JEE Main 2019 — Thermodynamics

When heat $Q$ is supplied to a diatomic gas of rigid molecules, at constant volume its temperature increases by $\Delta T$ . The heat required to produce the same change in temperature, at a constant pressure is:

An ideal gas is enclosed in a cylinder at pressure of $2$ atm and temperature, $300 K.$ The mean time between two successive collisions is $6\times {10}^{-8}s.$ If the pressure is doubled and temperature is increased to $500 K,$ the mean time between two successive collisions will be close to:

The specific heats, ${C}_{p}$ and ${C}_{v}$ of a gas of diatomic molecules, $A,$ are given (in units of $J mo{l}^{-1} {K}^{-1}$ ) by $29$ and $22,$ respectively. Another gas of diatomic molecules, $B,$ has the corresponding values $30$ and $21.$ If they are treated as ideal gases, then:

Ice at $-20^{\circ} \mathrm{C}$ is added to $50 \mathrm{~g}$ of water at $40^{\circ} \mathrm{C}$, When the temperature of the mixture reaches $0^{\circ} \mathrm{C},$ it is found that 20 $\mathrm{g}$ of ice is still unmelted. The amount of ice added to the water was close to (Specific heat of water $=4.2 \mathrm{~J} / \mathrm{g} /{ }^{\circ} \mathrm{C}$ Specific heat of Ice $=2.1 \mathrm{~J} / \mathrm{g} /{ }^{\circ} \mathrm{C}$ Heat of fusion of water at $\left.0^{\circ} \mathrm{C}=334 \mathrm{~J} / \mathrm{g}\right)$

The given diagram shows four processes i.e., isochoric, isobaric, isothermal and adiabatic. The correct assignment of the processes, in the same order is given by: 

A cylinder with fixed capacity of $67.2$ litre contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by $20^{\circ}C$ is: [Given that $R=8.31 J mo{l}^{-1} {K}^{-1}]$

Two materials having coefficients of thermal conductivity $3K$ and $K$ and thickness $d$ and $3\text{d}$ respectively, are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are ${\theta }_{2}$ and ${\theta }_{1}$ respectively, $({\theta }_{2}>{\theta }_{1}).$ The temperature at the interface is 

A $15g$ mass of nitrogen gas is enclosed in a vessel at a temperature, ${27}^{o}C$. The amount of heat transferred to the gas, so that $R.M.S.$ velocity of molecules is doubled, is about. $[R=8.3 J{(K \mathrm{mole})}^{-1}]$

For the given cyclic process $CAB$ as shown for a gas, the work done is: 

A metal ball of mass $0.1 \mathrm{~kg}$ is heated upto $500^{\circ} \mathrm{C}$ and dropped into a vessel of heat capacity $800 \mathrm{JK}^{-1}$ and containing $0.5 \mathrm{~kg}$ water. The initial temperature of water and vessel is $30^{\circ} \mathrm{C}$. What is the approximate percentage increment in the temperature of the water? [Specific Heat Capacities of water and metal are, respectively, $4200 \mathrm{Jkg}^{-}$ ${ }^{1} \mathrm{~K}^{-1}$ and $400 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}$ ]

A gas can be taken from $A$ to $B$ via two different processes $\mathrm{ACB}$ and $\mathrm{ADB}$.  When path $\mathrm{ACB}$ is used $60 J$ of heat flows into the system and $30 J$ of work is done by the system. If the path $\mathrm{ADB}$ is used then work done by the system is $10 J$, the heat flows into the system in the path $\mathrm{ADB}$ is:

A cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$. The thermal conductivity of the material of the inner cylinder is ${K}_{1}$ and that of the outer cylinder is ${K}_{2}$. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:

Two identical beakers $A$ and $B$ contain equal volumes of two different liquids at $60^{\circ}C$ each and left to cool down. Liquid in $A$ has density of $8\times {10}^{2} kg{m}^{-3}$ and specific heat of $2000 J k{g}^{-1}{K}^{-1}$ while the liquid in $B$ has density ${10}^{3} kg{ m}^{-3}$ and specific heat of $4000 J k{g}^{-1}{K}^{-1}$. Which of the following best describes their temperature versus time graph schematically? (assume the emissivity of both the beakers to be the same)

An $HCl$ molecule has rotational, translational and vibrational motions. If the rms velocity of $HCl$ molecules in its gaseous phase is $\overset{-}{\nu }$ , m is its mass and ${k}_{B}$ is Boltzmann's constant, then its temperature will be:

