} = 6 \times 10^{-6} \, \text{m}^{3}) \) to point \( C \, (P_{C} = 10^{4} \, \text{Pa}, \, V_{C} = 8 \times 10^{-6} \, \text{m}^{3}) \). A to B is an adiabatic path and B to C is an isothermal path.
The net heat absorbed per unit mole by the system is:

A copper block of mass \( 5.0 \, \text{kg} \) is heated to a temperature of \( 500^\circ \text{C} \) and is placed on large ice block. What is the maximum amount of ice that can melt?
[Specific heat of copper : \( 0.39 \, \text{J g}^{-1} {^\circ}\text{C}^{-1} \) and latent heat of fusion of water : \( 335 \, \text{J g}^{-1} \)]

An electric appliance supplies \( 6000 \, \text{J/min} \) heat to the system. If the system delivers a power of \( 90 \, \text{W} \). How long it would take to increase the internal energy by \( 2.5 \times 10^3 \, \text{J} \)?

Match the LIST-I with LIST-II:

Choose the correct answer from the options given below:

One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is \( \dfrac{\alpha^2}{4} R \, \text{J/mol K} \); then the value of \( \alpha \) will be __________ (Assume that the given diatomic gas has no vibrational mode).

A cylindrical tube \( AB \) of length \( l \), closed at both ends contains an ideal gas of \( 1 \, \text{mol} \) having molecular weight \( M \). The tube is rotated in a horizontal plane with constant angular velocity \( \omega \) about an axis perpendicular to \( AB \) and passing through the edge at end \( A \), as shown in the figure. If \( P_{A} \) and \( P_{B} \) are the pressures at \( A \) and \( B \) respectively, then: (Consider the temperature is same at all points in the tube)

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

Heat energy of \( 735 \, \text{J} \) is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but do not oscillate. The increase in the internal energy of the gas will be:

A sample of an ideal gas is taken through the cyclic process \( ABCA \) as shown in figure. It absorbs, \( 40 \, \text{J} \) of heat during the part \( AB \), no heat during \( BC \) and rejects \( 60 \, \text{J} \) of heat during \( CA \). A work of \( 50 \, \text{J} \) is done on the gas during the part \( BC \). The internal energy of the gas at \( A \) is \( 1560 \, \text{J} \). The workdone by the gas during the part \( CA \) is:

An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is \( S_{1} \) and that of the other part is \( S_{2} \). Given that \( S_{1} > S_{2} \). If the piston is removed then the total entropy of the system will be:

A Carnot engine whose heat sinks at \( 27^\circ \mathrm{C} \), has an efficiency of \( 25\% \). By how many degrees should the temperature of the source be changed to increase the efficiency by \( 100\% \) of the original efficiency?

What will be the average value of energy for a monoatomic gas in thermal equilibrium at temperature \( T \)?

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:

The total kinetic energy of \( 1 \, \text{mole} \) of oxygen at \( 27^\circ \text{C} \) is:
[Use universal gas constant \( (R) = 8.31 \, \text{J/mole K} \)]

A mixture of hydrogen and oxygen has volume \( 2000 \, \text{cm}^3 \), temperature \( 300 \, \text{K} \), pressure \( 100 \, \text{kPa} \) and mass \( 0.76 \, \text{g} \). The ratio of number of moles of hydrogen to number of moles of oxygen in the mixture will be:
[Take gas constant \( R = 8.3 \, \text{J} \, \text{K}^{-1} \, \text{mol}^{-1} \)]

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is \( 1 : 4 \). Then:
(I) The r.m.s. velocity of gas molecules in two vessels will be the same
(II) The ratio of pressure in these vessels will be \( 1 : 4 \)
(III) The ratio of pressure will be \( 1 : 1 \)
(IV) The r.m.s. velocity of gas molecules in two vessels will be in the ratio of \( 1 : 4 \)

Work done by a Carnot engine operating between temperatures \( 127^\circ \text{C} \) and \( 27^\circ \text{C} \) is \( 2 \, \text{kJ} \). The amount of heat transferred to the engine by the reservoir is:

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

If one mole of an ideal gas at \( (P_{1}, V_{1}) \) is allowed to expand reversibly and isothermally (A to B) its pressure is reduced to one-half of the original pressure (see figure). This is followed by a constant volume cooling till its pressure is reduced to one-fourth of the initial value (B \( \rightarrow \) C). Then it is restored to its initial state by a reversible adiabatic compression (C to A). The net work done by the gas is equal to:
 

The internal energy of a monoatomic gas is \( 3nRT \). One mole of helium is kept in a cylinder having internal cross section area of \( 17\,\text{cm}^{2} \) and fitted with a light movable frictionless piston. The gas is heated slowly by supplying \( 126\,\text{J} \) heat. If the temperature rises by \( 4^{\circ}\text{C} \), then the piston will move ______ cm.
\( \text{(atmospheric pressure } = 10^{5}\ \text{Pa)} \)

\( 7 \) mol of a certain monoatomic ideal gas undergoes a temperature increase of \( 40 \, \text{K} \) at constant pressure. The increase in the internal energy of the gas in this process is:
(Given \( R = 8.3 \, \text{J} \, \text{K}^{-1} \, \text{mol}^{-1} \))

For a diatomic gas, if \(\displaystyle \gamma_{1} = \left( \frac{C_{p}}{C_{v}} \right) \) for rigid molecules and \(\displaystyle \gamma_{2} = \left( \frac{C_{p}}{C_{v}} \right) \) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (\( C_{p} \) and \( C_{v} \) are specific heats of the gas at constant pressure and volume)

In the following \( p\! -\! V \) diagram the equation of state along the curved path is given by \( (V - 2)^{2} = 4 a p \) where \( a \) is a constant. The total work done in the closed path is:

The temperature of \( 3.00 \) mol of an ideal diatomic gas is increased by \( 40.0^\circ \text{C} \) without changing the pressure of the gas. The molecular in the gas rotate but do not oscillate. If the ratio of change in internal energy of the gas to the amount of work done by the gas is \( \frac{x}{10} \) . Then the value of \( x \) is:

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 \( 27\dfrac{P}{8} \). If initial volume of each compartment was \( 9 \) litres then the final volume in right-hand side compartment is ______ litres. (for this ideal gas \( C_{p}/C_{r}=1.5 \))

Read the following statements:
(I) When small temperature difference between a liquid and its surrounding is doubled, the rate of loss of heat of the liquid becomes twice
(II) Two bodies \( P \) and \( Q \) having equal surface areas are maintained at temperature \( 10^\circ \text{C} \) and \( 20^\circ \text{C} \). The thermal radiation emitted in a given time by \( P \) and \( Q \) are in the ratio \( 1 : 1 : 15 \)
(III) A cannot Engine working between \( 100 \, \text{K} \) and \( 400 \, \text{K} \) has an efficiency of \( 75\% \)
(IV) When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice.

Two ideal polyatomic gases at temperatures \( T_1 \) and \( T_2 \) are mixed so that there is no loss of energy. If \( F_1 \) and \( F_2 \), \( m_1 \) and \( m_2 \), \( n_1 \) and \( n_2 \) be the degrees of freedom, masses, number of molecules of the first and second gas respectively, the temperature of mixture of these two gases is;

P–T diagram of an ideal gas having three different densities \( \rho_{1}, \rho_{2}, \rho_{3} \) (in three different cases) is shown in the figure. Which of the following is correct:

An ideal gas undergoes a cyclic transformation starting from the point \( A \) and coming back to the same point by tracing the path \( A \rightarrow B \rightarrow C \rightarrow D \rightarrow A \) as shown in the three cases above. Choose the correct option regarding \( DU \):

If three moles of monoatomic gas \( \left( \gamma = \dfrac{5}{3} \right) \) is mixed with two moles of a diatomic gas \( \left( \gamma = \dfrac{7}{5} \right) \), the value of adiabatic exponent \( \gamma \) for the mixture is:

A vessel contains \( 14 \, \text{g} \) of nitrogen gas at a temperature of \( 27^\circ \text{C} \). The amount of heat to be transferred to the gas to double the r.m.s. speed of its molecules will be:
(Take \( R = 8.32 \, \text{J mol}^{-1} \, \text{K}^{-1} \).)

The work done in an adiabatic change in an ideal gas depends upon only:

A heat engine operates with the cold reservoir at temperature \( 324 \, \text{K} \). The minimum temperature of the hot reservoir, if the heat engine takes \( 300 \, \text{J} \) heat from the hot reservoir and delivers \( 180 \, \text{J} \) heat to the cold reservoir per cycle, is _______ \( \text{K} \).

Density of water at \( 4^\circ \text{C} \) and \( 20^\circ \text{C} \) are \( 1000 \, \text{kg/m}^3 \) and \( 998 \, \text{kg/m}^3 \) respectively. The increase in internal energy of \( 4 \, \text{kg} \) of water when it is heated from \( 4^\circ \text{C} \) to \( 20^\circ \text{C} \) is _____ \( \text{J} \).
(Specific heat capacity of water \( = 4.2 \, \text{J/kg} \) and \( 1 \) atmospheric pressure \( = 10^5 \, \text{Pa} \))

\( 0.056 \, \text{kg} \) of Nitrogen is enclosed in a vessel at a temperature of \( 127^\circ \text{C} \). The amount of heat required to double the speed of its molecules is ________ \( \text{k cal} \).
(Take \( R = 2 \, \text{cal mol}^{-1}\,\text{K}^{-1} \))

The number of air molecules per \( \text{cm}^3 \) increased from \( 3 \times 10^{19} \) to \( 12 \times 10^{19} \). The ratio of collision frequency of air molecules before and after the increase in number respectively is:

What will be the average value of energy along one degree of freedom for an ideal gas in thermal equilibrium at a temperature \( T \)? \( (k_{B} \) is Boltzmann constant\( ) \)

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 \( V_{\text{final}} = 27V \) is ______ \( \text{J} \).

The equation for real gas is given by \( \left(P + \frac{a}{V^{2}}\right)(V-b) = RT \), where \( P \), \( V \), \( T \) and \( R \) are the pressure, volume, temperature and gas constant, respectively. The dimension of \( ab^{-2} \) is equivalent to that of:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: If the average kinetic energy of \( \mathrm{H_2} \) and \( \mathrm{O_2} \) molecules, kept in two different sized containers are same, then their temperatures will be same.
Reason R: The r.m.s. speed of \( \mathrm{H_2} \) and \( \mathrm{O_2} \) molecules are same at same temperature.
Choose the correct answer from the options given below:

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

The root mean square speed of smoke particles of mass \( 5 \times 10^{-17} \, \text{kg} \) in their Brownian motion in air at NTP is approximately. [Given \( k = 1.38 \times 10^{-23} \, \text{J K}^{-1} \) ]

On the basis of kinetic theory of gases, the gas exerts pressure because its molecules:

A gas is compressed adiabatically, which one of the following statement is NOT true.

