Online Exam: Andrews' Experiment

1. In Andrews' experiment, the critical temperature is identified as the temperature at which:

(A) Pressure becomes zero (B) Volume becomes infinite (C) Distinction between gas and liquid vanishes (D) Isotherms intersect

2. The isotherms in Andrews’ experiment show a flat portion at certain temperatures. What physical process does this represent?

(A) Sublimation (B) Condensation of gas into liquid (C) Solidification (D) Heating at constant pressure

3. Using the van der Waals equation, derive the critical constants in terms of the constants \( a \) and \( b \). Which of the following correctly represents the critical volume \( V_c \)?

(A) \( V_c = b \) (B) \( V_c = 2b \) (C) \( V_c = 3b \) (D) \( V_c = 4b \)

4. Calculate the work done during the isothermal compression from point A to point B on an isotherm below the critical temperature using the van der Waals equation. Assume 1 mole of gas.

(A) \( W = nRT \ln(V_B/V_A) \) (B) \( W = nRT \ln\left( \frac{V_B - b}{V_A - b} \right) - a\left( \frac{1}{V_B} - \frac{1}{V_A} \right) \) (C) \( W = -a(V_B - V_A) \) (D) None of the above

5. For a van der Waals gas undergoing an isothermal expansion from \( V_1 \) to \( V_2 \), what is the condition for the work done to be positive?

(A) \( V_2 < V_1 \) (B) \( V_2 > V_1 \) (C) Work is always zero (D) Only at the critical point

6. Using the van der Waals equation, derive the critical pressure \( P_c \) in terms of \( a \) and \( b \). Which of the following is correct?

(A) \( P_c = \frac{a}{b^2} \) (B) \( P_c = \frac{a}{27b^2} \) (C) \( P_c = \frac{3a}{8b^2} \) (D) \( P_c = \frac{a}{9b^2} \)

7. Consider a gas that obeys the van der Waals equation. Which of the following graphs best explains the Andrews isotherms for temperatures below the critical temperature?

(A) Monotonically increasing \( P \) with \( V \) (B) Horizontal line for all \( V \) (C) A curve with a region of negative slope and a flat portion (D) An ellipse

8. Which of the following correctly expresses the critical compressibility factor \( Z_c = \frac{P_c V_c}{RT_c} \) for a van der Waals gas?

(A) \( Z_c = \frac{1}{3} \) (B) \( Z_c = \frac{3}{8} \) (C) \( Z_c = \frac{8}{27} \) (D) \( Z_c = 1 \)

9. At the critical point, the van der Waals isotherm has:

(A) A maximum (B) A minimum (C) An inflection point (D) A point of discontinuity

10. For a van der Waals gas, suppose at temperature \( T = 0.9 T_c \), the isotherm shows a flat horizontal segment between volumes \( V_l \) and \( V_g \). What does the area under this flat portion represent?

(A) The work done during isothermal compression (B) The latent heat (C) The coexistence pressure and phase transition (D) It has no physical meaning

11. According to the Zeroth Law of Thermodynamics, if system A is in thermal equilibrium with system B, and B is in thermal equilibrium with system C, then:

(A) System A is hotter than system C (B) System A is in thermal equilibrium with system C (C) System A and C exchange no heat (D) System B is cooler than A and C

12. Derive the condition for thermodynamic equilibrium considering thermal, mechanical, and chemical equilibrium.

(A) Temperature must be different throughout the system (B) Pressure should vary to balance heat flow (C) Temperature, pressure, and chemical potential must be uniform (D) Only thermal equilibrium is required

13. Derive the isothermal compressibility \( \kappa_T \) of an ideal gas. Use the ideal gas law \( PV = nRT \).

(A) \( \kappa_T = \frac{1}{P} \) (B) \( \kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T \) (C) \( \kappa_T = \left( \frac{\partial T}{\partial V} \right)_P \) (D) \( \kappa_T = -\left( \frac{\partial P}{\partial V} \right)_T \)

14. Derive the expression for work done during a reversible isothermal expansion of an ideal gas from volume \( V_1 \) to \( V_2 \).

(A) \( W = nRT \ln\left( \frac{V_1}{V_2} \right) \) (B) \( W = nRT \ln\left( \frac{V_2}{V_1} \right) \) (C) \( W = \frac{P_1 - P_2}{V} \) (D) \( W = \frac{1}{2} nRT \ln(V) \)

15. Derive the work done in a reversible adiabatic expansion of an ideal gas from \( V_1 \) to \( V_2 \).

(A) \( W = \frac{nR(T_2 - T_1)}{\gamma - 1} \) (B) \( W = \frac{P_1V_1 - P_2V_2}{\gamma - 1} \) (C) \( W = \frac{P_1V_1}{\gamma - 1} \left[ 1 - \left( \frac{V_1}{V_2} \right)^{\gamma - 1} \right] \) (D) All of the above

16. Show that for an ideal gas, the difference in specific heats is \( C_P - C_V = R \).

(A) \( C_P + C_V = R \) (B) \( C_P - C_V = R \) (C) \( C_P / C_V = R \) (D) \( C_P = C_V \)

17. A system absorbs 300 J of heat and does 100 J of work. What is the change in internal energy?

(A) 400 J (B) 200 J (C) 100 J (D) -200 J

18. Derive the efficiency of a Carnot engine operating between temperatures \( T_H \) and \( T_C \).

(A) \( \eta = \frac{T_H - T_C}{T_H} \) (B) \( \eta = \frac{T_C}{T_H} \) (C) \( \eta = 1 - \frac{T_C}{T_H} \) (D) \( \eta = 1 - \frac{T_H}{T_C} \)

19. A Carnot engine absorbs 800 J from a hot reservoir and expels 500 J to the cold reservoir. What is the work done per cycle?

(A) 1300 J (B) 300 J (C) 500 J (D) 800 J

20. Define the second law of thermodynamics using Clausius' statement.

(A) Heat can flow from cold to hot spontaneously (B) It is impossible to convert heat into work (C) No process is possible in which heat is transferred from a colder to a hotter body without work (D) Work done is always positive

21. Show how the Kelvin scale of temperature is derived from the efficiency of a Carnot engine.

(A) \( \frac{Q_C}{Q_H} = \frac{T_H}{T_C} \) (B) \( \frac{Q_C}{Q_H} = \frac{T_C}{T_H} \) (C) \( Q_H = Q_C \) (D) \( Q_C = 0 \)

22. Derive the expression for the coefficient of performance (COP) of a Carnot refrigerator.

(A) \( \text{COP} = \frac{T_H}{T_H - T_C} \) (B) \( \text{COP} = \frac{T_C}{T_H - T_C} \) (C) \( \text{COP} = \frac{Q_H}{Q_C} \) (D) \( \text{COP} = \frac{Q_C - Q_H}{T_C} \)

23. Define the Clausius-Clapeyron equation and write its form for a liquid-vapour phase transition.

(A) \( \frac{dT}{dP} = \frac{L}{T(V_v - V_l)} \) (B) \( \frac{dP}{dT} = \frac{L}{T(V_v - V_l)} \) (C) \( \frac{dP}{dT} = \frac{L}{V_v + V_l} \) (D) \( \frac{dP}{dT} = \frac{T}{L(V_v - V_l)} \)

24. A heat pump delivers 1500 J of heat to a room by consuming 500 J of work. What is its coefficient of performance?

(A) 1 (B) 2 (C) 3 (D) 4

25. Show that entropy change \( \Delta S \) for an ideal gas heated from \( T_1 \) to \( T_2 \) at constant volume is \( \Delta S = nC_V \ln\left( \frac{T_2}{T_1} \right) \).