For given gas at $1 atm$ pressure, $rms$ speed of the molecules is $200 m/s$ at $127^{\circ}C.$ At $2 atm$ pressure and at $227^{\circ} C,$ the $rms$ speed of the molecules will be:

A mixture of 2 moles of helium gas (atomic mass = 4u) $,$ and 1 mole of argon gas (atomic mass = 40 u) is kept at $300 K$ in a container. The ratio of their rms speeds $[\frac{{V}_{rms}(helium)}{{V}_{rms}(argon)}],$ is close to:

Following figure shows two processes $A$ and $B$ for a gas. If $\Delta {Q}_{A}$ and $\Delta {Q}_{B}$ are the amount of heat absorbed by the system in two cases, and $\Delta {U}_{A}$ and $\Delta {U}_{B}$ are changes in internal energies, respectively, then: 

Two moles of helium gas is mixed with three moles of hydrogen molecules (taken to be rigid). What is the molar specific heat of mixture at constant volume? $(R=8.3 J/mol K)$

A diatomic gas with rigid molecules does $10 J$ of work when expanded at constant pressure. What would be the heat energy absorbed by the gas, in this process?

A thermometer graduated according to a linear scale reads a value $x_{0}$ when in contact with boiling water, and $x_{0} / 3$ when in contact with ice. What is the temperature of an object in ${ }^{\circ} \mathrm{C}$, if this thermometer in the contact with the object reads $x_{0} / 2 ?$

A sample of an ideal gas is taken through the cyclic process $abca$ as shown in the figure. The change in the internal energy of the gas along the path $ca$ is $-180 J.$ The gas absorbs $250 J$ of heat along the path $ab$ and $60 J$ along the path $bc$ . The work done by the gas along the path $abc$ is: 

A massless spring $(k=800 N/m),$ attached with a mass $(500 g)$ is completely immersed in $1 kg$ of water. The spring is stretched by $2 cm$ and released so that it starts vibrating. What would be the order of magnitude of the change in the temperature of water when the vibrations stop completely? (Assume that the water container and spring receive negligible heat and specific heat of mass $=400 J/kg K,$ specific heat of water $=4184 J/kg K$ )

When ${M}_{1}$ gram of ice at $-{10}^{o}C$ (specific heat $=0.5 cal {g}^{-1} {℃}^{-1}$ ) is added to ${M}_{2}$ gram of water at ${50 }^{o}C,$ finally no ice is left and the water is at ${0 }^{o}C$ . The value of latent heat of ice, in $cal {g}^{-1}$ is:

Half mole of an ideal monoatomic gas is heated at a constant pressure of $1 \mathrm{atm}$ from $20^{\circ}C$ to $90^{\circ}C$. Work done by the gas is$($Gas constant,$R=8.21 J{\mathrm{mol}}^{-1}{K}^{-1}$)

The temperature, at which the root mean square velocity of hydrogen molecules equals their escape their escape velocity from the earth, is closest to: $[$ Boltzmann Constant ${k}_{B}=1.38\times {10}^{-23} J/K$ Avogadro number ${N}_{A}=6.02\times {10}^{26} /kg$ Radius of Earth: $6.4\times {10}^{6} m$ Gravitational acceleration on Earth $=10 m{s}^{-2}]$

A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T. Considering only translational and rotational modes, the total internal energy of the system is

Two rods $A$ and $B$ of identical dimensions are at temperature $30^{\circ} \mathrm{C}$. If $\mathrm{A}$ is heated upto $180^{\circ} \mathrm{C}$ and $\mathrm{B}$ upto $\mathrm{T}^{\circ} \mathrm{C},$ then the new lengths are the same. If the ratio of the coefficients of linear expansion of $\mathrm{A}$ and $\mathrm{B}$ is $4: 3,$ then the value of $T$ is

In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation $\mathrm{VT}=\mathrm{K},$ where $\mathrm{K}$ is a constant. In this process the temperature of the gas is increased by $\Delta \mathrm{T}$. The amount of heat absorbed by gas is (R is gas constant):

$2\mathrm{kg}$ of a monoatomic gas is at a pressure of $4\times {10}^{4}N{m}^{-2}$. The density of the gas is $8 \mathrm{kg}{m}^{-3}$. What is the order of energy of the gas due to its thermal motion?

