Thermodynamics - Class 11 Part 2

An ideal monatomic gas of molar mass $M$ undergoes a cyclic process. During the high-pressure isothermal expansion phase at temperature $T_0$, the volume increases from $V_0$ to $2V_0$ at a constant volumetric rate $\frac{dV}{dt} = C$. Simultaneously, a microscopic valve of area $A$ is opened, allowing gas to effuse into a vacuum. The valve is closed during the rest of the cycle. If the effusion rate is given by $\frac{dn}{dt} = -\frac{P A}{\sqrt{2\pi M R T}}$, what is the ratio of the work done by the gas during this expansion phase to the work that would have been done if the valve remained closed? (Let $\beta = \frac{A}{C} \sqrt{\frac{R T_0}{2\pi M}}$)

A non-ideal rubber cord is slowly stretched from its natural length $x = 0$ to an extension $x = 0.1 \text{ m}$ and then slowly relaxed back to $x = 0$. The magnitude of the restoring force exerted by the cord is $F_{s}(x) = kx + \alpha x^2$ during stretching and $F_{r}(x) = kx - \alpha x^2$ during relaxation, where $k = 100 \text{ N/m}$ and $\alpha = 300 \text{ N/m}^2$. Calculate the net work done by the restoring force of the cord over the complete cycle.

One mole of a monatomic ideal gas of molar mass $M$ (in $\text{kg mol}^{-1}$) undergoes a cyclic process consisting of four steps: (I) Reversible isothermal expansion at a state where the variance of its 1D Maxwell-Boltzmann velocity distribution is $\sigma_1^2$ (in $\text{m}^2\text{s}^{-2}$). (II) Irreversible isochoric cooling by placing it in direct thermal contact with a heat reservoir until the velocity variance becomes $\sigma_2^2$ ($\sigma_2 < \sigma_1$). (III) Reversible isothermal compression at the new state. (IV) Irreversible isochoric heating by placing it in direct thermal contact with a heat reservoir until the velocity variance returns to $\sigma_1^2$. What is the total entropy change of the universe ($\Delta S_{\text{univ}}$) per mole of the gas for one complete cycle? ($R$ is the universal gas constant in $\text{J K}^{-1}\text{mol}^{-1}$)

A stationary circular conducting loop of radius $a$ and resistance $R$ (with negligible self-inductance) is placed in a uniform magnetic field $\vec{B}$ perpendicular to its plane. The magnetic field is varied from $B_0$ to $-B_0$ in a fixed time interval $T$ through two different processes. In Process I, the magnetic field varies sinusoidally as $B(t) = B_0 \cos\left(\frac{\pi t}{T}\right)$. In Process II, the magnetic field varies linearly as $B(t) = B_0 \left(1 - \frac{2t}{T}\right)$. Let $W_I$ and $W_{II}$ be the total heat dissipated in the loop during Process I and Process II, respectively. Which of the following represents the ratio $\frac{W_I}{W_{II}}$?

An AC voltage source $V(t) = V_0 \sin(\omega t)$ is connected across a series LCR circuit. In the steady state, the current lags the voltage by a phase angle of $\frac{\pi}{4}$. Let $W$ be the total work done by the source and $\Delta U$ be the net change in the total energy stored in the inductor and capacitor during the time interval from $t=0$ to $t=\frac{\pi}{4\omega}$. Which of the following expressions correctly represents the dimensionless ratio $\frac{W - \Delta U}{V_0^2 / (\omega R)}$?

A cylindrical vessel of constant cross-sectional area $A$ contains a liquid of mass $M$. Initially, the liquid is at a uniform temperature $T_0$ and has height $H$. The density of the liquid varies with temperature as $\rho(T) = \rho_0[1 - \gamma(T - T_0)]$, where $\gamma$ is the volumetric expansion coefficient (assume $\gamma \Delta T \ll 1$). The liquid is subjected to two different heating processes:

Process I: The liquid is heated uniformly to a final temperature $T_0 + \Delta T$. Process II: The liquid is heated such that its temperature varies linearly with height $z$ from the bottom, with the bottom at $T_0$ and the top surface at $T_0 + 2\Delta T$.

Let $U_I$ and $U_{II}$ be the final gravitational potential energies of the liquid in Process I and Process II, respectively. Assuming the vessel does not expand, find the difference $U_I - U_{II}$.

The internal energy of a certain non-ideal gas is given by the relation $U = 3P(V + V_0)$, where $P$ is the pressure, $V$ is the volume, and $V_0$ is a positive constant. The gas is initially at a pressure $P_0$ and volume $7V_0$. It undergoes a reversible adiabatic expansion until its volume becomes $26V_0$. What is the work done BY the gas during this process?

A horizontal, thermally insulated cylinder of cross-sectional area $A$ contains an ideal gas confined by a frictionless piston of mass $M$. The external pressure acting on the piston from the outside varies with the piston's position $x$ as $P_{ext} = \alpha x^2$, where $\alpha$ is a positive constant. The closed end of the cylinder is at $x = 0$. Initially, the piston is held at rest at $x = L$. The piston is then released and moves outward, momentarily coming to rest at $x = 2L$. What is the change in the internal energy of the gas during this process?

A longitudinal compression pulse of small pressure amplitude $\Delta P$ ($\Delta P \ll P_0$) travels through an ideal diatomic gas ($\gamma = 1.4$) at equilibrium pressure $P_0$ and temperature $T_0$. The pulse strikes a perfectly rigid, thermally insulated wall at normal incidence and reflects completely. Let $\Delta T$ be the maximum transient increase in temperature of the infinitesimally thin gas layer immediately adjacent to the wall during the reflection. Which of the following expressions correctly represents $\Delta T$?

A cylinder contains one mole of a rigid diatomic ideal gas at equilibrium pressure $P_0$ and volume $V_0$. A piston drives the volume to oscillate as $V(t) = V_0 - \Delta V \cos(\omega t)$, where $\Delta V \ll V_0$. The translational degrees of freedom equilibrate instantaneously, but the rotational degrees of freedom have a relaxation time $\tau$. The effective molar heat capacity at constant volume is given by $C_V(\omega) = C_{V,trans} + \frac{C_{V,rot}}{1 + \omega^2 \tau^2}$. Assuming the process follows an effective adiabatic relation $P V^{\gamma_{eff}} = \text{constant}$ with $\gamma_{eff} = 1 + \frac{R}{C_V(\omega)}$, what is the amplitude of the pressure oscillations $\Delta P_0$ when the piston is driven at a frequency $\omega = \frac{1}{\tau}$?

A parallel plate capacitor has plates of length $L$, width $b$, and separation $d$. Let the left edge of the plates be at $x=0$ and the right edge at $x=L$. A rigid dielectric slab of width $b$, thickness $d$, and length greater than $L$ is inserted between the plates from the right. Its left face is at position $x$ ($0 < x < L$). The permittivity of the slab varies with the distance $s$ from its left face as $\epsilon(s) = \epsilon_0(1 + \alpha s)$. The region between the plates from $0$ to $x$ contains an ideal monatomic gas (permittivity $\epsilon_0$), confined by the slab which acts as a frictionless piston. The plates are maintained at a constant voltage $V$. If the gas expands reversibly from $x = L/4$ to $x = L/2$ while maintaining mechanical equilibrium, find the total heat $Q$ supplied to the gas. (Ignore fringing fields)

A closed cavity of variable volume $V$ contains one mole of an ideal monoatomic gas. The cavity also contains an electromagnetic radiation field in thermal equilibrium with the gas at temperature $T$. Due to the presence of the gas, the effective wave impedance of the cavity is $Z = \beta \sqrt{V}$, where $\beta$ is a positive constant. The radiation energy density is given by $u = \frac{\alpha T^4}{Z}$, where $\alpha$ is a positive constant. The radiation exerts a pressure $P_{rad} = \frac{u}{6}$ on the cavity walls. If the system undergoes a reversible adiabatic process, which of the following quantities remains constant? (Here $R$ is the universal gas constant)

A rigid, perfectly insulated container of volume $V$ is initially completely evacuated. It is connected via a valve to a large supply line containing an ideal diatomic gas maintained at a constant pressure $P_0$ and temperature $T_0$. The valve is opened, allowing gas to flow slowly into the container until the pressure inside reaches $P_0$, at which point the valve is closed. What is the final temperature $T_f$ of the gas inside the container?

One mole of an ideal monatomic gas is confined in a cylinder by a movable piston of area $A$. The piston contains a magnetic core of volume $V_p$. The system is placed in an external magnetic field $\mathbf{B} = kx \hat{i}$, where $x$ is the position of the piston. The gas undergoes a cyclic process consisting of four steps:

1. Isobaric expansion at pressure $P_0$ from $x_1$ to $x_2$. The core's magnetization is constant at $M_0 \hat{i}$.
2. Isochoric depressurization to $P_0/2$. The magnetization changes to $3M_0 \hat{i}$.
3. Isobaric compression at $P_0/2$ back to $x_1$. The magnetization is constant at $3M_0 \hat{i}$.
4. Isochoric pressurization to $P_0$. The magnetization returns to $M_0 \hat{i}$. Let $\Delta V = A(x_2 - x_1)$ and define a dimensionless parameter $\beta = \frac{V_p M_0 k}{P_0 A}$. Which of the following gives the effective molar heat capacity $C_{eff}$ of the system during the isobaric expansion (step 1), and the net heat $Q_{net}$ absorbed by the system over one complete cycle?

Two identical rigid bulbs of volume $V_0$ are connected by a thin tube of negligible volume. The system contains $N$ moles of an ideal monoatomic gas of molar mass $M$ and is placed in a zero-gravity environment. The entire assembly rotates with a constant angular velocity $\omega$ about an axis passing through the center of Bulb 1, such that Bulb 2 is at a distance $L$ from the axis. The gas is heated reversibly while a motor maintains the angular velocity $\omega$ constant. If $x = \frac{M \omega^2 L^2}{2 R T}$, what is the molar heat capacity $C$ of the gas during this process?

A uniform metal rod of mass $m$, length $L$, cross-sectional area $A$, specific heat capacity $c$ (assumed constant), and Young's modulus $Y$ is rigidly fixed between two unyielding walls at an absolute temperature $T_0$. The coefficient of linear expansion of the rod varies with temperature as $\alpha = \beta T$, where $\beta$ is a positive constant. The rod is heated to a temperature $2T_0$. Assuming the walls do not move and the rod does not buckle, what is the total heat $\Delta Q$ supplied to the rod?

A capacitor of capacitance $C$ with an initial charge $Q_0$ is connected in series with an inductor of inductance $L$, a switch, and a resistor of resistance $R$. The resistor is placed inside a rigid, thermally insulated cylinder containing $n$ moles of a monoatomic ideal gas. The switch is closed at $t=0$. The resistance $R$ is chosen such that the electrical circuit is critically damped. What is the rate of change of the temperature of the gas, $\frac{dT}{dt}$, at the exact instant the current in the circuit reaches its maximum magnitude? (Here $R_{gas}$ is the universal gas constant and $e$ is the base of the natural logarithm).

A horizontal cylindrical container of cross-sectional area $A$ has adiabatic walls and contains one mole of an ideal monatomic gas. A frictionless, adiabatic piston is pushed inwards at a constant speed $v_0$. Integrated into the piston is a conducting rod of length $L$ and electrical resistance $R_e$. The rod slides on two perfectly conducting, parallel rails that run along the inner length of the cylinder and are shorted at the closed end. A non-uniform magnetic field perpendicular to the plane of the rails exists in the region, given by $B(x) = B_0 x^{-1/2}$, where $x$ is the distance from the closed end. The entire electrical resistance $R_e$ is exposed to the gas, so all Joule heating is transferred to the gas. If the initial volume and temperature of the gas are $V_0$ and $T_0$, and the gas is compressed to a volume $V_0/8$, find the final temperature of the gas. (Given: $\frac{B_0^2 L^2 v_0}{R_e} = R T_0$, where $R$ is the universal gas constant).

A vertical cylinder of cross-sectional area $A$ contains a mixture of two non-reacting ideal gases with molar masses $M_1$ and $M_2$, having $n_1$ and $n_2$ moles respectively. The cylinder is closed by a movable, frictionless piston initially at a height $H$ from the base. The system is maintained at a constant temperature $T$ in a uniform downward gravitational field $g$. If the piston is slowly raised to an infinite height, what is the total work done by the gas mixture on the piston?

One mole of an ideal gas in a cylinder with a conducting piston of area $A$ undergoes a cyclic process $1 \to 2 \to 3 \to 1$. Process $1 \to 2$ is an isothermal expansion at temperature $T_0$ from volume $V_0$ to $2V_0$. Process $2 \to 3$ is isobaric compression at pressure $P_0$ back to volume $V_0$. Process $3 \to 1$ is isochoric heating back to the initial state 1. The initial pressure is $P_1 = 2P_0$. During the expansion $1 \to 2$, the piston moves through a magnetic field and experiences a retarding magnetic force $F = \frac{\alpha}{V}$, where $\alpha = P_0 V_0 A$. The electrical energy from the resulting eddy currents is entirely dissipated as heat into the gas. The magnetic field is zero during $2 \to 3$ and $3 \to 1$. What is the net heat absorbed by the gas from the external thermostat during the complete cycle?

A fixed horizontal cylinder of cross-sectional area $A$ contains an ideal gas sealed by a massless, frictionless piston. A solid block of mass $m$ slides on a rough horizontal floor parallel to the cylinder. A rigid, massless rod connects the piston to the block, maintaining a constant angle $\theta$ below the horizontal such that $\tan\theta = 0.5$. The coefficient of kinetic friction between the block and the floor is $\mu_k = 0.5$. Initially, the gas has volume $V_0 = A x_0$ and pressure $P_0 = \frac{2e}{3} \frac{mg}{A}$, where $e$ is the base of the natural logarithm. The system is released from rest and the gas expands isothermally. Assuming the block does not topple, what is the maximum kinetic energy of the block during the expansion?

An ideal monatomic gas ($\gamma = 5/3$) is confined in a cylindrical container of radius $R$ by a movable piston. The flat ends of the cylinder (including the piston) are perfectly thermally insulated and non-radiating. The curved walls of the cylinder behave as a perfectly black body ($\epsilon = 1$) and have negligible heat capacity, instantaneously taking the temperature of the gas $T$. Initially, the piston is at a distance $x_0$ from the closed end, and the gas temperature is $T_0$. The piston is then moved outward at a constant velocity $v$. Assuming the expansion is reversible and adiabatic (energy lost via radiation is negligible compared to the work done), what is the total energy radiated by the cylinder as the piston moves from $x = x_0$ to $x = 8x_0$? Let $P_0$ be the initial power radiated by the curved walls.

A thermally insulated rectangular box of length $L$, width $W$, and height $H$ contains $n$ moles of an ideal monoatomic gas of molar mass $M$. The box is placed in a uniform downward gravitational field $g$. Assume the gas is in thermal equilibrium at an initial temperature $T_0$, and the gravitational potential energy is small compared to the thermal energy ($MgH \ll RT_0$, where $R$ is the universal gas constant). The vertical wall at $x=L$ (of area $W \times H$) is flexible. An external agent quasi-statically pushes this wall inwards such that the inward displacement of the wall at a height $y$ from the bottom is given by $\delta(y) = \delta_0 \frac{y}{H}$, where $\delta_0 \ll L$. Find the change in temperature $\Delta T$ of the gas. (Keep terms up to first order in $\frac{MgH}{RT_0}$).

A vertical cylindrical container with adiabatic walls contains $n$ moles of a monatomic ideal gas. It is fitted with a frictionless, adiabatic piston of mass $M$ and cross-sectional area $A$. The atmospheric pressure is $P_0$, and it is given that $P_0 = \frac{Mg}{A}$. Initially, the container is placed in an elevator accelerating upwards with an acceleration $a = g$ (where $g$ is the acceleration due to gravity). The gas is in thermal and mechanical equilibrium with an initial volume $V_0$. Suddenly, the elevator's acceleration drops to zero and it continues to move with a constant upward velocity. The gas expands and eventually reaches a new equilibrium state with volume $V_f$. Find the ratio $\frac{V_f}{V_0}$.

A thermally insulated cylinder with a movable piston contains a solid semiconductor sample of constant heat capacity $C_s = 4R$ and a gas mixture in thermal equilibrium at an initial temperature $T_0$. The gas mixture consists of a total of $4$ moles of Helium (He) and Oxygen ($\text{O}_2$). According to Dalton's law of partial pressures, the initial partial pressure of He is equal to the partial pressure of $\text{O}_2$. The system undergoes a reversible adiabatic expansion from an initial volume $V_0$ to a final volume $8V_0$. The intrinsic carrier concentration of the semiconductor is given by $n_i(T) = A T^{3/2} e^{-E_g / 2k_B T}$, where the band gap energy is $E_g = 4 k_B T_0$. Assuming the volume of the semiconductor is negligible, what is the ratio of the final intrinsic carrier concentration to the initial intrinsic carrier concentration, $\frac{n_i(T_f)}{n_i(T_0)}$?

A heat engine operates on a cyclic process consisting of three steps:

1. Isochoric pressure increase from $P_2$ to $P_1$ at volume $V_1$.
2. Polytropic expansion following $P V^{3/2} = \text{constant}$ from $P_1$ to $P_2$.
3. Isobaric compression at $P_2$ back to volume $V_1$.

The measured values of the independent parameters are: $P_1 = 8.0 \times 10^5 \text{ Pa} \pm 1\%$ $V_1 = 1.0 \text{ m}^3 \pm 1\%$ $P_2 = 1.0 \times 10^5 \text{ Pa} \pm 2\%$

Assuming the polytropic exponent $3/2$ is exact, what is the maximum percentage error in the calculated net work done per cycle?

$n$ moles of an ideal monoatomic gas are trapped in a rectangular container of base area $A = b \times L$ (with $L$ along the $x$-axis) by a movable, frictionless, conducting piston. The bottom of the container is also a conducting plate, forming a capacitor with the piston. The gas acts as a dielectric whose dielectric constant varies spatially along the length $L$ as $K(x) = 1 + \frac{x}{L}$. Initially (State 1), the piston is at height $d$, the gas is at temperature $T_0$, and a battery of voltage $V_0$ is connected across the plates. The system undergoes the following cycle:

$1 \to 2$: Reversible isothermal expansion to height $2d$ with the battery connected. $2 \to 3$: Battery is disconnected, and the gas is cooled isochorically to $T_1$. $3 \to 4$: Reversible isothermal compression back to height $d$. $4 \to 1$: Gas is heated isochorically to $T_0$, and the battery is reconnected.

What is the net work done by the external agent on the piston during this complete cycle? (Ignore gravity)

A liquid-vapor mixture of total mass $m = 2 \text{ kg}$ undergoes a reversible thermodynamic cycle consisting of two isotherms at temperatures $T = 500 \text{ K}$ and $T - \Delta T = 495 \text{ K}$, and two isochores. At the higher temperature $T$, the cycle operates between states with vapor qualities $x_1 = 0.2$ and $x_2 = 0.8$. The latent heat of vaporization is $L = 1500 \text{ kJ/kg}$ (assumed constant). Assuming the specific volume of the liquid phase is negligible compared to the vapor phase and $\Delta T \ll T$, what is the net work done by the mixture during one complete cycle?

A refrigerator is placed inside an evacuated chamber whose walls are maintained at a constant temperature $T_0$. The refrigerator maintains its cold interior at a temperature $T_c$ and operates with a constant Coefficient of Performance $\beta$. Heat leaks into the cold interior at a constant rate $P$. The refrigerator rejects heat to the surroundings exclusively via radiation from its outer fins, which have a total surface area $A$ and emissivity $\epsilon$. Assuming steady-state operation, what is the temperature $T$ of the radiator fins? (Given $\sigma$ is the Stefan-Boltzmann constant).

A reversible refrigerator operates between a cold body and a heat sink. The cold body is a liquid undergoing a phase change at a constant temperature $T_L = 15 \text{ K}$. The heat sink has a temperature-dependent heat capacity given by $C(T) = \alpha T^3$, where $\alpha = 4 \times 10^{-4} \text{ J/K}^4$. Initially, the heat sink is at a temperature $T_0 = 20 \text{ K}$. The refrigerator extracts heat from the cold body and rejects it to the heat sink until the sink reaches a temperature $T_f = 40 \text{ K}$. What is the minimum work required to drive this refrigerator?

A uniform conducting rod of mass $m$, length $L$, and resistance $R$ is pivoted at one end and rotates in a horizontal plane with an initial angular velocity $\omega_0$. A uniform vertical magnetic field $B$ exists in the region. The pivot exerts a constant frictional torque $\tau_f$ on the rod. The electrical power generated due to the motional EMF is continuously and entirely used as the work input for an ideal Carnot refrigerator operating between a cold reservoir at temperature $T_C$ and a hot reservoir at $T_H$. Given that the system parameters satisfy the relation $B^2 L^4 \omega_0 = 8 R \tau_f$, what is the total heat extracted from the cold reservoir before the rod comes to a complete stop?

An ideal Carnot refrigerator operates between a cold reservoir and a hot reservoir. Both reservoirs contain the same ideal gas of molar mass $M$ in thermal equilibrium. The most probable speed of the gas molecules in the cold reservoir is $v_0$, while the root-mean-square (RMS) speed of the gas molecules in the hot reservoir is $2v_0$. What is the coefficient of performance (COP) of this refrigerator?

A vertical cylinder of cross-sectional area $A = 0.1 \, \text{m}^2$ is completely submerged in a fluid bath. The top of the cylinder is fixed at a depth $H_0 = 2.0 \, \text{m}$ below the fluid surface. The bottom of the cylinder is sealed by a movable piston of mass $M$ and volume $V_p = 0.05 \, \text{m}^3$. An external mechanism moves the piston to perform a cycle.

The cycle consists of four steps:

1. The piston is slowly pushed down, increasing the cylinder's internal volume from $V_1 = 0.1 \, \text{m}^3$ to $V_2 = 0.3 \, \text{m}^3$, while the fluid is maintained at $T_H = 350 \, \text{K}$.
2. The piston is held in place while the fluid is cooled to $T_C = 300 \, \text{K}$.
3. The piston is slowly pulled up, decreasing the volume from $V_2$ to $V_1$, while the fluid is maintained at $T_C$.
4. The piston is held in place while the fluid is heated back to $T_H$.

The fluid density is $\rho(300 \, \text{K}) = 1000 \, \text{kg/m}^3$ and $\rho(350 \, \text{K}) = 960 \, \text{kg/m}^3$. Calculate the magnitude of the net work done BY THE FLUID on the piston during one complete cycle. (Assume $g = 10 \, \text{m/s}^2$ and $P_{\text{atm}} = 10^5 \, \text{Pa}$)

Consider an ideal gas of non-interacting heteronuclear diatomic molecules. Each molecule consists of two atoms of masses $m$ and $2m$, separated by a bond length $r_0$ and bound by an effective spring of force constant $K$. The gas is maintained at a temperature $T = \frac{\hbar}{k_B \ln 2} \sqrt{\frac{3K}{2m}}$, where $k_B$ is the Boltzmann constant and $\hbar$ is the reduced Planck constant. Assuming the rotational modes are fully excited ($k_B T \gg \frac{\hbar^2}{2\mu r_0^2}$) but electronic transitions are negligible, what is the molar heat capacity at constant volume, $C_V$, of this gas?

A thermally insulated rigid container of volume $V$ initially contains $n_0$ moles of an ideal monatomic gas at temperature $T_0$. A tiny pinhole is created, allowing the gas to slowly effuse into a surrounding vacuum. The process continues until the number of moles of gas remaining in the container becomes $n_0/e^2$. What is the change in the molar entropy of the gas remaining inside the container? (Assume the hole is small enough that the Maxwell-Boltzmann velocity distribution is continuously maintained inside, and let $R$ be the universal gas constant).

One mole of an ideal monoatomic gas is confined in an adiabatic cylinder by a massless, frictionless piston subject to a constant external pressure $P_0$. The initial temperature of the gas is $T_0$. A heating element inside the gas is connected to a constant DC voltage source $V$. The electrical resistance of the heating element varies with the gas temperature $T$ as $r(T) = r_0 \left(\frac{T}{T_0}\right)$. Assuming the heat capacity of the heater is negligible, what is the time $t$ required for the volume of the gas to double? (Let $R$ be the universal gas constant).

An ideal $LC$ circuit with inductance $L$ and capacitance $C$ initially stores energy with a maximum charge $Q_0$ on the capacitor. The circuit is placed in a large vacuum chamber whose walls are maintained at a constant temperature $T$. Due to the emission of electromagnetic radiation, the circuit is not perfectly conservative and loses power at a rate $P = \alpha (\ddot{q})^2$, where $\alpha$ is a known positive constant and $q$ is the instantaneous charge on the capacitor. Assuming the radiative damping is extremely small ($\alpha \ll L \sqrt{LC}$), which of the following expressions correctly gives the effective radiation resistance $R_{rad}$ of the circuit and the entropy $\Delta S$ generated in the chamber walls during the first complete oscillation cycle?

Two identical solid spheres, A and B, each of mass $m$ and constant specific heat capacity $c$, are initially at temperatures $T_A$ and $T_B$ respectively, with $T_A > T_B > T_0$. They are placed inside a large evacuated enclosure maintained at a constant temperature $T_0$. The spheres exchange heat with each other via a conducting wire, and simultaneously lose heat to the enclosure exclusively via thermal radiation according to the Stefan-Boltzmann law, $\frac{dQ}{dt} = -\epsilon \sigma A (T^4 - T_0^4)$. After a long time, both spheres reach thermal equilibrium with the enclosure. What is the total change in entropy of the universe ($\Delta S_{\text{univ}}$) during this entire process?

A Carnot refrigerator operates to maintain a cold reservoir at a constant temperature $T_C$. It extracts heat from the cold reservoir at a constant rate $P_C$. The refrigerator rejects heat into an ambient sealed room containing an ideal gas of mass $M$ and specific heat capacity at constant pressure $C_p$. The room is designed to expand isobarically to maintain a constant pressure $P_0$. Initially, the ambient room air has a temperature $T_0$ and density $\rho_0$. Assuming the room is perfectly thermally insulated from the outside environment, what is the time $t$ required for the density of the ambient room air to decrease to $\frac{\rho_0}{e}$?

One mole of an ideal monatomic gas is confined in a horizontal cylinder by a movable piston of mass $m$ and cross-sectional area $A$. The piston is subjected to a conservative force field such that, if it were free, it would experience a non-uniform acceleration $a(x) = \alpha x^2$ directed towards the negative x-direction, where $\alpha$ is a positive constant and $x$ is the position of the piston. The gas expands reversibly, pushing the piston from $x = L$ to $x = 2L$. If the initial pressure and volume of the gas are $P_0$ and $V_0$ respectively, and the space outside the cylinder is evacuated, what is the total heat supplied to the gas during this process?

A rigid container holds an ideal diatomic gas. The container loses heat to the surrounding environment (at a constant temperature $T_{\text{env}} = 300\text{ K}$) at a rate $\frac{dQ_{\text{loss}}}{dt} = H(T - T_{\text{env}})$, where $H = 1\text{ W/K}$ and $T$ is the instantaneous gas temperature.

A semiconductor heater inside the container is powered by a constant voltage source $V$. The resistance of the heater depends on the gas temperature as $R(T) = R_0 e^{\beta/T}$, where $\beta = 600 \ln 2\text{ K}$ and $\frac{V^2}{R_0} = 600\text{ W}$. Assuming the heater is always in thermal equilibrium with the gas, and the gas is initially at $300\text{ K}$, what is the final steady-state temperature of the gas?

A vertical, perfectly thermally insulated cylinder of cross-sectional area $A$ contains a monatomic ideal gas. The cylinder is closed by a frictionless, electrically conducting but thermally insulating piston of mass $m$. The piston is constrained to move along a frictionless helical groove of pitch $p$ (vertical displacement $p$ per revolution). A uniform vertical magnetic field $B$ exists throughout the region. The center and the rim of the piston are connected via zero-resistance sliding contacts to a heating coil of resistance $R$ and negligible heat capacity located entirely inside the gas. Initially, the system is in equilibrium with the piston at height $h_0$ and external atmospheric pressure $P_0$. Let $P_1 = P_0 + \frac{mg}{A}$. A block of mass $M$ is suddenly placed on the piston. The piston descends, and the system eventually comes to rest at a new equilibrium height $h_f$. Let $P_2 = P_0 + \frac{(m+M)g}{A}$. Which of the following expressions gives $h_f$? (Ignore the induced magnetic field).

A perfectly insulated cylinder of cross-sectional area $S$ contains $n$ moles of an ideal monoatomic gas at initial pressure $P_0$ and volume $V_0$. A wire of unstretched length $L$, cross-sectional area $A$, Young's modulus $Y$, and breaking stress $\sigma_b$ connects the piston to the rigid bottom of the cylinder. The wire is completely enclosed within the gas. The piston is slowly pulled outward by an external force until the wire breaks. Immediately after breaking, the piston is locked in place. Assuming all the elastic potential energy of the wire is dissipated as heat into the gas, which of the following expressions gives the final pressure $P_f$ of the gas in equilibrium? (Let $V_1 = V_0 + \frac{S \sigma_b L}{Y}$ be the volume of the gas just before the wire breaks. Neglect the volume and heat capacity of the wire.)

A thermally insulated horizontal cylinder of cross-sectional area $A$ contains an ideal rigid diatomic gas. A frictionless, insulating piston of mass $M$ confines the gas to a column of length $x$. The fixed end of the cylinder and the piston are coated with conducting plates, forming an isolated parallel plate capacitor with a constant charge $Q$. Due to the gas density shifts, its dielectric constant varies with the column length as $K = \frac{L_0}{x}$ (assume $x \ll L_0$). If the piston is slightly displaced from its equilibrium position, what is the angular frequency $\omega$ of the resulting adiabatic oscillations?

A horizontal, thermally insulated cylinder of cross-sectional area $A$ contains an ideal gas with adiabatic exponent $\gamma = 1.5$. A massless piston encloses the gas and is attached to the closed end of the cylinder by a linear elastic wire. Initially, the gas has volume $V_0 = 1 \text{ m}^3$ and pressure $P_0 = 8 \times 10^5 \text{ Pa}$, and the wire is unstretched. The external pressure $P_{\text{ext}}$ on the piston is slowly reduced from $P_0$ to zero, causing the gas to expand reversibly and adiabatically. The wire breaks exactly when $P_{\text{ext}}$ reaches zero. Given that the breaking force of the wire is $F_b = 10^5 A$, what is the total work done by the gas strictly against the external pressure $P_{\text{ext}}$ during this process?

One mole of an ideal rigid diatomic gas $A_2$ is enclosed in a cylinder with a massless, frictionless piston. The external pressure is maintained at a constant value $P_0$. The initial temperature of the gas is $T_0$. Heat is slowly supplied to the system, causing the gas to expand. As the temperature increases, the gas $A_2$ partially dissociates into a monatomic gas $A$. The degree of dissociation $\alpha$ (fraction of initial moles of $A_2$ that has dissociated) varies with temperature $T$ as $\alpha = \frac{T - T_0}{T_0}$ for $T_0 \le T \le 2T_0$. Assuming the bond dissociation energy of $A_2$ is negligible, what is the total heat supplied to the system to raise its temperature from $T_0$ to $2T_0$?

One mole of an ideal diatomic gas is confined in a horizontal cylinder by a movable piston of cross-sectional area $A$. The external pressure is constant at $P_0$. Initially, the gas is in thermal and mechanical equilibrium with volume $V_0$. A heater slowly supplies heat to the gas, causing it to expand. During the expansion, the piston experiences a non-conservative resistive force $f = kx$ (where $x$ is the displacement of the piston from its initial position) in addition to the external pressure. The work done against this resistive force is entirely dissipated as heat into the gas. The heating is stopped when the volume of the gas reaches $2V_0$ and its pressure reaches $2P_0$. What is the total heat $Q$ supplied by the heater during this process?

An ideal monatomic gas ($\gamma = 5/3$) flows steadily through a horizontal, thermally insulated pipe of constant cross-sectional area. At section 1, the gas velocity is $v_0$, pressure is $P_0$, and density is $\rho_0$. Due to viscous dissipation, the pressure drops to $P_0/4$ at section 2. If the initial state satisfies $\frac{\gamma}{\gamma-1} \frac{P_0}{\rho_0} = 16 v_0^2$, what is the gas velocity $v$ at section 2?

A thermally insulated cylinder contains an ideal monoatomic gas at initial pressure $P_0$ and volume $V_0$. A piston of cross-sectional area $A$ is connected to a rack-and-pinion mechanism. The pinion gear has radius $R$ and is hinged inside the cylinder such that any heat generated by friction at the hinge is entirely transferred to the gas. The hinge provides a constant frictional torque $\tau$ opposing any rotation. An external agent slowly rotates the gear, compressing the gas to a final volume $\frac{V_0}{8}$. The linear displacement $x$ of the piston is related to the angular displacement $\theta$ of the gear by $x = R\theta$. Which of the following expressions gives the final pressure of the gas?

A uniform spherical cloud of an ideal gas of total mass $M$ and initial radius $R_0$ undergoes a quasi-static, isothermal compression in deep space to a final radius $\frac{R_0}{2}$. The compression is driven by the cloud's own gravitational self-attraction, and an external constraint ensures the cloud maintains a uniform mass density profile throughout the process. What is the magnitude of the heat rejected by the gas cloud to the surrounding space?