(A) \( \Delta S = nR \ln\left( \frac{T_2}{T_1} \right) \) (B) \( \Delta S = nC_P \ln\left( \frac{T_2}{T_1} \right) \) (C) \( \Delta S = nC_V \ln\left( \frac{T_2}{T_1} \right) \) (D) \( \Delta S = nT \ln\left( \frac{C_P}{C_V} \right) \)

26. Using Clausius' and Kelvin's statements, explain why all heat engines must reject some heat to a cold reservoir and cannot be 100% efficient.

(A) Because temperature is always changing (B) Because heat flow always occurs from hot to cold spontaneously (C) Because Clausius' statement forbids perfect heat-to-work conversion (D) Both B and C

27. A Carnot engine operates between \( T_H = 600\,\mathrm{K} \) and \( T_C = 300\,\mathrm{K} \). Derive its efficiency and calculate how much work is done if it absorbs 900 J of heat from the hot reservoir.

(A) 300 J (B) 450 J (C) 600 J (D) 900 J

28. Derive the expression for the coefficient of performance (COP) of a Carnot heat pump operating between \( T_H \) and \( T_C \).

(A) \( \frac{T_H}{T_H - T_C} \) (B) \( \frac{T_C}{T_H - T_C} \) (C) \( \frac{T_C}{T_H} \) (D) \( \frac{T_H - T_C}{T_C} \)

29. Derive the Clausius-Clapeyron equation for a first-order phase transition and explain its physical significance.

(A) \( \frac{dT}{dP} = \frac{L}{T(V_v - V_l)} \) (B) \( \frac{dP}{dT} = \frac{L}{T(V_v - V_l)} \) (C) \( \frac{dP}{dT} = \frac{T}{L(V_v - V_l)} \) (D) \( \frac{dP}{dT} = \frac{L}{V_v + V_l} \)

30. A Carnot refrigerator operates between 277 K and 300 K. Calculate its coefficient of performance (COP).

(A) 11.1 (B) 12.0 (C) 13.5 (D) 14.3

31. Define entropy and derive its expression for a reversible process. Use the second law to explain the physical meaning of entropy.

(A) \( dS = \frac{dQ}{T} \) for all processes (B) \( dS = \frac{dQ_{\text{rev}}}{T} \) for reversible processes only (C) \( dS = TdQ \) (D) \( dS = \frac{dQ}{T^2} \)

32. State and prove the principle of increase of entropy for an isolated system. Which of the following is true for irreversible processes?

(A) \( \Delta S = 0 \) (B) \( \Delta S < 0 \) (C) \( \Delta S > 0 \) (D) Entropy is undefined

33. Derive the expression for the change in entropy during heat conduction between two bodies at temperatures \( T_1 \) and \( T_2 \), where \( T_1 > T_2 \).

(A) \( \Delta S = 0 \) (B) \( \Delta S = Q \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \) (C) \( \Delta S = Q \left( \frac{T_2 - T_1}{T_1 T_2} \right) \) (D) \( \Delta S = \ln\left( \frac{T_2}{T_1} \right) \)

34. Calculate the change in entropy for an ideal gas when it is reversibly heated from \( T_1 \) to \( T_2 \) at constant volume. Let \( C_V \) be the specific heat at constant volume.

(A) \( \Delta S = C_P \ln\left( \frac{T_2}{T_1} \right) \) (B) \( \Delta S = nR \ln\left( \frac{T_2}{T_1} \right) \) (C) \( \Delta S = nC_V \ln\left( \frac{T_2}{T_1} \right) \) (D) \( \Delta S = nT \ln\left( \frac{C_P}{C_V} \right) \)

35. Using a TS diagram, derive the efficiency of a Carnot engine and relate it to entropy changes of the system and reservoirs.

(A) \( \eta = \frac{T_C}{T_H} \) (B) \( \eta = 1 - \frac{T_C}{T_H} \) (C) \( \eta = \frac{T_H}{T_C} \) (D) \( \eta = \frac{T_H - T_C}{T_H + T_C} \)

36. Derive any one of Maxwell’s thermodynamic relations starting from a thermodynamic potential and explain the physical significance of the relation.