When $100 \mathrm{~g}$ of a liquid $\mathrm{A}$ at $100^{\circ} \mathrm{C}$ is added to $50 \mathrm{~g}$ of a liquid $B$ at temperature $75^{\circ} \mathrm{C}$, the temperature of the mixture becomes $90^{\circ} \mathrm{C}$. The temperature of the mixture, if $100 \mathrm{~g}$ of liquid $A$ at $100^{\circ} \mathrm{C}$ is added to $50 \mathrm{~g}$ of liquid $\mathrm{B}$ at $50^{\circ} \mathrm{C}$, will be:

A thermally insulated vessel contains $150 g$ of water at $0^{\circ}C$ . Then the air from the vessel is pumped out adiabatically. A fraction of water turns into ice and the rest evaporates at $0^{\circ}C$ itself. The mass of evaporated water will be closest to: (Latent heat of vaporization of water $=2.10\times {10}^{6} J k{g}^{-1}$ and Latent heat of Fusion of water $=3.36\times {10}^{5} J k{g}^{-1}$ )

A rigid diatomic ideal gas undergoes an adiabatic process at room temperature. The relation between temperature and volume for this process is $\mathrm{TV}^{\mathrm{x}}=$ constant, then $\mathrm{x}$ is:

An unknown metal of mass $192 g$ heated to a temperature of $100^{\circ}C$ was immersed into a brass calorimeter of mass $128 g$ containing $240 g$ of water at a temperature of ${8.4}^{o}C.$ Calculate the specific heat of the unknown metal if water temperature stabilizes at ${21.5}^{o}C.$ $($ Specific heat of brass is $394J k{g}^{-1}{K}^{-1})$

One mole of an ideal gas passes through a process where pressure and volume obey the relation $P={P}_{o}[1-\frac{1}{2}{(\frac{{V}_{o}}{V})}^{2}]$ . Here ${P}_{o}$ and ${V}_{o}$ are constants. Calculate the change in the temperature of the gas if its volume changes from ${V}_{o}$ to $2{V}_{o}$ .

An ideal gas occupies a volume of $2{m}^{3}$ at a pressure of $3\times {10}^{6}Pa.$ The energy of the gas is:

$1\mathrm{kg}$ of water, at $20 ^{\circ}C$ is heated in an electric kettle whose heating element has a mean (temperature averaged) resistance of $20 \Omega .$ The rms voltage in the mains is $200 V$. Ignoring heat loss from the kettle, time taken for water to evaporate fully is close to [ Specific heat of water $=4200 J{\mathrm{kg}}^{-1}^{\circ}{C}^{-1}$ Latent heat of water $=2260 kJk{g}^{-1}$ ]

$n$ moles of an ideal gas with constant volume heat capacity ${C}_{v}$ undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is:

A $25\times {10}^{-3} {m}^{3}$ volume cylinder is filled with $1$ mol of ${O}_{2}$ gas at room temperature $(300 K)$ . The molecular diameter of ${O}_{2}$ , and its root mean square speed, are found to be $0.3 nm$ and $200 m/s$ , respectively. What is the average collision rate (per second) for an ${O}_{2}$ molecule?

A heat source at $T={10}^{3}K$ is connected to another heat reservoir at $T={10}^{2}K$ by a copper slab which is $1 m$ thick. Given that the thermal conductivity of copper is $0.1 W{K}^{-1} {m}^{-1}$, the energy flux through it in the steady-state is:

At $40^{\circ}C,$ a brass wire of $1 mm$ radius is hung from the ceiling. A small mass, $M$ is hung from the free end of the wire. When the wire is cooled down from $40^{\circ}C$ to $20^{\circ}C$ it regains its original length of $0.2 m.$ The value of $M$ is close to: (Coefficient of linear expansion and Young’s modulus of brass are ${10}^{-5}{/}^{^{\circ}}C$ and ${10}^{11} N/{m}^{2},$ respectively; $g=10 m {s}^{-2}$ )

A uniform cylindrical rod of length $L$ and radius r, is made from a material whose Young’s modulus of Elasticity equals $Y.$ When this rod is heated by temperature T and simultaneously subjected to a net longutudinal compressional force F, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equal to:

The number density of molecules of a gas depends on their distance r from the origin as, $n(r)={n}_{0}{e}^{-\alpha {r}^{4}}.$ Then the numer of molecules is proportional to:

Temperature difference of ${120}^{o}C$ is maintained between two ends of a uniform rod $AB$ of length $2L.$ Another bent rod $PQ,$ of same cross-section as $AB$ and length $\frac{3L}{2},$ is connected across $AB$ (See figure). In steady state, temperature difference between $P$ and $Q$ will be close to: