Kinetic Theory - Class 11 Part 2

A rigid, insulated container holds $N$ atoms of a monoatomic gas, each of mass $m$, at an initial temperature $T_0$. A catalyst is introduced, causing all atoms to undergo a fusion process where they pair up to form $\frac{N}{2}$ rigid diatomic molecules. Each fusion event (the formation of one molecule) releases a binding energy $\epsilon = 7 k_B T_0$, which is entirely absorbed by the gas. If $v_1$ is the initial root-mean-square speed of the monoatomic gas and $v_2$ is the final root-mean-square speed of the diatomic gas, what is the ratio $\frac{v_2}{v_1}$?

A macromolecule of mass $m$ with $f$ internal degrees of freedom is attached to a spring of force constant $k$ and undergoes weakly damped oscillations along the $x$-axis. It experiences a drag force $F_d = -b v$, where the damping coefficient $b$ is directly proportional to the molecule's instantaneous internal energy $U$, such that $b = \beta U$ ($\beta$ is a positive constant). Initially ($t=0$), the molecule is in thermal equilibrium with the surrounding fluid at temperature $T_0$, and is given an initial macroscopic oscillation energy $E_0$. Assuming the energy dissipated by the drag force is entirely absorbed by the molecule's internal degrees of freedom (with negligible heat transfer to the fluid), which of the following expressions gives the time $t_{1/2}$ required for the oscillation energy to decrease to $E_0/2$? (Assume $k_B$ is the Boltzmann constant).

A closed cylindrical container of height $H$ is filled with an ideal gas of molar mass $M$ at a constant temperature $T$. The container is placed in an elevator moving upwards with a constant acceleration $a$. Two identical small orifices are punctured simultaneously at the bottom ($y=0$) and the top ($y=H$) of the container, allowing the gas to effuse into a surrounding vacuum. Which of the following expressions represents the ratio of the initial mass leakage rate at the bottom orifice to that at the top orifice? (Assume $g$ is the acceleration due to gravity and $R$ is the universal gas constant)

As part of a thermodynamic cycle, one mole of a gas undergoes a reversible adiabatic expansion. The gas obeys the ideal gas equation of state $PV = RT$, but its internal energy is temperature-dependent and given by $U(T) = \frac{3}{2}RT + \frac{R}{2T_0}T^2$, where $R$ is the universal gas constant and $T_0$ is a positive reference temperature. During the expansion, the root-mean-square (RMS) speed of the gas molecules drops to exactly half of its initial value. If the expansion begins at a temperature $T_1 = 2T_0$, what is the expansion ratio $\frac{V_2}{V_1}$?

A large number of non-interacting electrons (mass $m$, charge $-e$) are constrained to move in separate, identical circular non-conducting rings of radius $r$ lying in the $xy$-plane. Initially, all electrons revolve counterclockwise, and their speeds follow the standard Maxwell-Boltzmann distribution of speeds for a 3D ideal gas at temperature $T$. A uniform time-varying magnetic field $\vec{B}(t) = B(t) \hat{k}$ is applied, increasing slowly from $0$ to $B_0$. Which of the following expressions represents the magnitude of the final average magnetic dipole moment per electron? (Here $k_B$ is the Boltzmann constant)

A large chamber is filled with an ideal gas of molecule mass $m$ and uniform number density $n$ at temperature $T$. A thin, flat plate of area $A$ and mass $M_p$ is placed inside the chamber. The chamber is then accelerated with a constant acceleration $a$ in a gravity-free space, in a direction normal to the plate. Assume the acceleration $a$ is small enough that the local gas density $n$ around the plate remains uniform. All collisions between the gas molecules and the plate are perfectly elastic. Eventually, the plate reaches a steady terminal velocity $u$ relative to the chamber ($u \ll \bar{v}$, where $\bar{v}$ is the mean speed of the gas molecules). Which of the following expressions gives the correct value of $u$? (Ignore any edge effects of the plate, and let $k$ be the Boltzmann constant).