The average kinetic energy of a molecule of the gas is:

In \( 1^{\text{st}} \) case, Carnot engine operates between temperature \( 300 \, \text{K} \) and \( 100 \, \text{K} \). In \( 2^{\text{nd}} \) case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in \( 2^{\text{nd}} \) case) will be:

An insulated cylinder of volume \( 60 \, \text{cm}^{3} \) is filled with a gas at \( 27^\circ \text{C} \) and \( 2 \) atmospheric pressure. Then the gas is compressed making the final volume as \( 20 \, \text{cm}^{3} \) while allowing the temperature to rise to \( 77^\circ \text{C} \). The final pressure is ______ atmospheric pressure.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Statement I: Change in internal energy of a system containing \( n \) mole of ideal gas can be written as \( \Delta U = nC_v (T_f - T_i) - \frac{nR}{\gamma - 1} (T_f - T_i) \), where \( \gamma = \frac{C_p}{C_v} \), \( T_i \) = initial temperature, \( T_f \) = final temperature.
Statement II: Relation between degree of freedom \( f \) and \( \gamma(= \frac{C_p}{C_v}) \) is \( \left( \gamma = 1 + \frac{2}{f} \right) \).
Choose the correct answer from the options given below

A cylinder container of volume \( 4.0 \times 10^{-3} \, \text{m}^3 \) contains \( 1 \) mole of hydrogen and \( 2 \) moles of carbon dioxide. Assume the temperature of the mixture is \( 400 \, \text{K} \). The pressure of the mixture of gases is:
[Take gas constant as \( 8.3 \, \text{J mol}^{-1} \text{K}^{-1} \)]

For a particular ideal gas which of the following graphs represents the variation of mean squared velocity of the gas molecules with temperature?

An engine operating between the boiling and freezing points of water will have.
A. efficiency more than \( 27\% \).
B. efficiency less than the efficiency of a Carnot engine operating between the same two temperatures.
C. efficiency equal to \( 27\% \).
D. efficiency less than \( 27\% \).
Choose the correct answer from the options given below:

Consider a sample of oxygen behaving like an ideal gas. At \( 300 \, \text{K} \), the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be:
(Molecular weight of oxygen is \( 32 \, \text{g/mol} \); \( R = 8.3 \, \text{J} \, \text{K}^{-1} \, \text{mol}^{-1} \))

A real gas within a colored chamber at \( 27^\circ\text{C} \) 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 \( R=8 \,\text{J/mol K} \)):

One mole of an ideal gas is taken through an adiabatic process where the temperature rises from \( 27^\circ \text{C} \) to \( 37^\circ \text{C} \). If the ideal gas is composed of polyatomic molecule that the \( 4 \) vibrational modes, which of the following is true?
[\( R = 8.314 \, \text{J mol}^{-1} \text{K}^{-1} \)]

A monoatomic gas at pressure \( P \) and volume \( V \) is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be:

\( 1 \, \text{kg} \) of water at \( 100^\circ \text{C} \) is converted into steam at \( 100^\circ \text{C} \) by boiling at atmospheric pressure. The volume of water changes from \( 1.00 \times 10^{-3} \, \text{m}^3 \) as a liquid to \( 1.671 \, \text{m}^3 \) as steam. The change in internal energy of the system during the process will be:
(Given latent heat of vaporization \( = 2257 \, \text{kJ}\!/\!\text{kg} \), Atmospheric pressure \( = 1 \times 10^5 \, \text{Pa} \))

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:

According to law of equipartition of energy the molar specific heat of a diatomic gas at constant volume where the molecule has one additional vibrational mode is:

An ideal gas initially at \( 0^\circ \text{C} \) temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is \( \tfrac{3}{2} \), the change in temperature due to the thermodynamic process is __________ \( \text{K} \).

A sample of gas at temperature \( T \) is adiabatically expanded to double its volume. Adiabatic constant for the gas is \( \gamma = \dfrac{3}{2} \).
The work done by the gas in the process is:
\( (\mu = 1 \text{ mole}) \)

A gas has \( n \) degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be:

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 an adiabatic process.
In the light of the above statements, choose the correct answer from the options given below:

A Carnot's engine working between \( 400 \, \text{K} \) and \( 800 \, \text{K} \) has a work output \( 1200 \, \text{J} \) per cycle. The amount of heat energy supplied to the engine from the source in each cycle is:

\( 10 \) mole of an ideal gas is undergoing the process shown in the figure. The heat involved in the process from \( P_1 \) to \( P_2 \) is \( a \) Joule \( ( P_1 = 21.7 \, \text{Pa} \text{ and } P_2 = 30 \, \text{Pa}, \, C_v = 21 \, \text{J} / \text{K} \cdot \text{mol} \, R = 8.3 \, \text{J} / \text{molK} ) \). The value of \( a \) is ________ .
 

