Kinetic Theory of Gases

As shown in the figure, an insulated container is fitted with a thermally conducting but immovable partition $(P_1)$ and a freely movable but thermally insulated piston $(P_2)$. The partition $P_1$ with thermal conductivity $K$, cross sectional area $A$ and width $x$ divides the container into two sections, $S_1$ and $S_2$, each containing one mole of a monoatomic gas. The piston $P_2$ moves freely such that the gas in $S_2$ is always at the atmospheric pressure. Initially, the difference between the temperatures of $S_1$ and $S_2$ is $\Delta T_0$. The time it takes for the temperature difference to become $\dfrac{\Delta T_0}2$ is $nxR/KA$, where $R$ is the universal gas constant. The value of $n$ is:
[Given: $\ln2\approx0.7$]

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of $\Delta Q:\Delta U:\Delta W$ is ________.

A mixture of carbon dioxide and oxygen has volume 8310 cm³, temperature 300 K, pressure 100 kPa and mass 13.2 g. The number of moles of carbon dioxide and oxygen gases in the mixture respectively are ________.
(Assume both carbon dioxide and oxygen gases behave like ideal gases) [$R=8.31$ J/mol.K]

5 moles of unknown gas is heated at constant volume from 10 °C to 20 °C. The molar specific heat of this gas at constant pressure $c_p=8$ cal/mol.°C and $R=8.36$ J/mol.°C. The change in the internal energy of the gas is ________ calorie.

One gas of $n_1$ mole of molecules at temperature $T_1$, volume $V_1$, and pressure $P_1$, and another gas of $n_2$ mole of molecules at temperature $T_2$, volume $V_2$, and pressure $P_2$, are mixed resulting in pressure $P$ and volume $V$ of the mixture. The temperature of the mixture is ________.

An ideal gas undergoes a process maintaining relation between pressure ($P$) and volume ($V$) as $P=P_0\left(1+\left(\dfrac{V_0}V\right)^2\right)^{-1}$, where $P_0$ and $V_0$ are constants. If two samples $A$ and $B$ (two moles each) with initial volumes $V_0$ and $3V_0$ respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, $T_B-T_A$ is ________.
($R$ = gas constant)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion A: If the average kinetic energy of $\ce{H2}$ and $\ce{O2}$ molecules, kept in two different sized containers are same, then their temperatures will be same.

Reason R: The r.m.s. speed of $\ce{H2}$ and $\ce{O2}$ molecules are same at same temperature.

Choose the correct answer from the options given below

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Statement I: Change in internal energy of a system containing $n$ mole of ideal gas can be written as $\Delta U=nC_v\left(T_f-T_i\right)=\dfrac{nR}{\gamma-1}\left(T_f-T_i\right)$, where $\gamma=\dfrac{C_p}{C_v}$, $T_i$ = initial temperature, $T_f$ = final temperature.

Statement II: Relation between degree of freedom $f$ and $\gamma\ (=C_p/C_v)$ is $\left(\gamma=1+\dfrac2f\right)$

Choose the correct answer from the options given below

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3$ = constant. The coefficient of volume expansion of the gas is ________.

Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure 90 kPa and temperature 400 K. Keeping the temperature of one vessel constant at 400 K the second vessel temperature is raised to 500 K. The final pressure in the vessels is ________ kPa.

If 2 mole of an ideal monoatomic gas at temperature $T$, is mixed with 6 mole of another ideal monoatomic gas at temperature $2T$ then the temperature of mixture is :

One mole of diatomic gas having rotational modes only is kept in a cylinder with a piston system. The cross-section area of the cylinder is 4 cm². The gas is heated slowly to raise the temperature by 1.2 °C during which the piston moves by 25 mm. The amount of heat supplied to the gas is ________ J.

(Atmospheric pressure = 100 kPa, R = 8.3 J/mol. K) (Neglect mass of the piston)

An ideal monatomic gas of $n$ moles is taken through a cycle $WXYZW$ consisting of consecutive adiabatic and isobaric quasi-static processes, as shown in the schematic $V$-$T$ diagram. The volume of the gas at $W$, $X$ and $Y$ points are, $64\ \mathrm{cm^3}$, $125\ \mathrm{cm^3}$ and $250\ \mathrm{cm^3}$, respectively. If the absolute temperature of the gas $T_W$ at the point $W$ is such that $nRT_W = 1$ J ($R$ is the universal gas constant), then the amount of heat absorbed (in J) by the gas along the path $XY$ is _____.

The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\dfrac32$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\dfrac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\dfrac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \dfrac{kL}{A}\alpha$, then the value of $\alpha$ is _____.

A spherical soap bubble inside an air chamber at pressure $P_0 = 10^5\,\mathrm{Pa}$ has a certain radius so that the excess pressure inside the bubble is $\Delta P = 144\,\mathrm{Pa}$. Now, the chamber pressure is reduced to $8P_0/27$ so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure $\Delta P$ in both the cases to be much smaller than the chamber pressure. The new excess pressure $\Delta P$ in Pa is _____.

A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas ($\gamma = 5/3$) and one mole of an ideal diatomic gas ($\gamma = 7/5$). Here, $\gamma$ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. The change in its internal energy is _____ Joule.

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $\mathrm v_n$. At a fixed temperature and pressure, which of the following is the correct option?

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$ to $\dfrac{100}{101}V_0$ with the movable piston. Considering air as an ideal gas, the height $(h)$ of the liquid column in the capillary above the liquid level in cm is _____.

[Given: Surface tension of the liquid is $0.075\ \mathrm{N\,m^{-1}}$, atmospheric pressure is $10^5\ \mathrm{N\,m^{-2}}$, acceleration due to gravity $(g)$ is $10\ \mathrm{m\,s^{-2}}$, density of the liquid is $10^3\ \mathrm{kg\,m^{-3}}$ and contact angle of capillary surface with the liquid is zero]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm\,s^{-1}}$.

When the chimney is closed using a cap at the top, a pressure difference $\Delta P$ develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $\Delta P$ is _____ $\mathrm{N\,m^{-2}}$.