Consider an intrinsic semiconductor with a band gap energy $E_g$. The charge carriers behave as a classical ideal gas, and their mean free path $\lambda$ is independent of temperature. The carrier number density varies with temperature $T$ as $n(T) = C T^{3/2} e^{-E_g / 2k_B T}$, where $C$ is a constant and $k_B$ is the Boltzmann constant. When a constant uniform electric field $E$ is applied, the current density is $J$. Assuming the effective mass of the carriers is constant and the drift velocity is governed by the Drude model, what is the ratio of the current density at temperature $T$ to the current density at temperature $2T$?

An ideal gas flows through a cylindrical pipe of diameter $D$ with a constant uniform velocity $v$ and at a constant pressure $P$. At an initial temperature $T_0$, the mean free path of the gas molecules is $\lambda_0$ and the Reynolds number of the flow is $Re_0$. The gas is then heated to a higher temperature such that the mean free path of the molecules becomes $4\lambda_0$. Assuming the flow velocity $v$ and pressure $P$ remain unchanged, what is the new Reynolds number of the flow? (Assume the gas molecules are hard spheres and follow the standard kinetic theory of gases).

A parallel beam of ideal gas atoms, each of mass $m$, strikes a freely movable, initially stationary flat wall of mass $M$ at a constant rate of $n$ atoms per second. The atoms move with a speed $v$ perpendicular to the wall. Assume all collisions between the atoms and the wall are perfectly elastic and independent. Let $F_{\text{true}}$ be the actual initial average force exerted on the wall. Let $F_{\text{KTG}}$ be the theoretical force calculated using the standard Kinetic Theory of Gases assumption (which treats the wall as infinitely massive and static). What is the value of the relative difference $\frac{F_{\text{KTG}} - F_{\text{true}}}{F_{\text{true}}}$?

One mole of a real gas obeys the modified equation of state $\left(P + \frac{a}{V^2}\right)V = RT$, where $a$ is a positive constant, $P$ is pressure, $V$ is molar volume, $T$ is absolute temperature, and $R$ is the universal gas constant. The gas undergoes a reversible isothermal expansion from an initial volume $V_0$ to a final volume $2V_0$ at a constant temperature $T_0$. Which of the following correctly represents the heat absorbed $Q$ and the work done $W$ by the gas during this process?

A vertical cylindrical tank of height $H$ contains an ideal gas of molar mass $M$ in hydrostatic equilibrium under a uniform gravitational field $g$. The temperature of the gas decreases linearly with height. The RMS speed of the gas molecules at the bottom ($y=0$) is $v_0$, and at the top ($y=H$) is $v_H$. The pressure at the top of the tank is $P_H$. A small orifice is opened at the bottom of the tank, allowing the gas to escape into an external environment maintained at pressure $P_H$. Assuming the efflux velocity is given by $v_e = \sqrt{\frac{2 \Delta P}{\rho_0}}$, where $\Delta P$ is the pressure difference across the orifice and $\rho_0$ is the density of the gas at the bottom, which of the following expressions correctly represents $v_e$?

Consider a cylindrical conductor of length $L$ and uniform cross-sectional area carrying a steady current. The conductor has a steady-state temperature profile $T(x) = T_0 \left(1 + \frac{3x}{L}\right)$ from $x=0$ to $x=L$. According to the classical Drude-Lorentz model, the free electrons behave as an ideal gas. Due to temperature-dependent scattering, the electron mean free path $\lambda$ is inversely proportional to the local absolute temperature $T$. Assuming the free electron density remains uniform, what is the ratio of the local electric field at $x=L$ to the electric field at $x=0$, $\frac{E(L)}{E(0)}$?

A dilute classical gas consists of particles of mass $m$ at temperature $T$ and pressure $P$. Instead of being hard spheres, the particles interact exclusively via a repulsive central potential $V(r) = \frac{K}{r^2}$ ($K > 0$). A collision is registered if the scattering angle in the center-of-mass frame exceeds a fixed threshold $\theta_0$. If the gas is heated to a temperature $2T$ while maintaining a constant pressure $P$, the mean free path of the gas particles changes from $\lambda_0$ to $\lambda'$. What is the ratio $\frac{\lambda'}{\lambda_0}$?

A thermally insulated container holds a mixture of two ideal gases, Gas $A$ and Gas $B$, in thermal equilibrium at temperature $T$. The molecules of Gas $A$ have mass $m$ and diameter $d$, while the molecules of Gas $B$ have mass $4m$ and diameter $2d$. The number density of Gas $A$ is $n$ and that of Gas $B$ is $2n$. Let $Z_{AA}$ be the average collision frequency of a single Gas $A$ molecule with other Gas $A$ molecules. Let $Z_{AB}$ be the average collision frequency of a single Gas $A$ molecule with Gas $B$ molecules. Assuming the molecules behave as rigid spheres, what is the ratio $\frac{Z_{AB}}{Z_{AA}}$?

A steady, parallel beam of gas particles, each of mass $m$, is injected at $x=0$ with an initial number density $n_0$ and a uniform velocity $v_0$ in the $+x$ direction. The beam travels towards a perfectly elastic wall. At a given instant, the wall is located at $x=d$ and is moving with a constant velocity $u$ ($u < v_0$) in the $+x$ direction. The mean free path of the particles in the region $0 \le x \le d$ is $\lambda$. Assuming any particle that undergoes a collision is scattered out of the beam and does not reach the wall, what is the pressure exerted on the wall by the unscattered beam particles at this instant?

A perfectly absorbing wall of work function $\phi$ ($\phi \gg kT$) separates a vacuum from an ideal monoatomic gas (mass $m$, number density $n$, temperature $T$). Gas molecules striking the wall with normal kinetic energy $E_x = \frac{1}{2}mv_x^2 \ge \phi$ are absorbed, transferring their momentum to the wall.

Simultaneously, the vacuum side of the wall is illuminated by a photon beam of intensity $I$, which is completely absorbed. The radiation pressure from the photons exactly balances the partial pressure exerted only by the absorbed gas molecules.

Assuming the gas follows the Maxwell-Boltzmann velocity distribution $f(v_x) = n \sqrt{\frac{m}{2\pi kT}} e^{-mv_x^2/2kT}$, which of the following represents the intensity $I$?

(Hint: For large $v_0$, use the approximation $\int_{v_0}^{\infty} v^2 e^{-\alpha v^2} dv \approx \frac{v_0}{2\alpha} e^{-\alpha v_0^2}$)

A closed cylindrical tube of length $L$ and uniform cross-sectional area contains an ideal gas of molar mass $M$. The tube is accelerated uniformly with an acceleration $a$ along its longitudinal axis. The gas is maintained at a constant thermodynamic temperature $T$. If $u_f$ and $u_r$ represent the translational kinetic energy per unit volume of the gas at the front (leading) and rear (trailing) ends of the tube respectively, which of the following expressions gives the ratio $\frac{u_r}{u_f}$? (Assume $R$ is the universal gas constant).

One mole of an ideal diatomic gas is heated to a high temperature such that its vibrational modes become partially active. The internal energy of the gas is modeled empirically as $U(T) = \frac{5}{2}RT + \frac{RT^2}{2T_0}$, where $R$ is the universal gas constant and $T_0$ is a characteristic constant temperature. If the gas undergoes a reversible adiabatic expansion, which of the following relationships between temperature $T$ and volume $V$ is valid throughout the process?

One mole of an ideal monatomic gas undergoes a cyclic process $\mathcal{C}$ in the $P-V$ plane described by the equation $(P - 3P_0)^2 + \alpha^2 (V - 3V_0)^2 = P_0^2$, where $\alpha = P_0/V_0$, $P_0 = 10^5 \text{ Pa}$, and $V_0 = 2 \times 10^{-3} \text{ m}^3$. What is the net change in the total internal translational kinetic energy of the gas after completing one full cycle?

A vessel contains a mixture of two ideal gases, A and B, with molecular masses $m_A = m$ and $m_B = 2m$, maintained at a constant temperature $T$. The number densities of both gases are equal ($n_A = n_B = n$). Assuming the collision diameters of both molecules are identical ($d_A = d_B = d$), what is the ratio of the collision frequency per unit volume between unlike molecules ($Z_{AB}$) to that between like molecules of gas A ($Z_{AA}$)?

Consider an ideal gas consisting of identical particles, each of mass $m$, in thermal equilibrium at absolute temperature $T$. Let $\langle \lambda \rangle$ be the true average de Broglie wavelength of the gas molecules, calculated by averaging over the Maxwell-Boltzmann speed distribution. Let $\lambda_{rms}$ be the de Broglie wavelength of a molecule moving exactly with the root-mean-square speed of the gas. What is the ratio $\frac{\langle \lambda \rangle}{\lambda_{rms}}$?

A planetary atmosphere consists of an ideal monatomic gas of molar mass $M$. Due to a specific mixing mechanism, the atmosphere maintains a temperature profile such that the sum of the average microscopic kinetic energy per molecule and its macroscopic gravitational potential energy remains constant at all heights $h$. Assuming a uniform gravitational acceleration $g$, surface temperature $T_0$, and surface pressure $P_0$, what is the atmospheric pressure $P$ at a height $h = \frac{RT_0}{2Mg}$? (Here $R$ is the universal gas constant).

A rigid, adiabatic container of total volume $V$ is divided by a thin, insulating partition. One side contains $4$ moles of a monatomic ideal gas at $240\text{ K}$, and the inner walls in contact with this gas have a heat capacity of $C_{cal} = 4R$. The other side contains $2$ moles of a diatomic ideal gas at $900\text{ K}$ (the walls on this side have negligible heat capacity). The partition is removed, allowing the gases to mix and reach thermal equilibrium. For the diatomic gas, vibrational degrees of freedom are completely frozen below $600\text{ K}$ and fully active at or above $600\text{ K}$ (assume a sharp transition). What is the final partial pressure of the diatomic gas? ($R$ is the universal gas constant).

In a Young's Double Slit Experiment, a transparent tube of length $L$ is placed in front of the upper slit. The tube is initially divided into two compartments of lengths $L_1$ and $L_2$ ($L_1 + L_2 = L$) by a thin, transparent partition. Compartment 1 contains Gas A (molar mass $M_1$) at pressure $P_1$, with a refractive index $n_1$. Compartment 2 contains Gas B (molar mass $M_2$) at pressure $P_2$, with a refractive index $n_2$. Both gases are ideal, non-reacting, and maintained at a constant temperature $T$. The refractive index $n$ of each gas satisfies the Gladstone-Dale relation $(n-1) \propto \rho$, where $\rho$ is the density of the gas. The partition is suddenly removed, allowing the gases to mix completely. Let $\mu = \frac{M_1 M_2}{M_1 + M_2}$ be the reduced mass of the gas pair. If $\lambda$ is the wavelength of the light used, what is the magnitude of the shift in the central interference maximum (in number of fringes) after the mixing is complete?

A closed organ pipe of length $L$ and radius $r$, made of a material with a coefficient of linear expansion $\alpha$, is placed in a gas chamber. Initially, the chamber contains pure Helium gas (molar mass $4 \text{ g/mol}$) at temperature $T_0$, and the pipe resonates at its fundamental frequency $f_0$. Neon gas (molar mass $20 \text{ g/mol}$) is then injected into the chamber until its partial pressure equals the partial pressure of Helium. Simultaneously, the chamber's temperature is raised to $3T_0$. Assuming both gases are ideal and monoatomic, and $\alpha T_0 \ll 1$, what is the new fundamental frequency of the pipe?

A solid block with a coefficient of linear thermal expansion $\alpha$ contains a spherical cavity of volume $V_0$ at an initial temperature $T_0$. The cavity is filled with a mixture of $2$ moles of a monatomic ideal gas and $1$ mole of a diatomic ideal gas. At $T_0$, the diatomic gas has one fully active vibrational mode. The total internal energy of the gas mixture at $T_0$ is $U_0$. Assuming the solid expands isotropically and the gases obey Dalton's law of partial pressures, what is the rate of change of the total gas pressure with respect to temperature, $\left. \frac{dP}{dT} \right|_{T_0}$, at $T = T_0$?

A cylindrical spacecraft of length $L$ and cross-sectional area $A$ accelerates uniformly at $a$ along its longitudinal axis in deep space. It contains an equimolar ideal gas mixture of Gas 1 (molecular mass $m_1$) and Gas 2 (molecular mass $m_2 = 4m_1$) maintained at a constant temperature $T$. A microscopic leak develops at the flat rear wall of the spacecraft (the trailing end). Given that the acceleration satisfies $a = \frac{kT}{m_1 L} \ln 2$, where $k$ is the Boltzmann constant, what is the initial ratio of the mass escape rate of Gas 2 to Gas 1, $\frac{dM_2/dt}{dM_1/dt}$?

An ionized ideal gas mixture is contained within a flexible, thermally insulated, non-conducting spherical shell of radius $R$. The shell carries a uniform total charge $Q$. Due to electrostrictive effects of the surrounding medium, the shell exerts a net inward pressure on the gas given by $P = \frac{\sigma^2}{2\epsilon_0}$, where $\sigma$ is the surface charge density. The gas mixture consists of $1$ mole of a monoatomic gas and $1$ mole of a rigid diatomic gas.

Initially, the system is in mechanical and thermal equilibrium. If the total charge $Q$ undergoes a very small fractional increase such that $\frac{\Delta Q}{Q} = \alpha$, the system reaches a new equilibrium via a reversible adiabatic process. What is the corresponding fractional change in the gas pressure, $\frac{\Delta P}{P}$?

A small solid sphere of density $\rho_s$ and radius $r$ is placed in a tall closed cylinder filled with an isothermal mixture of two ideal gases, A and B, at a constant temperature $T$. The molar masses of the gases are $M_A$ and $M_B$. The sphere is in static equilibrium at $z=0$, where the partial pressures of the gases are $P_A$ and $P_B$. The sphere is slightly displaced vertically. Assuming the gas viscosity $\eta$ is uniform and the drag force is given by Stokes' law, what is the critical radius $r_c$ of the sphere for which its perturbed motion is critically damped? (Assume the universal gas constant is $R$ and acceleration due to gravity is $g$ in the $-z$ direction)

A vertical, perfectly insulated cylinder of cross-sectional area $A$ contains a gas mixture consisting of $n$ moles of a monoatomic gas and $n$ moles of a rigid diatomic gas at an initial pressure $P_0$ and volume $V_0$. A frictionless, insulated piston of mass $M$ rests in equilibrium on the gas. A sudden downward impulse $J$ is imparted to the piston. Assuming the initial velocity of the piston is much less than the speed of sound in the gas mixture, what is the maximum additional compression $\Delta x$ of the gas?

An ideal gas of molecular mass $m$ is maintained at a constant pressure $P$ and a measured temperature $T$ inside a large container. The gas effuses through a small hole of area $A$ into a vacuum. The temperature measurement has a small uncertainty $\Delta T$. Assuming the gas follows the Maxwell-Boltzmann velocity distribution, which of the following expressions correctly represents the absolute uncertainty $\Delta \Phi$ in the effusion rate $\Phi$ (defined as the total number of molecules escaping per unit time)? (Here $k$ is the Boltzmann constant)

A closed cylindrical container of total mass $M$ (including the liquid inside) is being pulled vertically upwards by a constant external force $F$, such that it moves with an initial upward acceleration $a_0$. The container is filled with an ideal incompressible liquid of density $\rho$ up to a height $h$. At time $t=0$, a small plug of cross-sectional area $s$ ($s \ll$ container's base area) at the bottom of the container is suddenly removed, allowing the liquid to effuse downwards. Assume ideal effusion (coefficient of contraction $C_c = 1$). Let $m_0 = \rho s h$ be the mass of the liquid column directly above the orifice. Which of the following expressions gives the instantaneous upward acceleration $a$ of the container immediately after the plug is removed?

Two rigid vessels $A$ and $B$, each of volume $V$, contain an ideal monatomic gas. They are connected by a porous tube of length $L$. The pore diameter of the tube is much smaller than the mean free path of the gas molecules. Initially, the entire system is at a uniform temperature $T_0$ and pressure $P_0$. Vessel $A$ is maintained at $T_0$, while vessel $B$ is heated and maintained at $4T_0$. A steady-state non-linear temperature gradient $T(x)$ is established along the connecting tube. Neglecting the volume of the tube, what is the steady-state pressure in vessel $A$?

A thermally insulated rigid vessel contains an ideal diatomic gas (with active rotational but frozen vibrational degrees of freedom) at temperature $T$. A microscopic hole is opened, allowing the gas to effuse into a vacuum. Let $N$ be the number of gas molecules remaining in the vessel. Which of the following expressions correctly relates the instantaneous fractional rate of change of temperature $\left(\frac{1}{T}\frac{dT}{dt}\right)$ to the fractional rate of change of the number of molecules $\left(\frac{1}{N}\frac{dN}{dt}\right)$?

One mole of a non-ideal diatomic gas has a temperature-dependent molar heat capacity at constant volume given by $C_v(T) = \frac{R}{2} \left[ 5 + 2 \left(\frac{T_0}{T}\right)^2 e^{-T_0/T} \right]$, where $T_0$ is a characteristic vibrational activation temperature and $R$ is the universal gas constant. The gas is heated at constant volume from an initial temperature $T_1 = \frac{T_0}{\ln 4}$ to a final temperature $T_2 = \frac{T_0}{\ln 2}$. What is the change in the internal energy of the gas?

A uniform solid disk of mass $m$ and radius $R$ is pivoted at a frictionless hinge located on its rim. The disk is initially at rest in its stable equilibrium position. It is then given an initial angular velocity $\omega_0 = \sqrt{\frac{2g}{R}}$. Let $K(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - k^2 \sin^2\phi}}$ denote the complete elliptic integral of the first kind. Which of the following expressions represents the exact time period $T$ of the resulting anharmonic oscillation?

Consider a non-degenerate semiconductor plasma with an electron number density $n$. The electrons are in thermal equilibrium at temperature $T$ and follow a 3D Maxwell-Boltzmann velocity distribution. The collision frequency of an electron with the crystal lattice depends on its thermal speed $v$ as $\nu(v) = \alpha v$, where $\alpha$ is a positive constant. A high-frequency alternating electric field $\vec{E}(t) = \vec{E}_0 \cos(\omega t)$ is applied to the semiconductor. Assuming the frequency is extremely high such that $\omega \gg \nu(v)$ for all statistically significant electron speeds, determine the exact expression for the time-averaged power dissipated per unit volume, $\langle P \rangle$, due to Joule heating.

(Given: $m$ is the mass of an electron, $e$ is the elementary charge, and $k_B$ is the Boltzmann constant)

An ideal inductor of inductance $L = 2.0 \text{ H}$ carries an initial steady current $I_0 = 10 \text{ A}$. At time $t=0$, a switch is thrown such that the inductor begins to discharge through a resistor. The resistor has an initial resistance $R_0 = 5.0 \ \Omega$ at $t=0$, mass $m = 10 \text{ g}$, and specific heat capacity $s = 500 \text{ J kg}^{-1}\text{K}^{-1}$. The temperature coefficient of resistance of the resistor material is $\alpha = 4.0 \times 10^{-3} \text{ K}^{-1}$. Assuming all Joule heating is entirely retained by the resistor (adiabatic process), calculate the value of $\frac{d^2 I}{dt^2}$ at $t=0$.