At \( 300\,\text{K} \), the rms speed of oxygen molecules is \( \sqrt{\frac{\alpha+5}{\alpha}} \) times to that of its average speed in the gas. Then, the value of \( \alpha \) will be: (used \( \pi=\frac{22}{7} \))

Following statements are given :
I. The average kinetic energy of a gas molecule decreases when the temperature is reduced.
II. The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature
III. The average kinetic energy of a gas molecule decreases with increase in volume
IV. Pressure of a gas increases with increase in temperature at constant pressure
V. The volume of gas decreases with increase in temperature
Choose the correct answer from the options given below :

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to: (Where \( \gamma \) is the ratio of specific heats).

Consider a mixture of gas molecule of types \( A \), \( B \) and \( C \) having masses \( m_A < m_B < m_C \). The ratio of their root mean square speeds at normal temperature and pressure is:

Let \( \eta_1 \) is the efficiency of an engine at \( T_1 = 447^\circ \text{C} \) and \( T_2 = 147^\circ \text{C} \) while \( \eta_2 \) is the efficiency at \( T_1 = 947^\circ \text{C} \) and \( T_2 = 47^\circ \text{C} \). The ratio \( \dfrac{\eta_1}{\eta_2} \) will be:

The temperature of a gas having \( 2.0 \times 10^{25} \) molecules per cubic meter at \( 1.38 \, \text{atm} \). (Given, \( k = 1.38 \times 10^{-23} \, \text{J} \, \text{K}^{-1} \))

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

Sound travels in a mixture of two moles of helium and \( n \) moles of hydrogen. If rms speed of gas molecules in the mixture is \( \sqrt{2} \) times the speed of sound, then the value of \( n \) will be:

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:

A mixture of hydrogen and oxygen has volume \( 500 \, \text{cm}^3 \), temperature \( 300 \, \text{K} \), pressure \( 400 \, \text{kPa} \) and mass \( 0.76 \, \text{g} \). The ratio of masses of oxygen to hydrogen will be:

The translational degrees of freedom \( (f_t) \) and rotational degrees of freedom \( (f_r) \) of \( CH_4 \) molecule are:

The initial pressure and volume of an ideal gas are \( P_{o} \) and \( V_{o} \). The final pressure of the gas when the gas is suddenly compressed to volume \( \dfrac{V_{o}}{4} \) will be: (Given \( \gamma \) = ratio of specific heats at constant pressure and at constant volume)

The correct relation between \( \gamma=\frac{C_p}{C_v} \) and temperature \( T \) is:

If \( 2 \) mole of an ideal monoatomic gas at temperature \( T \), is mixed with \( 6 \) mole of another ideal monoatomic gas at temperature \( 2T \) then the temperature of mixture is:

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

Match the LIST-I with LIST-II.

Choose the correct answer from the options given below:

The temperature of an ideal gas is increased from \( 200 \, \text{K} \) to \( 800 \, \text{K} \). If r.m.s. speed of gas at \( 200 \, \text{K} \) is \( v_0 \). Then, r.m.s. speed of the gas at \( 800 \, \text{K} \) will be:

Nearly \( 10\% \) of the power of a \( 110 \, \text{W} \) light bulb is converted to visible radiation. The change in average intensities of visible radiation, at a distance of \( 1 \, \text{m} \) from the bulb to a distance of \( 5 \, \text{m} \) is \( a \times 10^{-2} \, \text{W} / \text{m}^{2} \). The value of ' \( a \) ' will be ________.

The kinetic energy of translation of the molecules in \( 50 \, \text{g} \) of \( \text{CO}_2 \) gas at \( 17^\circ \text{C} \) is:

The mean free path and the average speed of oxygen molecules at \( 300 \, \text{K} \) and \( 1 \, \text{atm} \) are \( 3 \times 10^{-7} \, \text{m} \) and \( 600 \, \text{m/s} \), respectively. Find the frequency of its collisions.

A heat engine has an efficiency of \( \dfrac{1}{6} \). When the temperature of sink is reduced by \( 62^{\circ}\text{C} \), its efficiency get doubled. The temperature of the source is:

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

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:

A flask contains Hydrogen and Argon in the ratio \( 2:1 \) by mass. The temperature of the mixture is \( 30^\circ \, \text{C} \). The ratio of average kinetic energy per molecule of the two gases \( (K \, \text{argon}/K \, \text{hydrogen}) \) is:
(Given Atomic Weight of Ar \( = 39.9 \))

If the r.m.s speed of chlorine molecule is \( 490 \, \text{m/s} \) at \( 27^{\circ}\text{C} \), the r.m.s speed of argon molecules at the same temperature will be (Atomic mass of argon \( = 39.9 \, u \), molecular mass of chlorine \( = 70.9 \, u \) )

\( 0.08 \, \mathrm{kg} \) 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 \( J = 4.18 \, \mathrm{joule/cal} \). The change in its internal energy is approximately

The P–V diagram of a diatomic ideal gas system going under cyclic process as shown in figure. The work done during an adiabatic process \( CD \) is (use \( \gamma = 1.4 \))

Pressure of an ideal gas, contained in a closed vessel, is increased by \( 0.4\% \) when heated by \( 1^\circ \text{C} \). Its initial temperature must be:

A container of fixed volume contains a gas at $27^\circ\text{C}$. To double the pressure of the gas, the temperature of gas should be raised to _________ $^\circ\text{C}$.

A sample of gas at temperature \( T \) is adiabatically expanded to double its volume. The work done by the gas in the process is: (given, \( \gamma = \frac{3}{2} \))

A source supplies heat to a system at the rate of \( 1000 \, \text{W} \). If the system performs work at a rate of \( 200 \, \text{W} \). The rate at which internal energy of the system increases is:

For a gas \( C_p - C_v = R \) in a state P and \( C_p - C_v = 1.10\,R \) in a state Q, \( T_P \) and \( T_Q \) are the temperatures in two different states P and Q respectively. Then

A mixture of carbon dioxide and oxygen has volume \( 8310 \, \text{cm}^{3} \), temperature \( 300 \, \text{K} \), pressure \( 100 \, \text{kPa} \) and mass \( 13.2 \, \text{g} \). The number of moles of carbon dioxide and oxygen gases in the mixture respectively are ________.
(Assume both carbon dioxide and oxygen gases behave like ideal gases) \([ R = 8.31 \, \text{J}/(\text{mol}\cdot\text{K}) ]\)

\( N \) moles of a polyatomic gas (\( f=6 \)) must be mixed with two moles of a monoatomic gas so that the mixture behaves as a diatomic gas. The value of \( N \) is:

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 ______ \( \text{J} \). (Neglect mass of the piston) [Atmospheric pressure = \( 100 \, \text{kPa} \), \( R = 8.3 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1} \)]

Which of the following graphs represent the behavior of an ideal gas? Symbols have their usual meaning.

A monatomic gas having \( \gamma = \frac{5}{3} \) is stored in a thermally insulated container and the gas is suddenly compressed to \( \left(\frac{1}{8}\right)^{\text{th}} \) 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]

The average translational kinetic energy of \( \mathrm{N_2} \) gas molecules at _____ \( ^\circ \text{C} \) becomes equal to the K.E. of an electron accelerated from rest through a potential difference of \( 0.1 \) volt. (Given \( k_B = 1.38 \times 10^{-23} \, \text{J} / \text{K} \) ).

A monoatomic ideal gas, initially at temperature \( T_1 \) is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \( T_2 \) by releasing the piston suddenly. If \( l_1 \) and \( l_2 \) are the lengths of the gas column, before and after the expansion respectively, then the value of \( \dfrac{T_1}{T_2} \) will be:

A diatomic gas \( (\gamma = 1.4) \) does \( 100 \; \text{J} \) of work when it is expanded isobarically. Then the heat given to the gas __________ \( \text{J} \).

Which of the following figure represents the relation between Celsius and Fahrenheit temperatures?

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

Calculate the value of mean free path \(( \lambda )\) for oxygen molecules at temperature \( 27^\circ \text{C} \) and pressure \( 1.01 \times 10^{5} \, \text{Pa} \). Assume the molecular diameter \( 0.3 \, \text{nm} \) and the gas is ideal.
\( (k = 1.38 \times 10^{-23} \, \text{J K}^{-1} )\)

The pressure and volume of an ideal gas are related as \( PV^{3/2} = K \) (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 mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature \( 27^{\circ}\text{C} \). The ratio of specific heat of gases at constant volume respectively is:

One gas of \( n_1 \) mole of molecules at temperature \( T_1 \), volume \( V_1 \), and pressure \( P_1 \) and another gas of \( n_2 \) mole of molecules at temperature \( T_2 \), volume \( V_2 \) and pressure \( P_2 \), are mixed resulting in pressure \( P \) and volume \( V \) of the mixture. The temperature of the mixture is _______.

Two moles a monoatomic gas is mixed with six moles of diatomic gas. The molar specific heat of the mixture at constant volume is:

If one mole of the polyatomic gas is having two vibrational modes and \( \beta \) is the ratio of molar specific heats for polyatomic gas \( \left( \beta = \dfrac{C_{P}}{C_{V}} \right) \) then the value of \( \beta \) is:

An air bubble of volume \( 2.9 \, \text{cm}^3 \) rises from the bottom of a swimming pool of \( 5 \, \text{m} \) deep. At the bottom of the pool water temperature is \( 17^\circ\text{C} \). The volume of the bubble when it reaches the surface, where the water temperature is \( 27^\circ\text{C} \), is ______ \( \text{cm}^3 \).
\( (g = 10 \, \text{m/s}^2 , \ \text{density of water} = 10^3 \, \text{kg/m}^3 , \text{ and } 1 \, \text{atm pressure is } 10^5 \, \text{Pa}) \)

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

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 _______ \( \text{K} \).

Two Carnot engines A and B operate in series such that engine A absorbs heat at \( T_1 \) and rejects heat to a sink at temperature \( T \). Engine B absorbs half of the heat rejected by Engine A and rejects to the sink at \( T_3 \). When workdone in both the cases is equal, the value of \( T \) is:

A Carnot engine with efficiency \( 50\% \) takes heat from a source at \( 600 \, \text{K} \). In order to increase the efficiency to \( 70\% \), keeping the temperature of sink same, the new temperature of the source will be:

A polyatomic ideal gas has \( 24 \) vibrational modes. What is the value of \( \gamma \) ?

At which temperature the r.m.s velocity of a hydrogen molecule equal to that of an oxygen molecule at \( 47^\circ \mathrm{C} \)?