(A) \( \left( \frac{\partial T}{\partial V} \right)_S = \left( \frac{\partial P}{\partial S} \right)_V \) (B) \( \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V \) (C) \( \left( \frac{\partial S}{\partial P} \right)_T = -\left( \frac{\partial V}{\partial T} \right)_P \) (D) All of the above

37. Derive the two TdS equations from the first and second laws of thermodynamics for a simple compressible system.

(A) \( TdS = dU + PdV \) (B) \( TdS = dH - VdP \) (C) Both A and B (D) None of these

38. Show that \( C_P - C_V = T \left( \frac{\partial P}{\partial T} \right)_V \left( \frac{\partial V}{\partial T} \right)_P \) using Maxwell relations and the TdS equations.

(A) \( C_P - C_V = T\left( \frac{\partial V}{\partial T} \right)_P^2 \left( \frac{\partial P}{\partial V} \right)_T \) (B) \( C_P - C_V = T \left( \frac{\partial P}{\partial T} \right)_V \left( \frac{\partial V}{\partial T} \right)_P \) (C) \( C_P - C_V = R \) (D) None of the above

39. Starting from appropriate thermodynamic potentials, derive the expressions for enthalpy \( H \), Helmholtz free energy \( F \), and Gibbs free energy \( G \).

(A) \( H = U + PV, F = U - TS, G = H - TS \) (B) \( H = U - PV, F = TS - U, G = PV - TS \) (C) \( H = TS + PV, F = U + TS, G = H + TS \) (D) \( H = U + TS, F = U + PV, G = F - TS \)

40. State the third law of thermodynamics and explain its consequences on the behavior of entropy at absolute zero temperature.

(A) Entropy at absolute zero is zero for all substances (B) Entropy at zero temperature is undefined (C) Entropy approaches a constant as \( T \to 0 \) (D) Entropy becomes infinite as \( T \to 0 \)

41. Derive the equation for steady-state one-dimensional heat conduction through a slab. Define thermal conductivity in terms of heat flux.

(A) \( q = -k \frac{dT}{dx} \) (B) \( q = - \frac{dT}{dx} \) (C) \( q = -kT \) (D) \( q = -k \frac{dx}{dT} \)

42. In Lee’s disc experiment, derive the expression for the thermal conductivity \( k \) of a bad conductor using steady state condition.

(A) \( k = \frac{m s (\Delta T)}{A t} \) (B) \( k = \frac{m s \frac{dT}{dt} \cdot x}{A(T_1 - T_2)} \) (C) \( k = \frac{A(T_1 - T_2)}{x \cdot t} \) (D) \( k = \frac{x \cdot (T_1 - T_2)}{m s \cdot t} \)

43. Define thermal resistance and derive its expression for a conducting slab of thickness \( x \), area \( A \), and thermal conductivity \( k \).

(A) \( R_{th} = \frac{x}{kA} \) (B) \( R_{th} = \frac{k}{Ax} \) (C) \( R_{th} = kAx \) (D) \( R_{th} = \frac{1}{kAx} \)

44. Define energy flux, intensity, and radiant emittance. Which of the following correctly expresses Stefan's law of radiation?

(A) \( Q = \sigma A T \) (B) \( Q = \sigma A T^4 \) (C) \( Q = \sigma T^4 \) (D) \( Q = A T^4 \)

45. A black body emits thermal radiation with a power \( Q \). If its absolute temperature is doubled, what happens to the power emitted?

(A) It doubles (B) It increases four times (C) It increases eight times (D) It increases sixteen times

46. Define microstates and macrostates. Derive the relation between entropy and the number of microstates \( \Omega \) using Boltzmann’s principle.

(A) \( S = k \Omega \) (B) \( S = k \ln(\Omega) \) (C) \( S = \ln(\Omega^k) \) (D) \( S = \frac{1}{\Omega} \)

47. Explain the concept of phase space. For a system of \( N \) particles, how many dimensions does the phase space have? Distinguish between \( \mu \)-space and \( \Gamma \)-space.

(A) \( 6N \) and \( \mu \)-space is 6D (B) \( 6N \) and \( \mu \)-space is single-particle space (C) \( \mu \)-space for N particles is same as \( \Gamma \)-space (D) None of the above

48. State and explain the principle of equal a priori probability. Derive its implication in microcanonical ensemble.

(A) All states have unequal probability (B) Probability \( p_i = \frac{1}{\Omega} \) in microcanonical ensemble (C) Only ground state is accessible (D) Probabilities are time-dependent

49. Explain the ergodic hypothesis. How is it related to statistical equilibrium and time averages?

(A) Ergodicity implies all microstates are visited equally (B) Time average equals ensemble average (C) System evolves to statistical equilibrium (D) All of the above

50. Derive the expression for the number of microstates \( \Omega(E) \) for a microcanonical ensemble confined to energy shell \( [E, E + \delta E] \).

(A) \( \Omega = \frac{1}{h^{3N} N!} \int \delta(E - H(p,q)) \, d^{3N}p \, d^{3N}q \) (B) \( \Omega = \sum \delta(E + H(p,q)) \) (C) \( \Omega = \int e^{-\beta E} \, dE \) (D) \( \Omega = \frac{1}{Z} \)

51. Define the canonical ensemble. Derive the expression for the canonical partition function \( Z \).

(A) \( Z = \int e^{-\beta H} \, d\Gamma \) (B) \( Z = \sum e^{-\beta E_i} \) (C) Both A and B (D) \( Z = e^{\beta E} \)

52. Using the canonical ensemble, derive an expression for the average energy \( \langle E \rangle \) in terms of the partition function \( Z \).

(A) \( \langle E \rangle = -\frac{\partial \ln Z}{\partial \beta} \) (B) \( \langle E \rangle = \frac{1}{Z} \sum E_i \) (C) \( \langle E \rangle = \frac{\partial Z}{\partial \beta} \) (D) \( \langle E \rangle = -\frac{\partial Z}{\partial T} \)

53. Derive the equipartition theorem and show that each quadratic degree of freedom contributes \( \frac{1}{2}kT \) to the average energy.

(A) \( \langle E \rangle = \frac{1}{2}kT \) per quadratic term (B) \( \langle E \rangle = kT \) per particle (C) \( \langle E \rangle = \frac{3}{2}kT \) per ensemble (D) Energy is constant

54. Define the grand canonical ensemble. Derive the grand partition function \( \mathcal{Z} \) and the average particle number.

(A) \( \mathcal{Z} = \sum_{N} \sum_{i} e^{\beta (\mu N - E_i)} \) (B) \( \langle N \rangle = \frac{\partial \ln \mathcal{Z}}{\partial \mu} \) (C) Both A and B (D) \( \mathcal{Z} = \int \Omega(E) \, dE \)

55. Explain how the partition function encodes the thermodynamic properties of a system. Which of the following quantities can be derived from it?

(A) Average energy (B) Entropy (C) Helmholtz free energy (D) All of the above

56. Derive the Maxwell-Boltzmann distribution function for a classical ideal gas.

(A) \( f(\epsilon) = \frac{1}{e^{(\epsilon - \mu)/kT} + 1} \) (B) \( f(\epsilon) = \frac{1}{e^{(\epsilon - \mu)/kT} - 1} \) (C) \( f(\epsilon) = A e^{-\epsilon/kT} \) (D) \( f(\epsilon) = A \epsilon^2 e^{-\epsilon/kT} \)

57. Derive the Fermi-Dirac distribution function for indistinguishable fermions and explain its significance at \( T = 0 \).

(A) \( f(\epsilon) = \frac{1}{e^{(\epsilon - \mu)/kT} + 1} \) (B) \( f(\epsilon) = \frac{1}{e^{(\epsilon - \mu)/kT} - 1} \) (C) \( f(\epsilon) = e^{-\epsilon/kT} \) (D) \( f(\epsilon) = A \epsilon^2 e^{-\epsilon/kT} \)

58. Derive the Bose-Einstein distribution law for indistinguishable bosons and explain the conditions under which Bose-Einstein condensation occurs.

(A) \( f(\epsilon) = \frac{1}{e^{(\epsilon - \mu)/kT} - 1} \) (B) \( f(\epsilon) = \frac{1}{e^{(\epsilon - \mu)/kT} + 1} \) (C) \( f(\epsilon) = e^{-\epsilon/kT} \) (D) \( f(\epsilon) = \frac{1}{e^{-\mu/kT}} \)

59. Using Maxwell-Boltzmann distribution, derive the expression for the average energy of a particle in a classical ideal gas.

(A) \( \langle E \rangle = \frac{1}{2}kT \) (B) \( \langle E \rangle = \frac{3}{2}kT \) (C) \( \langle E \rangle = \frac{5}{2}kT \) (D) \( \langle E \rangle = kT \)

60. Show that the Fermi energy is the energy of the highest occupied level at absolute zero. Derive an expression for Fermi energy \( \epsilon_F \) of a 3D free electron gas.

(A) \( \epsilon_F = \frac{\hbar^2}{2m} \left( 3\pi^2 n \right)^{2/3} \) (B) \( \epsilon_F = \frac{1}{2}mv^2 \) (C) \( \epsilon_F = \frac{3}{2}kT \) (D) \( \epsilon_F = \frac{h}{2\pi} \)

61. A classical gas of \( N \) particles occupies volume \( V \). Using MB statistics, derive the expression for the number of particles with energy between \( \epsilon \) and \( \epsilon + d\epsilon \).

(A) \( n(\epsilon) = A e^{-\epsilon/kT} \) (B) \( n(\epsilon) = A \epsilon^2 e^{-\epsilon/kT} \, d\epsilon \) (C) \( n(\epsilon) = A \epsilon e^{-\epsilon/kT} \, d\epsilon \) (D) \( n(\epsilon) = A e^{\epsilon/kT} \, d\epsilon \)

62. Compare the three distributions — MB, BE, and FD — and determine their limits when \( (\epsilon - \mu) \gg kT \).

(A) All reduce to Bose-Einstein form (B) All reduce to \( f(\epsilon) = A e^{-\epsilon/kT} \) (C) All reduce to Maxwell-Boltzmann form (D) None of the above

63. At what temperature does Bose-Einstein condensation begin? Derive an expression for the critical temperature \( T_c \) for an ideal Bose gas in 3D.

(A) \( T_c = \frac{2\pi \hbar^2}{mk} \left( \frac{n}{\zeta(3/2)} \right)^{2/3} \) (B) \( T_c = \frac{n}{\zeta(3/2)} \) (C) \( T_c = \frac{1}{2}kT \) (D) \( T_c = \frac{3}{2}kT \)

64. For a 3D Fermi gas at \( T = 0 \), derive the total energy and show that the average energy per particle is \( \frac{3}{5} \epsilon_F \).

(A) \( \langle E \rangle = \epsilon_F \) (B) \( \langle E \rangle = \frac{3}{5} \epsilon_F \) (C) \( \langle E \rangle = \frac{1}{2} \epsilon_F \) (D) \( \langle E \rangle = \frac{5}{3} \epsilon_F \)

65. Derive the expression for pressure of a Fermi gas at \( T = 0 \) and show that it is related to the Fermi energy.

(A) \( P = \frac{2}{5} n \epsilon_F \) (B) \( P = n \epsilon_F \) (C) \( P = \frac{3}{5} n \epsilon_F \) (D) \( P = \epsilon_F \)

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