Two moles of a non-ideal diatomic gas undergo an adiabatic expansion from an initial temperature $T_1 = 600 \text{ K}$ to a final temperature $T_2 = 300 \text{ K}$. In this temperature range, the effective number of degrees of freedom of the gas, $f(T)$, varies due to the partial activation of vibrational modes and is phenomenologically given by $f(T) = 5 + 2 \left( \frac{T}{T_0} \right)^2$, where $T_0 = 300 \text{ K}$. Calculate the work done by the gas during this expansion. (Assume the numerical coefficients in the options carry the appropriate units of $\text{mol} \cdot \text{K}$, and $R$ is the universal gas constant).

A horizontal, perfectly insulated cylinder contains an ideal monatomic gas at an initial pressure $P_0$ and volume $V_0$. The cylinder is fitted with a massless, perfectly insulated piston of cross-sectional area $A$. The piston expands into a vacuum ($P_{ext} = 0$) but experiences a velocity-dependent frictional force $f = \alpha v$ from the cylinder walls, where $v$ is the instantaneous velocity of the piston and $\alpha$ is a positive constant. Assume that all the heat generated by this friction is continuously and entirely absorbed by the gas. What is the time taken for the volume of the gas to double from its initial value?

An ideal gas of molar mass $M$ flows steadily and isothermally at a constant temperature $T$ through a horizontal converging-diverging nozzle. At the inlet, the cross-sectional area is $A_1$ and the gas velocity is $v_1$. At the outlet, the cross-sectional area is $A_2$ and the gas velocity is $v_2$. Assuming ideal gas behavior and neglecting viscous effects, which of the following equations correctly relates the velocities and areas? (Here, $R$ is the universal gas constant).

A thermally insulated cylinder equipped with a movable piston contains $N_0$ molecules of an ideal monatomic gas at an initial temperature $T_0$ and volume $V_0$. A small hole is punctured in the cylinder, allowing the gas to slowly effuse into a surrounding vacuum. Simultaneously, the piston is moved outward quasi-statically such that the volume of the gas increases. When the number of gas molecules remaining in the cylinder drops to $\frac{N_0}{8}$ and the volume becomes $8V_0$, what is the final temperature $T$ of the gas remaining in the cylinder? (Assume the average kinetic energy carried away by each effusing monatomic molecule is $2kT$, where $k$ is the Boltzmann constant).

A large vessel contains an ideal gas of molar mass $M$ at pressure $P$ and temperature $T$. The vessel is connected to a vacuum chamber via a perfectly absorbing conical capillary. The entrance to the capillary is a small flat aperture of area $A$ located on the vessel's wall. The capillary expands outward in the shape of a cone of half-angle $\theta_0$, with its apex centered at the entrance aperture and its axis perpendicular to the aperture. Any gas molecule that strikes the inner wall of the capillary condenses and is permanently absorbed. What is the net mass of gas that successfully exits the capillary into the vacuum per unit time? ($R$ is the universal gas constant).

An ionised gas (plasma) is trapped between two parallel capacitor plates of area $A$ located at $x = 0$ and $x = d$. The plasma is maintained at a constant temperature $T$. Under steady-state confinement, the plasma behaves as a dielectric medium whose relative permittivity varies spatially as $\epsilon_r(x) = 1 + \alpha x$ (where $\alpha > 0$). A constant potential difference $V_0$ is maintained across the plates. The electrostatic force density acting on the plasma due to the dielectric gradient is given by $f = \frac{1}{2}\epsilon_0 E^2 \frac{d\epsilon_r}{dx}$. In mechanical equilibrium, this force density is balanced by the pressure gradient of the gas ($\frac{dP}{dx} = f$). Assuming the plasma behaves as an ideal gas, if the number density of particles at $x = 0$ is $n_0$, what is the number density $n(d)$ at $x = d$? (Here $k_B$ is the Boltzmann constant).

A rigid vessel contains a mixture of two non-interacting ideal gases, $A$ and $B$, in thermal equilibrium at absolute temperature $T$. The mass of a single molecule of gas $A$ is $m$, and the mass of a single molecule of gas $B$ is $4m$. The collision cross-section area between a molecule of $A$ and a molecule of $B$ is $\sigma$. If the number density of gas $B$ is $n_B$ and the mean speed of gas $A$ molecules is $v_0 = \sqrt{\frac{8kT}{\pi m}}$ (where $k$ is the Boltzmann constant), what is the collision frequency $Z_{AB}$ of a single molecule of gas $A$ with the molecules of gas $B$?

A quantity of $n = 3 \text{ moles}$ of an ideal non-linear polyatomic gas with molar heat capacity at constant volume $C_v = 3R$ undergoes a quasi-static cyclic process. The path of the cycle in the $P-V$ plane is given by the parametric equations $P(t) = P_0 + \alpha \sin(\omega t)$ and $V(t) = V_0 + \beta \cos(\omega t)$ for time $t \in [0, \frac{2\pi}{\omega}]$. The constants are given as $P_0 = 2 \times 10^5 \text{ Pa}$, $V_0 = 0.05 \text{ m}^3$, $\alpha = 0.5 \times 10^5 \text{ Pa}$, and $\beta = 0.01 \text{ m}^3$. What is the net change in the total translational kinetic energy of the gas molecules over one complete cycle?

A thin spherical shell of radius $R$ and thickness $t$ ($t \ll R$) is made of a material with Young's modulus $Y$ and coefficient of linear thermal expansion $\alpha$. Assume the Poisson's ratio of the material is zero. The shell contains an ideal gas having $f$ degrees of freedom. Initially, the system is in thermal equilibrium at temperature $T_0$, the shell is completely unstressed (internal pressure equals the external atmospheric pressure $P_0$). A total heat $Q$ is slowly supplied to the gas. Assume the heat capacity of the shell is negligible, and the work done by the gas during expansion is negligible compared to the heat supplied. Assuming the change in radius $\Delta R \ll R$, which of the following expressions correctly represents the tensile hoop stress $\sigma$ developed in the shell?

A vertical, thermally conducting cylinder of cross-sectional area $A$ is closed at the bottom and fitted with a frictionless, airtight piston of mass $M_p$. The cylinder contains an ideal gas of total mass $M_g$ and molar mass $\mu$. The entire system is maintained at a constant temperature $T$. Due to the large height of the gas column, the hydrostatic pressure gradient within the gas cannot be ignored. The piston is slightly displaced downwards from its equilibrium position and released. Assuming the gas continuously maintains hydrostatic equilibrium and its kinetic energy is negligible during the motion, what is the angular frequency $\omega$ of the resulting vertical oscillations?

(Given: Acceleration due to gravity is $g$, Universal gas constant is $R$, and the space above the piston is a vacuum)

A rigid, thermally insulated container holds a total mass $m \text{ (in kg)}$ of an ideal gas having an adiabatic exponent $\gamma$ (dimensionless). A paddle wheel inside the container is driven by an external agent, doing non-PV work on the gas. The initial average speed of the gas molecules is $v_0 \text{ (in m/s)}$. The stirring continues until the final root-mean-square (RMS) speed of the gas molecules becomes $v_f \text{ (in m/s)}$. Assuming no heat exchange with the surroundings, which of the following expressions represents the total work $W \text{ (in J)}$ done by the paddle wheel?

A spacecraft contains a highly rarefied ideal gas of molar mass $M_0$, uniform mass density $\rho$, and temperature $T$. Inside, a thin, perfectly reflecting free-floating partition of mass $M$ and area $A$ is oriented perpendicular to the spacecraft's longitudinal axis. Immediately after the spacecraft begins to accelerate at a constant rate $a$ along its axis, the partition reaches a quasi-steady terminal velocity $u$ ($u \ll v_{rms}$) relative to the spacecraft. Assume the gas remains uniform and its velocity distribution is Maxwellian in the spacecraft's frame. Which of the following expressions correctly represents $u$?