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\( 1 \, \text{g} \) of a liquid is converted to vapour at \( 3\times10^{5} \, \text{Pa} \) pressure. If \( 10\% \) of the heat supplied is used for increasing the volume by \( 1600 \, \text{cm}^{3} \) during this phase change, then the increase in internal energy in the process will be:

A body cools from \( 60^\circ\text{C} \) to \( 40^\circ\text{C} \) in \( 6 \, \text{minutes} \). If, temperature of surroundings is \( 10^\circ\text{C} \). Then, after the next \( 6 \, \text{minutes} \), its temperature will be ______ \( ^\circ\text{C} \).

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:

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 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 \( k_{B} = 1.4 \times 10^{-23} \, \text{J K}^{-1} \))

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:

The root mean square velocity of molecules of gas 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:

10.

A hypothetical gas expands adiabatically such that its volume changes from \( 08 \, \text{litres} \) to \( 27 \, \text{litres} \). If the ratio of final pressure of the gas to initial pressure of the gas is \( \dfrac{16}{81} \). Then the ratio of \( \dfrac{C_P}{C_V} \) will be:

11.

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:

12.

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:

13.

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:

14.

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

15.

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

16.

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:

17.

\( 1 \, \text{kg} \) 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 \( = 1 \times 10^5 \, \text{Pa} \))

18.

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:

19.

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 \( \pi=\frac{22}{7} \))

20.

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)

21.

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

22.

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:

23.

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 \( = 39.9 \))

24.

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 \( = 70.9 \, u \) )

25.

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, \( \gamma = \frac{3}{2} \))

26.

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:

27.

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:

28.

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:

29.

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 \( \pi=\dfrac{22}{7} \))

30.

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:

31.

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:

32.

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:

33.

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.

34.

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:

35.

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

36.

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 \( 1:5 \).

37.

Match List I with List II.

Choose the correct answer from the option given below:

38.

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 :

39.

Match List I with List II:

List I	List II
(A) \( 3 \) Translational degrees of freedom	(I) Monoatomic gases
(B) \( 3 \) Translational, \( 2 \) rotational degrees of freedoms	(II) Polyatomic gases
(C) \( 3 \) Translational, \( 2 \) rotational and \( 1 \) vibrational degrees of freedom	(III) Rigid diatomic gases
(D) \( 3 \) Translational, \( 3 \) rotational and more than one vibrational degrees of freedom	(IV) Nonrigid diatomic gases

Choose the correct answer from the options given below:

40.

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:

41.

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:

42.

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:

43.

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.

44.

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

45.

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

46.

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 :

47.

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

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JEE Main 2023 31st Jan Morning Shift

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

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