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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:

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 \( = 10^{3} \, \text{kg} / \text{m}^{3} \))

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

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

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:

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: \( R = 8.3 \, \text{J mol}\,\text{K} \), \( \ln 2 = 0.6931 \)]

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:

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 \( 2.5 \times 10^3 \, \text{J} \)?

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:

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

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 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:

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;

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\( ) \)

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

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

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 \( 32 \, \text{g/mol} \); \( R = 8.3 \, \text{J} \, \text{K}^{-1} \, \text{mol}^{-1} \))

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 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:

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:

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:

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 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 \( \gamma = 1.4 \))

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

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

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:

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

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:

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

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

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)

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 .

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 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?

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 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)

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 ______ \( \times 10^{-2} \).

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JEE Main 2021 27th July Morning Shift

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: