Q261JEE Main 2020MCQ

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

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Q262JEE Main 2020NAT

Q263JEE Main 2020MCQ

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

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📖 Explanation

${n}=\frac{{PV}}{ {RT}}, \frac{3}{2} \ {kT}=4 \\times 10^{-14}$ ${N}=\frac{{PV}}{ {RT}} \times {Na}$ $= \frac{2 \\times 13.6 \\times 980 \\times 4}{ \frac{8}{3} \\times 10^{-14}}$ $=3.99 \\times 10^{18}$

Q264JEE Main 2020MCQ

Consider a mixture of n moles of helium gas and 2 n moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its $C _P/C_V$ value will be :

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📖 Explanation

$C_P/C_V=\frac{n_1C_{P_1}+n_2C_{P_2}}{n_1C_{V_1}+n_2C_{V_2}}$ $=\frac{n\times {5R}/2+2n\times {7R}/2}{n\times {3R}/2+2n\times {5R}/2}=19/13$

Q265JEE Main 2020MCQ

A balloon filled with helium $(32°{C}$ and1.7 atm.) bursts. Immediately afterwards the expansion of helium can be considered as:

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Q266JEE Main 2020MCQ

Three rods of identical cross-section and lengths 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 $100°{C}$ and the other at $0°{C}$ (see figure). If the joints of the rod are at $70° {C}$ and $20°{C}$ in steady state 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 :

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📖 Explanation

Rods are identical have same length $(l)$ and area of cross-section (A) Combination are in series, so heat current is same for all Rods $(\frac{\DeltaQ}{\Deltat})_{{\text{AB}}}=(\frac{\DeltaQ}{\Deltat})_{\text{BC}}$ $=(\frac{\DeltaQ}{\Deltat})_{\text{CD}}=$ Heat current ${(100-70) {\text{K}}_{1} {\text{A}}}/{l}$ $={(70-20) {\text{K}}_{2} {\text{A}}}/{l}$ $={(20-0) {\text{K}}_{3} {\text{A}}}/{l}$ $30 {\text{K}}_{1}=50 {\text{K}}_{2}=20 {\text{K}}_{3}$ $3 {\text{K}}_{1}=2 {\text{K}}_{3}$ $\frac{K_{1}}{K_{3}}=\frac{2}{3}=2: 3$ $5 {\text{K}}_{2}=2 {\text{K}}_{3}$ ${{\text{K}}_{2}}{{\text{K}}_{3}}=\frac{2}{5}=2: 5$

Q267JEE Main 2020MCQ

When the temperature of a metal wire isincreased from $0°{C}$ to $10°{C},$ its length increasesby $0.02 \% .$ The percentage change in its massdensity will be closest to:

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📖 Explanation

Given $\frac{\Delta{L}}{ {L}}=0.02 \%$ $\therefore \Delta{L}= {L} \alpha \Delta{T}$ $\Rightarrow \frac{\Delta{L}}{ {L}}=\alpha \Delta{T}=0.02 \%$ $\therefore \beta =2 \alpha$ (Areal coefficient of expansion) $\Rightarrow \beta \Delta{T}=2 \alpha \Delta{T}=0.04 \%$ Volume $=$ Area $\\times$ Length Density $(\rho )= \frac{\text{Mass }}{\text{Volume}}$ $= \frac{\text{ Mass }}{\text{ Area} \\times \text{ Length }}=\frac{{M}}{ {AL}}$ $\Rightarrow \frac{\Delta \rho }{\rho }= \frac{\DeltaM}{ {M}}- \frac{\Delta{A}}{ {A}}- \frac{\Delta{L}}{ {L}}$ ( Mass remains constant $)$ $\Rightarrow ( \frac{\Delta \rho }{\rho } )= \frac{\Delta{A}}{ {A}}+ \frac{\Delta{L}}{ {L}}$ $=\beta \Delta{T}+\alpha \Delta{T}$ $=0.04 \%+0.02 \%$ $=0.06 \%$

Q268JEE Main 2020NAT

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

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📖 Explanation

$P V=n R T$ $P_{1} V_{1}=n R 250$ $P_{2} (2 V_{1} )= \frac{5 n}{4} R \\times 2000$ Divide $\frac{P_{1}}{2 P_{2}}= \frac{4 \\times 250}{5\times 2000}$ $\frac{P_{1}}{P_{2}}= \frac{1}{5}$ $\frac{P_{2}}{P_{1}}=5$

Q269JEE Main 2020MCQ

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 remainsunchanged.

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📖 Explanation

The mean free path of molecules of an ideal gas is given as: $\lambda=\frac{{V}}{\\sqrt{2} \pi {d}^{2} {N}}$ where : $\ {V}=$ Volume of container $\ {N}=$ No of molecules Hence with increasing temp since volume of container does not change (closed container), so mean free path is unchanged. Average collision time $= {\text{ mean free path }}/{ {V}_{ {av}}}= \frac{\lambda}{(\text{avg speed of molecules})}$ $\because$ avg speed $\alpha \sqrt{ {T}}$ $\therefore$ Avg coll. time $\alpha {1}/{\\sqrt{ {T}}}$ Hence with increase in temperature the average collision time decreases.

Q270JEE Main 2020NAT

A Carnot engine operates between two reservoirs of temperatures 900 K and 300 K. The engine performs 1200 J of work per cycle. The heat energy (in J) delivered by the engine to the low temperature reservoir, in a cycle, is____. [7-Jan-2020 Shift 1]

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📖 Explanation

for carnot engine $Q_1/Q_2=T_1/T_2$ ${Q+1200}/Q=900/300$ $Q + 1200 = 3Q$ $Q = 600 J.$

Q271JEE Main 2019MCQ

The specific heats, $C_P$ and $C_V$ of a gas of diatomic molecules, A, are given (in units of $Jmol^{-1} K ^{-1}$ ) 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 :

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📖 Explanation

$l=2 m$ $3^{\ { rd }}$ harmonic mode $\Rightarrow f=\ \frac{3}{2 l} \\sqrt{\ \frac{T}{\mu}}$ $f=3 f_{0}$ $f_{0}=\ \frac{f}{3}=80 H z$

Q272JEE Main 2019MCQ

The gas mixture constists of 3 moles of oxygen and 5 moles of argon at temperature T. Considering only translational and rotational modes, the total inernal energy of the system is:

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📖 Explanation

U = $f_1/n n_1$ RT + $f_2/2 n_2$ RT = $5/2$ (3RT) + $3/2$ × 5RT U = 15RT

Q273JEE Main 2019MCQ

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 thermalconductivity of the system for heat flowing along the length of the cylinder is:

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📖 Explanation

$K_{\text{eq}}$ = $\frac{K_1A_1+K_2A_2}{A_1+A_2}$ = $\frac{K_1(\pi R^2)+K_2(3\pi R^2)}{4 \pi R^2}$ = ${K_1+3K_2}/4$

Q274JEE Main 2019MCQ

A mixture of 2 moles of helium gas (atomic mass= 4 u), and 1 mole of argon gas (atomic mass= 40 u) is kept at 300 K in a container. The ratio of their rms speeds $[\frac{V_{rms}(\text{helium})}{V_{rms}(\text{argon})}]$ , is close to [9-Jan-2019 Shift 1]

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📖 Explanation

$\frac{V_{rms}(He)}{V_{rms}(Ar)}$ = $\sqrt{M_{Ar}/M_{He}}$ = $\sqrt{{40}/4}$ = 3.16

Q275JEE Main 2019MCQ

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)

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📖 Explanation

$(C_{V} )_{ {mix}}= {n_{1} C_{V_{1}}+n_{2} C_{V_{2}}}/{n_{1}+n_{2}}$ $= \frac{2\times \frac{3 R}{2}+3\times \frac{5 R}{2}}{2+3}$ $(C_{V}\ )_{ {mix}}= \frac{R}{2} ( \frac{6+15}{5} )$ $(C_{V}\ )_{ {mix}} = \frac{21 R}{10}= \frac{21\times 8.314}{10}$ = 17.4 J/mol K

Q276JEE Main 2019MCQ

Two materials having coefficients of thermal conductivity '3K' and 'K' and thickness 'd' and '3d', respectively, are joined to forma 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:

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📖 Explanation

$H_{1}=H_{2}$ $\frac{(3 K) A\ (\theta _{2}- \theta )}{d}=\ \frac{K A\ (\theta -\theta _{1}\ )}{3 d}$ $9 \theta _{2}-9 \theta =\theta -\theta _{1}$ $\ \Rightarrow \theta =\ \frac{9}{10} \theta _{2}+\ \frac{1}{10} \theta _{1}$

Q277JEE Main 2019MCQ

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

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📖 Explanation

$\frac{-d T}{d t}= \frac{\sigma A (T^{4}-T_{0}^{4} )}{m s}= \frac{\sigma A (T^{4}-T_{0}^{4} )}{V(\rho s)}$ ρs is more for B.

Q278JEE Main 2019MCQ

An ideal gas occupies a volume of 2m3 at a pressure of 3 × 106 Pa. The energy of the gas is:

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📖 Explanation

Energy = $1/2$ nRT = $f/2$ PV = $f/2 (3\times 10^6)$ (2) = f × 3 × $10^6$ Considering gas is monoatomic i.e. f = 3 E. = 9 × $10^6$ J

Q279JEE Main 2019MCQ

Three Carnot engines operate in series between a heat source at a temperature $T_1$ and a heat sink at temperature $T_4$ (see figure). There are two other reservoirs at temperature $T_2$ , and $T_3$ , as shown, with $T_2$ > $T_2$ > $T_3$ > $T_4$ . The three engines are equallyefficient if:

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📖 Explanation

$t_1$ = 1 - $T_2/T_1$ = 1 - $T_2/T_2$ = 1 - $T_4/T_3$ \Rightarrow $T_2/T_1$ = $T_3/T_4$ = $T_4/T_3$ \Rightarrow $T_2$ = $\sqrt{T_1T_3}$ = $\sqrt{T_1\sqrt{T_2T_4}}$ $T_3$ = $\sqrt{T_2T_4}$ $T_2^{3/2}$ = $\sqrt{T_1^{1/2}} T_4^{1/4}$ $T_2$ = $T_1^{2/3}T_4^{1/3}$

Q280JEE Main 2019MCQ

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:

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📖 Explanation

Given $Q=n C_{v} \ \Delta T$ $\RightarrowQ^{\1 }=n C_{p} \Delta T=?$ $\Rightarrow Q^{\1}=\ \frac{C_{p}}{C_{v}} Q=\ \frac{7}{5} Q$