The temperature of a body in air falls from \( 40^\circ \text{C} \) to \( 24^\circ \text{C} \) in \( 4 \) minutes. The temperature of the air is \( 16^\circ \text{C} \). The temperature of the body in the next \( 4 \) minutes will be:

A vessel contains \( 16 \) \( \text{g} \) of hydrogen and \( 128 \) \( \text{g} \) of oxygen at standard temperature and pressure. The volume of the vessel in \( \text{cm}^3 \) is:

Two vessels \( A \) and \( B \) are of the same size and are at same temperature. \( A \) contains \( 1 \, \text{g} \) of hydrogen and \( B \) contains \( 1 \, \text{g} \) of oxygen. \( P_A \) and \( P_B \) are the pressure of the gases in \( A \) and \( B \) respectively, then \( \frac{P_A}{P_B} \) is:

For an ideal heat engine, the temperature of the source is \( 127^\circ \text{C} \). In order to have \( 60\% \) efficiency the temperature of the sink should be __________ \( ^\circ \text{C} \). (Round off to the Nearest Integer)

The r.m.s. speed of oxygen molecules at \( 47^\circ \text{C} \) is equal to that of the hydrogen molecules kept at _______ \( ^\circ \text{C} \). (Mass of oxygen molecule/mass of hydrogen molecule = \( 32/2 \))

On a temperature scale ‘\( X \)’, the boiling point of water is \( 65^\circ X \) and the freezing point is \( -15^\circ X \). Assume that the \( X \) scale is linear. The equivalent temperature corresponding to \( -95^\circ X \) on the Farenheit scale would be:

The rms speed of oxygen molecule in a vessel at particular temperature is \( \left(1+\dfrac{5}{x}\right)^{\dfrac{1}{2}} v \), where \(v\) is the average speed of the molecule. The value of \(x\) will be: (Take \( \pi=\dfrac{22}{7} \))

In a carnot engine, the temperature of reservoir is \( 527^\circ \text{C} \) and that of sink is \( 200 \, \text{K} \). If the workdone by the engine when it transfers heat from reservoir to sink is \( 12000 \, \text{kJ} \), the quantity of heat absorbed by the engine from reservoir is ______ \( \times 10^{6} \, \text{J} \).

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: If \( dQ \) and \( dW \) represent the heat supplied to the system and the work done on the system respectively. Then according to the first law of thermodynamics \( dQ = dU - dW \).
Reason R: First law of thermodynamics is based on law of conservation of energy.
In the light of the above statements, choose the correct answer from the options given below:

A diatomic gas \( (\gamma = 1.4) \) does \( 400 \, \text{J} \) of work when it is expanded isobarically. The heat given to the gas in the process is _______ \( \text{J} \).

The rms speeds of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are \( V_H \), \( V_O \) and \( V_C \) respectively then:

Three vessels of equal volume contain gases at the same temperature and pressure. The first vessel contains neon (monoatomic), the second contains chlorine (diatomic) and third contains uranium hexafluoride (polyatomic). Arrange these on the basis of their root mean square speed \( (v_{\text{rms}}) \) and choose the correct answer from the options given below:

For an ideal gas the instantaneous change in pressure \( p \) with volume \( v \) is given by the equation \( \frac{dp}{dV} = -ap \). If \( p = p_{0} \) at \( v = 0 \) is the given boundary condition, then the maximum temperature one mole of gas can attain is: (Here \( R \) is the gas constant)

The total internal energy of two mole monoatomic ideal gas at temperature \( T = 300 \, \text{K} \) will be ______ \( \text{J} \). (Given \( R = 8.31 \, \text{J/mol}\,\text{K} \))

The ratio of vapour densities of two gases at the same temperature is \(\displaystyle \frac{4}{25} \), then the ratio of r.m.s. velocities will be:

The mean free path of molecules of a certain gas at STP is \( 1500d \), where \( d \) is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at \( 373 \, \text{K} \) is approximately:

A sample of gas with \( \gamma = 1.5 \) is taken through an adiabatic process in which the volume is compressed from \( 1200 \, \text{cm}^3 \) to \( 300 \, \text{cm}^3 \). If the initial pressure is \( 200 \, \text{kPa} \). The absolute value of the workdone by the gas in the process = ______ J .

Which statements are correct about degrees of freedom?
\( (A) \) A molecule with \( n \) degrees of freedom has \( n^{2} \) different ways of storing energy.
\( (B) \) Each degree of freedom is associated with \( \dfrac{1}{2} RT \) average energy per mole.
\( (C) \) A monatomic gas molecule has \( 1 \) rotational degree of freedom where as diatomic molecule has \( 2 \) rotational degrees of freedom.
\( (D) \) \( \mathrm{CH}_{4} \) has a total of \( 6 \) degrees of freedom.
Choose the correct answer from the options given below:

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

Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure \( 90 \, \text{kPa} \) and temperature \( 400 \, \text{K} \). Keeping the temperature of one vessel constant at \( 400 \, \text{K} \) the second vessel temperature is raised to \( 500 \, \text{K} \). The final pressure in the vessels is _____ \( \text{kPa} \).

A gas of certain mass filled in a closed cylinder at a pressure of \( 3.23 \, \text{kPa} \) has temperature \( 50^\circ \text{C} \). The gas is now heated to double its temperature. The modified Pressure is ______ \( \text{Pa} \).

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

Given below are two statements:
Statement-I: When \( \mu \) amount of an ideal gas undergoes adiabatic change from state \( (P_{1}, V_{1}, T_{1}) \) to state \( (P_{2}, V_{2}, T_{2}) \), then work done is \( W = \dfrac{\mu R (T_{2}-T_{1})}{1-\gamma} \), where \( \gamma = \dfrac{C_{p}}{C_{v}} \), and \( R \) = universal gas constant.
Statement-II: In the above case, when work is done on the gas, the temperature of the gas would rise.
Choose the correct answer from the options given below:

Given below are two statements:
Statement-I: If heat is added to a system, its temperature must increase.
Statement-II: If positive work is done by a system in a thermodynamic process, its volume must increase. In the light of the above statements, choose the correct answer from the options given below.

A block of mass \(1\text{kg}\) attached to a spring is made to oscillate with an initial amplitude of \(12 \text{cm}\). After \(2\) minutes the amplitude decreases to \( 6 \, \text{cm} \). Determine the value of the damping constant for this motion. (Take \( \ln 2 = 0.693 \))

A certain amount of gas of volume \( V \) at \( 27^\circ \text{C} \) temperature and pressure \( 2 \times 10^7 \, \text{N m}^{-2} \) expands isothermally until its volume gets doubled. Later it expands adiabatically until its volume gets redoubled. The final pressure of gas will be (Use \( \gamma = 1.5 \) )

A cylinder of fixed capacity of \( 44.8 \, \text{litres} \) contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by \( 20.0^\circ \text{C} \) will be: (Given gas constant \( R = 8.3 \, \text{J K}^{-1} \, \text{mol}^{-1} \) )

A Carnot engine operating between two reservoirs has efficiency \( \frac{1}{3} \). When the temperature of cold reservoir raised by \( x \), its efficiency decreases to \( \frac{1}{6} \). The value of \( x \), if the temperature of hot reservoir is \( 99^\circ \text{C} \), will be:

An ideal gas has undergone through the cyclic process as shown in the figure. Work done by the gas in the entire cycle is ________ \( \times 10^{-1} \) \( \text{J} \) (Take \( \pi = 3.14 \) )

The temperature of \( 1 \) mole of an ideal monoatomic gas is increased by \( 50^\circ \text{C} \) 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 ________.

A thermodynamic system is taken through cyclic process. The total work done in the process is:

Given below are two statements:
Statement I: The average momentum of a molecule in a sample of an ideal depends on temperature.
Statement II: The rms speed of oxygen molecules in a gas is \( v \). If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become \( 2v \).
In the light of the above statements, choose the correct answer from the options given below:

Heat energy of \( 184 \, \text{kJ} \) is given to i.e. of mass \( 600 \, \text{g} \) at \( -12^\circ \text{C} \). Specific heat of ice is \( 2222.3 \, \text{J} \, \text{kg}^{-1} \, {}^\circ\text{C}^{-1} \) and latent heat of ice in \( 336 \, \text{kJ} / \text{kg}^{-1} \).

(A) Final temperature of system will be \( 0^\circ \text{C} \).
(B) Final temperature of the system will be greater than \( 0^\circ \text{C} \).
(C) The final system will have a mixture of ice and water in the ratio of \( 5:1 \).
(D) The final system will have a mixture of ice and water in the ratio of \( 1:5 \).

The average kinetic energy of a monoatomic molecule is \( 0.414 \, \text{eV} \) at temperature: (Use \( K_B = 1.38 \times 10^{-23} \, \text{J} / \text{mol} - \text{K} \) )

Match List I with List II.

Choose the correct answer from the option given below:

If the root mean square velocity of hydrogen molecule at a given temperature and pressure is \( 2 \, \text{km/s} \), the root mean square velocity of oxygen at the same condition in km/s is:

Other is labelled as Reason R
Assertion A : Efficiency of a reversible heat engine will be highest at \( -273^\circ \text{C} \) temperature of cold reservoir.
Reason R : The efficiency of Carnot’s engine depends not only on temperature of cold reservoir but it depends on the temperature of hot reservoir too and is given as \( n = \left( 1 - \frac{T_2}{T_1} \right) \).
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 \( 473 \, \text{K} \) and \( 273 \, \text{K} \). In a new system two engines - engine \( E_1 \) works between \( 473 \, \text{K} \) to \( 373 \, \text{K} \) and engine \( E_2 \) works between \( 373 \, \text{K} \) to \( 273 \, \text{K} \). If \( \eta_{12} \), \( \eta_1 \) and \( \eta_2 \) are the efficiencies of the engines \( E \), \( E_1 \) and \( E_2 \), respectively, then:

\( 5 \) moles of unknown gas is heated at constant volume from \( 10^{\circ}\text{C} \) to \( 20^{\circ}\text{C} \). The molar specific heat of this gas at constant pressure \( C_{p}=8\ \text{cal}/\text{mol}\cdot{}^{\circ}\text{C} \) and \( R=8.36\ \text{J}/\text{mol}\cdot{}^{\circ}\text{C} \). The change in the internal energy of the gas is ______ calorie.

Match List I with List II:

Choose the correct answer from the options given below:

A thermally insulated vessel contains an ideal gas of molecular mass \( M \) and ratio of specific heats \( 1.4 \). Vessel is moving with speed \( v \) and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by: (\( R \) = universal gas constant)

Under isothermal condition, the pressure of a gas is given by \(P=aV^{-3}\), where \(a\) is a constant and \(V\) is the volume of the gas. The bulk modulus at constant temperature is equal to:

A gas mixture consists of \( 2 \) moles of oxygen and \( 4 \) moles of neon at temperature \( T \). Neglecting all vibrational modes, the total internal energy of the system will be:

Consider two containers A and B containing monoatomic gases at the same Pressure (\( P \)), Volume (\( V \)) and Temperature (\( T \)). The gas in A is compressed isothermally to \( \dfrac{1}{8} \) of its original volume while the gas in B is compressed adiabatically to \( \dfrac{1}{8} \) of its original volume. The ratio of final pressure of gas in B to that of gas in A is:

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

The volume of an ideal gas increases \( 8 \) times and temperature becomes \(( \frac{1}{4} \))th of initial temperature during a reversible change. If there is no exchange of heat in this process \( (\Delta Q = 0) \) then identify the gas from the following options (Assuming the gases given in the options are ideal gases):

If the collision frequency of hydrogen molecules in a closed chamber at \( 27^\circ \text{C} \) is \( Z \), then the collision frequency of the same system at \( 127^\circ \text{C} \) is:

A gas mixture consists of \( 8 \) moles of argon and \( 6 \) moles of oxygen at temperature \( T \). Neglecting all vibrational modes, the total internal energy of the system is:

According to kinetic theory of gases,
A. The motion of the gas molecules freezes at \( 0^\circ \text{C} \).
B. The mean free path of gas molecules decreases if the density of molecules is increased.
C. The mean free path of gas molecules increases if temperature is increased keeping pressure constant.
D. Average kinetic energy per molecule per degree of freedom is \( \frac{3}{2} k_B T \) (for monoatomic gases).
Choose the most appropriate answer from the options given below:

Heat is given to an ideal gas in an isothermal process.
A. Internal energy of the gas will decrease.
B. Internal energy of the gas will increase.
C. Internal energy of the gas will not change.
D. The gas will do positive work.
E. The gas will do negative work.

A refrigerator consumes an average \( 35 \, \text{W} \) power to operate between temperature \( -10^\circ \text{C} \) to \( 25^\circ \text{C} \). If there is no loss of energy then how much average heat per second does it transfer?

A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample is:

In the reported figure, there is a cyclic process \( ABCDA \) on a sample of \( 1 \) mole of a diatomic gas. The temperature of the gas during the process \( A \rightarrow B \) and \( C \rightarrow D \) are \( T_{1} \) and \( T_{2} \) (\( T_{1} > T_{2} \)) respectively.

Choose the correct option out of the following for work done if processes \( BC \) and \( DA \) are adiabatic.

The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:

The temperature at which the kinetic energy of oxygen molecules becomes double then its value at \( 27^\circ \text{C} \) is:

At a certain temperature, the degrees of freedom per molecule for gas is \( 8 \). The gas performs \( 150 \, \text{J} \) of work when it expands under constant pressure. The amount of heat absorbed by the gas will be _____ \( \text{J} \).

The amount of heat needed to raise the temperature of \( 4 \) moles of a rigid diatomic gas from \( 0^\circ \mathrm{C} \) when no work is done is ______________. ( \( R \) is the universal gas constant)

If \( n \) is the number: density and \( d \) is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by:

The correct relation between the degrees of freedom \( f \) and the ratio of specific heat \( \gamma \) is:

A system consists of two types of gas molecules A and B having same number density \( 2 \times 10^{25} \, /\text{m}^{3} \). The diameter of A and B are \( \displaystyle 10\overset{\circ}{A}\) and \( \displaystyle 5\overset{\circ}{A} \) respectively. They suffer collision at room temperature. The ratio of average distance covered by the molecule A to that of B between two successive collision is ______ \( \times 10^{-2} \).

Consider two boxes containing ideal gases \( A \) and \( B \) such that their temperatures, pressures and number densities are same. The molecular size of \( A \) is half of that of \( B \) and mass of molecule \( A \) is four times that of \( B \). If the collision frequency in gas \( B \) is \( 32 \times 10^{18} / \text{s} \) then collision frequency in gas \( A \) is ________ \( / \text{s} \).

\( 300 \, \text{cal.} \) of heat is given to a heat engine and it rejects \( 225 \, \text{cal.} \) of heat. If source temperature is \( 227^\circ \text{C} \), then the temperature of sink will be __________ \( ^\circ \text{C} \).

A flask contains argon and oxygen in the ratio of \( 3 : 2 \) in mass and the mixture is kept at \( 27^\circ \text{C} \). The ratio of their average kinetic energy per molecule respectively will be:

A flask contains hydrogen and oxygen in the ratio of \( 2 : 1 \) by mass at temperature \( 27^\circ \text{C} \). The ratio of average kinetic energy per molecule of hydrogen and oxygen respectively is :

There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at \( 8\,\text{kPa} \) at \( 1000\,\text{K} \) while the smaller vessel contains the gas at \( 7\,\text{kPa} \) at \(500\,\text{K} \). If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at \( 600\,\text{K} \), at steady state the pressure in the vessels will be (in \( \text{kPa} \)).

The thermodynamic process, in which internal energy of the system remains constant is:

A vessel contains \( 0.15 \, \text{m}^3 \) of a gas at pressure \( 8 \) bar and temperature \( 140^\circ \text{C} \) with \( c_p = 3R \) and \( c_v = 2R \). It is expanded adiabatically till pressure falls to \( 1 \) bar. The work done during this process is ______ \( \text{kJ} \). (\( R \) is gas constant)