JEE Main 2020 — Thermodynamics

A bakelite beaker has volume capacity of $500\mathrm{cc}$ at $30^{\circ}C$. When it is partially filled with ${V}_{m}$ volume (at $30^{\circ}C$) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If ${\gamma }_{\text{bester }}=6\times {10}^{-6}0^{\circ}{C}^{-1},$ where $\gamma$ is the coefficient of volume expansion, then ${V}_{m}$ (in $\mathrm{cc}$ ) is close to $\ldots$

A balloon filled with helium $(32^{\circ}C \mathrm{and} 1.7\mathrm{atm})$ bursts. Immediately after wards the expansion of helium can be considered as :

A bullet of mass $5\mathrm{gram}$, travelling with a speed of $210m{s}^{-1}$ strikes a fixed wooden target. One half of its kinetic energy is converted into heat in the wood. The rise of temperature of the bullet if the specific heat of its material is $0.030{(\mathrm{gram}^{\circ}C)}^{-1}(1\mathrm{calorie}=4.2\times {10}^{7}\mathrm{ergs})$ close to :

A calorimeter of water equivalent $20g$ contains $180g$ of water at $25^{\circ}C$. $m''$' grams of steam at $100^{\circ}C$ is mixed in it till the temperature of the mixture is $31^{\circ}C$. The value of $m''$ is close to (Latent heat of water$=540\mathrm{cal}{g}^{-1}$, specific heat of water$=1\mathrm{cal}{g}^{-1^{\circ}}{C}^{-1}$)

A closed vessel contains $0.1$ mole of a monoatomic ideal gas at $200\text{ K}$. If $0.05$ mole of the same gas at $400K$ is added to it, the final equilibrium temperature (in $\text{K}$) of the gas in the vessel will be close to ______

A gas mixture consists of $3$ moles of oxygen and $5$ moles of argon at temperature $T$. Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of $RT$) of the mixutre is :

A litre of dry air at STP expands adiabatically to a volume of $3$ litres. If $\gamma =1.40,$ the work done by air is: $({3}^{1.4}=4.6555)$ [Take air to be an ideal gas]

A non-isotropic solid metal cube has coefficients of linear expansion as: $5\times { 10}^{ -5} / {}^{\text{o}} \text{C}$ along the x-axis and $5\times { 10}^{ -6} / {}^{\text{o}} \text{C}$ along the y and the z-axis. If the coefficient of volume expansion of the solid is $\text{C}\times { 10}^{ -6} / {}^{\text{o}} \text{C}$ then the value of $C$ is _____________

A thermodynamic cycle $xyzx$ is shown on a $V$ - $T$ diagram.  The $P$ - $V$ diagram that best describes this cycle is: (Diagrams are schematic and not to scale)

An engine takes in $5$ moles of air at $20^{\circ}C$ and $1\mathrm{atm}$, and compresses it adiabatically to $1/{10}^{\mathrm{th}}$ of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be $\mathrm{XkJ}$. The value of $X$ to the nearest integer is:

An ideal gas in a closed container is slowly heated. As its temperature increases, which of the following statements are true ? (A) the mean free path of the molecules decreases (B) the mean collision time between the molecules decreases. (C) the mean free path remains unchanged. (D) the mean collision time relations unchanged.

Assuming the nitrogen molecule is moving with $r.m.s$. velocity at $400K$, the de-Broglie wave length of nitrogen molecule is close to : (Given : nitrogen molecule weight : $4.64\times {10}^{-26} \mathrm{kg}$, Boltzman constant : $1.38\times {10}^{-23} J{K}^{-1}$, Planck constant : $6.63\times {10}^{-34} Js$)

Consider a mixture of $n$ moles of helium gas and $2n$ moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its $\frac{{C}_{P}}{{C}_{V}}$ value will be:

Consider two ideal diatomic gases $A$ and $B$ at some temperature $T$ . Molecules of the gas $A$ are rigid, and have a mass $m$ . Molecules of the gas $B$ have an additional vibrational mode and have a mass $\frac{m}{4}$ . The ratio of the specific heats ${({C}_{V}^{})}_{A}\text{and }{({C}_{V}^{})}_{B}$ of gas $A$ and $B,$ respectively is:

$M$ grams of steam at ${100}^{o}C$ is mixed with $200g$ of ice at its melting point in a thermally insulated container. If it produces liquid water at ${40}^{o}C$ [heat of vaporization of water is $540cal/g$ and heat of fusion of ice is $80cal/g$ ], the value of $M$ is ________

In a dilute gas at pressure $P$ and temperature '$t$', the time between successive collision of a molecule varies with $T$ as :

In an adiabatic process, the density of a diatomic gas becomes $32$$n$ times its initial value. The final pressure of the gas is found to be $n$ times the initial pressure. The value of $n$ is:

Initially a gas of diatomic molecules is contained in a cylinder of volume ${V}_{1}$ at a pressure ${P}_{1}$ and temperature $250K$. Assuming that $25%$ of the molecules get dissociated causing a change in number of moles. The pressure of the resulting gas at temperature $2000K,$ when contained in a volume $2{V}_{1}$ is given by ${P}_{2}$. The ratio ${P}_{2}/{P}_{1}$ is -

 Consider a gas of triatomic molecules. The molecules are assumed to be triangular and made of massless rigid rods whose vertices are occupied by atoms. The internal energy of a mole of the gas at temperature T is:

Two moles of an ideal monoatomic gas occupy a volume 2V at temperature 300 K...

Match the $\frac{{\text{C}}_{\text{p}}}{{\text{C}}_{\text{v}}}$ ratio for ideal gases with different type of molecules: <table class="pyq-table"><tbody><tr><td>Molecule Type</td><td>${\text{C}}_{\text{p}}/{\text{C}}_{\text{v}}$</td></tr><tr><td>(A) Monoatomic</td><td>(I) $7/5$</td></tr><tr><td>(B) Diatomic rigid molecules</td><td>(II) $9/7$</td></tr><tr><td>(C) Diatomic non-rigid molecules</td><td>(III) $4/3$</td></tr><tr><td>(D) Triatomic rigid molecules</td><td>(IV) $5/3$</td></tr></tbody></table>

Match the thermodynamics processes taking place in a system with the correct conditions. In the table : $\Delta Q$is the heat supplied, $\Delta W$ is the work done and $\Delta U$ is change in internal energy of the system. <table class="pyq-table"><tbody><tr><td></td><td>Process</td><td></td><td>Condition</td></tr><tr><td>(I)</td><td>Adiabatic</td><td>(A)</td><td>$\Delta W=0$</td></tr><tr><td>(II)</td><td>Isothermal</td><td>(B)</td><td>$\Delta Q=0$</td></tr><tr><td>(III)</td><td>Isochoric</td><td>(C)</td><td>$\Delta U\neq 0,\Delta W\neq 0,\Delta Q\neq 0$</td></tr><tr><td>(IV)</td><td>Isobaric</td><td>(D)</td><td>$\Delta U=0$</td></tr></tbody></table>

Molecules of an ideal gas are known to have three translational degrees of freedom. The gas is maintained at a temperature of T. The total internal energy, U of a mole of this gas, and the value of $\gamma =(\frac{{C}_{p}}{{C}_{v}})$ are given, respectively, by

Nitrogen gas is at $300^{\circ}C$ temperature. The temperature (in $K$ ) at which the rms speed of a ${H}_{2}$ molecule would be equal to the rms speed of a nitrogen molecule, is $\ldots \ldots \ldots .$ (Molar mass of ${N}_{2}$ gas $28g$ ).

Number of molecules in a volume of $4{\mathrm{cm}}^{3}$ of a perfect monoatomic gas at some temperature T and at a pressure of $2\mathrm{cm}$ of mercury is close to? (Given, mean kinetic energy of a molecule (at T) is $4\times {10}^{-14}\mathrm{erg},g=980\mathrm{cm}{s}^{-2}$ density of mercury $=13.6g{\mathrm{cm}}^{-3}$)

Starting at temperature $300K,$ one mole of an ideal diatomic gas $(\gamma =1.4)$ is first compressed adiabatically from volume ${V}_{1}$ to ${V}_{2}=\frac{{V}_{1}}{16}.$ It is then allowed to expand isobarically to volume $2{V}_{2}$ . If all the processes are the quasi-static then the final temperature of the gas (in ${}^{o}K$ ) is (to the nearest integer) ___________.

The change in the magnitude of the volume of an ideal gas when a small additional pressure $\Delta P$ is applied at a constant temperature, is the same as the change when the temperature is reduced by a small quantity $\Delta T$ at constant pressure. The initial temperature and pressure of the gas were $300K$and $2\mathrm{atm}$ respectively. If $|\Delta T|=C|\Delta P|$ then value of $C$ in $(K/\mathrm{atm})$ is __________

The plot that depicts the behavior of the mean free time $\tau$ (time between two successive collisions) for the molecules of an ideal gas, as a function of temperature $(T),$ qualitatively, is: (Graphs are schematic and not drawn to scale)

The specific heat of water $=4200\text{ J}$ ${\text{kg}}^{–1}{\text{ K}}^{–1}$ and the latent heat of ice $=3.4\times {10}^{5}\text{ J}$ k${\text{g}}^{–1}$. $100$ grams of ice at $0{ }^{\text{o}}\text{C}$ is placed in $200\text{ g}$ of water at $25{ }^{\text{o}}\text{C}$. The amount of ice that will melt as the temperature of water reaches $0{ }^{\text{o}}\text{C}$ is close to (in grams)

Three containers ${C}_{1},{C}_{2}$ and ${C}_{3}$ have water at different temperatures. The table below shows the final temperature $T$ when different amounts of water (given in liters) are taken from each container and mixed (assume no loss of heat during the process) <table class="pyq-table"><tbody><tr><th>${C}_{1}$</th><th>${C}_{2}$</th><th>${C}_{3}$</th><th>$T$</th></tr><tr><td>$1l$</td><td>$2l$</td><td>$--$</td><td>${60}^{o}C$</td></tr><tr><td>$--$</td><td>$1l$</td><td>$2l$</td><td>${30}^{o}C$</td></tr><tr><td>$2l$</td><td>$--$</td><td>$1l$</td><td>${60}^{o}C$</td></tr><tr><td>$1l$</td><td>$1l$</td><td>$1l$</td><td>$\theta$</td></tr></tbody></table> The value of $\theta$ (in ${}^{o}C$ to the nearest integer) is___________

Three different processes that can occur in an ideal monoatomic gas are shown in the $P$ vs $V$ diagram. The paths are labelled as $A\rightarrow B,A\rightarrow C$ and $A\rightarrow D$. The change in internal energies during these process are taken as ${E}_{AB},{E}_{AC}$ and ${E}_{AD}$ and the work done as ${W}_{AB},{W}_{AC}$ and ${W}_{AD}$. The correct relation between these parameters are: 

Three rods of identical cross-section and length are made of three different materials of thermal conductivity ${K}_{1},{K}_{2}$ and ${K}_{3}$, respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at $100^{\circ}C$ and the other at $0^{\circ}C$ (see figure). If the joints of the rod are at $70^{\circ}C$ and $20^{\circ}C$ in steady and there is no loss of energy from the surface of the rod, the correct relationship between ${K}_{1},{K}_{2}$ and ${K}_{3}$ is : 

To raise the temperature of a certain mass of gas by $50^{\circ}C$ at a constant pressure,$160$ calories of heat is required. When the same mass of gas is cooled by $100^{\circ}C$ at constant volume,$240$ calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)?

Two different wires having lengths ${L}_{1}$ and ${L}_{2}$ and respective temperature coefficient of linear expansion ${\alpha }_{1}$ and ${\alpha }_{2},$ are joined end-to-end. Then the effective temperature coefficient of linear expansion is :

Two gases - argon (atomic radius $0.07nm,$ atomic weight $40$ ) and xenon (atomic radius $0.1nm,$ atomic weight $140$ ) have the same number density and are at the same temperature. The ratio of their respective mean free times is closest to:

Two moles of an ideal gas, with $\frac{{C}_{P}}{{C}_{V}}=\frac{5}{3}$, are mixed with three moles of another ideal gas $\frac{{C}_{P}}{{C}_{V}}=\frac{4}{3}$. The value of $\frac{{C}_{P}}{{C}_{V}}$ for the mixture is

Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently, the mean collision time between the gas molecule changes from ${\tau }_{1}$ to ${\tau }_{2}$ . If $\frac{{C}_{P}}{{C}_{v}}=\gamma$ for this gas then a good estimate for $\frac{{\tau }_{2}}{{\tau }_{1}}$ is given by

When the temperature of a metal wire is increased from $0ºC\mathrm{to}10ºC$, its length increases by $0.02%$.The percentage change in its mass density will be closed to:

Which of the following is an equivalent cyclic process corresponding to the thermodynamic cyclic given in the figure? Where, $1\rightarrow 2$ is adiabatic. (Graphs are schematic and are not to scale)