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

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

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.

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

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]

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

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 : \( 335 \, \text{J g}^{-1} \)]

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 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:

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?

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 \( 1 : 4 \)

\( 7 \) 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 \( R = 8.3 \, \text{J} \, \text{K}^{-1} \, \text{mol}^{-1} \))

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.

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 \( R = 8.32 \, \text{J mol}^{-1} \, \text{K}^{-1} \).)

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 _______ \( \text{K} \).

\( 0.056 \, \text{kg} \) 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 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 \( k = 1.38 \times 10^{-23} \, \text{J K}^{-1} \) ]

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:

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:

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:

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 :

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:

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 ________.

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:

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 ______ \( \times 10^{6} \, \text{J} \).

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

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:

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:

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 \( R = 8.3 \, \text{J K}^{-1} \, \text{mol}^{-1} \) )

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:

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)

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:

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 _____ \( \text{J} \).

\( 300 \, \text{cal.} \) 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: