A Carnot engine $(\mathrm{E})$ is working between two temperatures 473 K and 273 K . In a new system two engines - engine $E_1$ works between 473 K to 373 K and engine $E_2$ works between 373 K to 273 K . If ${\eta }_{12},{\eta }_1$ and ${\eta }_2$ are the efficiencies of the engines $E,E_1$ and $E_2$, respectively, then

A monoatomic gas having $\gamma =\frac{5}{3}$ is stored in a thermally insulated container and the gas is suddenly compressed to ${\left(\frac{1}{8}\right)}^{\text{th}}$ of its initial volume. The ratio of final pressure and initial pressure is: ($\gamma$ is the ratio of specific heats of the gas at constant pressure and at constant volume)

Water falls from a height of 200 m into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool.(Take g = 10 m/s2, specific heat of water = 4200 J/(kg K))

In an adiabatic process, which of the following statements is true?

The equation for real gas is given by $\left(\mathrm{P}+\frac{\mathrm{a}}{{\mathrm{V}}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where $\mathrm{P},\mathrm{V},\mathrm{T}$ and R are the pressure, volume, temperature and gas constant, respectively. The dimension of ${\mathrm{ab}}^{-2}$ is equivalent to that of :

Match the List I with List II .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List - I List - II (A) Triatomic rigid gas (I) $\frac{C_p}{C_v}=\frac{5}{3}$ (B) Diatomic non-rigid gas (II) $\frac{C_p}{C_v}=\frac{7}{5}$ (C) Monoatomic gas (III) $\frac{C_p}{C_v}=\frac{4}{3}$ (D) Diatomic rigid gas (IV) $\frac{C_p}{C_v}=\frac{9}{7}$ Choose the correct answer from the options given below:

$$\text{ Match List - I with List - II. }$$ $$\begin{array}{lll}& \text{ List - I }& List-II\\ & & \\ \text{ (A) }& \text{ Heat capacity of body }& \text{ (I) }\mathrm{J}{\mathrm{kg}}^{-1}\\ \text{ (B) }& \text{ Specific heat capacity of body }& \text{ (II) }\mathrm{J}{\mathrm{K}}^{-1}\\ \text{ (C) }& \text{ Latent heat }& \text{ (III) }\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\\ \text{ (D) }& \text{ Thermal conductivity }& \text{ (IV) }\mathrm{J}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}\end{array}$$ $$\text{ Choose the correct answer from the options given below : }$$

Identify the characteristics of an adiabatic process in a monoatomic gas. (A) Internal energy is constant. (B) Work done in the process is equal to the change in internal energy. (C) The product of temperature and volume is a constant. (D) The product of pressure and volume is a constant. (E) The work done to change the temperature from ${\mathrm{T}}_1$ to ${\mathrm{T}}_2$ is proportional to $\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)$. Choose the correct answer from the options given below :

The helium and argon are put in the flask at the same room temperature (300 K). The ratio of average kinetic energies (per molecule) of helium and argon is: (Give: Molar mass of helium = 4 g/mol, Molar mass of argon = 40 g/mol)

A gas is kept in a container having walls which are thermally non-conducting. Initially the gas has a volume of $800{\mathrm{cm}}^3$ and temperature $27^{\circ }\mathrm{C}$. The change in temperature when the gas is adiabatically compressed to $200{\mathrm{cm}}^3$ is: (Take $\gamma =1.5;\gamma$ is the ratio of specific heats at constant pressure and at constant volume)

During the melting of a slab of ice at 273 K at atmospheric pressure :

A piston of mass $M$ is hung from a massless spring whose restoring force law goes as $F=-k{x}^3$, where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' $n$ ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height ${\mathrm{L}}_0$ to ${\mathrm{L}}_1$, the total energy delivered by the filament is:(Assume spring to be in its natural length before heating)

Match List - I with List - II. List - I List - II (A) Isothermal (I) ΔW (work done) = 0 (B) Adiabatic (II) ΔQ (supplied heat) = 0 (C) Isobaric (III) ΔU (change in internal energy) ≠ 0 (D) Isochoric (IV) ΔU = 0 Choose the correct answer from the options given below :

Consider a rectangular sheet of solid material of length $l=9\mathrm{cm}$ and width $\mathrm{d}=4\mathrm{cm}$. The coefficient of linear expansion is $\alpha =3.1\times 10^{-5}{\mathrm{K}}^{-1}$ at room temperature and one atmospheric pressure. The mass of sheet $m=0.1\mathrm{kg}$ and the specific heat capacity $C_{\mathrm{v}}=900\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$. If the amount of heat supplied to the material is $8.1\times 10^2\mathrm{J}$ then change in area of the rectangular sheet is :

There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K . If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K , at steady state the pressure in the vessels will be (in kPa ).

Match List - I with List - II. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List - I List - II (A) Isobaric (I) $\Delta Q=\Delta W$ (B) Isochoric (II) $\Delta Q=\Delta U$ (C) Adiabatic (III) $\Delta Q=$ zero (D) Isothermal (IV) $\Delta Q=\Delta U+P\Delta V$ $\Delta Q=$ Heat supplied $\Delta W=$ Work done by the system $\Delta \mathrm{U}=$ Change in internal energy $\mathrm{P}=$ Pressure of the system $\Delta \mathrm{V}=$ Change in volume of the system Choose the correct answer from the options given below :

Consider the sound wave travelling in ideal gases of $\mathrm{He},{\mathrm{CH}}_4$, and ${\mathrm{CO}}_2$. All the gases have the same ratio $\frac{P}{\rho }$, where $P$ is the pressure and $\rho$ is the density. The ratio of the speed of sound through the gases ${\mathrm{V}}_{\mathrm{He}}:{\mathrm{V}}_{{\mathrm{CH}}_4}:{\mathrm{V}}_{{\mathrm{CO}}_2}$ is given by

The mean free path and the average speed of oxygen molecules at 300 K and 1 atm are $3\times 10^{-7}\mathrm{m}$ and $600\mathrm{m}/\mathrm{s}$, respectively. Find the frequency of its collisions.

An ideal gas exists in a state with pressure $P_0$, volume $V_0$. It is isothermally expanded to 4 times of its initial volume $\left({\mathrm{V}}_0\right)$, then isobarically compressed to its original volume. Finally the system is heated isochorically to bring it to its initial state. The amount of heat exchanged in this process is

Pressure of an ideal gas, contained in a closed vessel, is increased by $0.4 \%$ when heated by $1^{\circ }\mathrm{C}$. Its initial temperature must be: