1. What is the primary goal of statistical mechanics?
a) To study the behavior of individual particles
b) To predict macroscopic properties from microscopic interactions
c) To solve differential equations
d) To design experimental setups
Answer: b) To predict macroscopic properties from microscopic interactions
2. Which quantity represents the number of microstates corresponding to a given macrostate?
a) Partition function
b) Boltzmann factor
c) Entropy
d) Density of states
Answer: d) Density of states
3. In the canonical ensemble, which function provides the probability of a system being in a particular state?
a) Partition function
b) Free energy
c) Boltzmann factor
d) Partition coefficient
Answer: c) Boltzmann factor
4. Which statistical distribution describes the number of particles in various energy states in a system of non-interacting, indistinguishable particles?
a) Maxwell-Boltzmann distribution
b) Fermi-Dirac distribution
c) Bose-Einstein distribution
d) Poisson distribution
Answer: a) Maxwell-Boltzmann distribution
5. What is the key characteristic of a system described by the Fermi-Dirac distribution?
a) Particles are indistinguishable and follow classical statistics
b) Particles are distinguishable and follow quantum statistics
c) Particles obey the Pauli exclusion principle
d) Particles can occupy the same energy state without restriction
Answer: c) Particles obey the Pauli exclusion principle
6. In the grand canonical ensemble, which quantity is fixed?
a) Volume
b) Temperature
c) Chemical potential
d) Energy
Answer: c) Chemical potential
7. Which quantity measures the disorder or randomness in a system?
a) Enthalpy
b) Entropy
c) Free energy
d) Helmholtz free energy
Answer: b) Entropy
8. What does the Gibbs free energy
𝐺
G measure?
a) The total energy of the system
b) The maximum reversible work done by the system
c) The entropy change of the system
d) The chemical potential
Answer: b) The maximum reversible work done by the system
9. In the microcanonical ensemble, which quantities are fixed?
a) Temperature and volume
b) Energy and volume
c) Energy and number of particles
d) Temperature and chemical potential
Answer: c) Energy and number of particles
10. Which distribution describes particles in a system where indistinguishable particles follow quantum statistics and do not obey the Pauli exclusion principle?
a) Bose-Einstein distribution
b) Fermi-Dirac distribution
c) Maxwell-Boltzmann distribution
d) Poisson distribution
Answer: a) Bose-Einstein distribution
11. The concept of microstates and macrostates is fundamental to which theory?
a) Thermodynamics
b) Quantum mechanics
c) Statistical mechanics
d) Classical mechanics
Answer: c) Statistical mechanics
12. In the grand canonical ensemble, what is the role of the chemical potential
𝜇
μ?
a) It fixes the energy of the system
b) It fixes the temperature of the system
c) It fixes the number of particles in the system
d) It fixes the average number of particles in the system
Answer: d) It fixes the average number of particles in the system
13. Which ensemble is used to describe systems with fixed temperature and volume but varying number of particles?
a) Microcanonical ensemble
b) Canonical ensemble
c) Grand canonical ensemble
d) Isothermal ensemble
Answer: c) Grand canonical ensemble
14. In statistical mechanics, the concept of equipartition of energy states that:
a) All energy levels are equally occupied
b) Each degree of freedom contributes equally to the total energy
c) Energy is equally distributed among all particles
d) Energy levels are quantized and discrete
Answer: b) Each degree of freedom contributes equally to the total energy
15. The Boltzmann constant
𝑘
k is used to relate which two quantities?
a) Energy and temperature
b) Pressure and volume
c) Entropy and volume
d) Free energy and number of particles
Answer: a) Energy and temperature
16. Which distribution describes the probability of occupancy of energy states by fermions?
a) Bose-Einstein distribution
b) Maxwell-Boltzmann distribution
c) Fermi-Dirac distribution
d) Poisson distribution
Answer: c) Fermi-Dirac distribution
17. In statistical mechanics, the partition function
𝑍
Z is a function of:
a) Temperature only
b) Volume only
c) Energy levels and temperature
d) Chemical potential only
Answer: c) Energy levels and temperature
18. In the canonical ensemble, which quantity is minimized to find the equilibrium state?
a) Energy
b) Helmholtz free energy
c) Gibbs free energy
d) Entropy
Answer: b) Helmholtz free energy
19. In the microcanonical ensemble, what quantity is not fixed?
a) Energy
b) Number of particles
c) Volume
d) Temperature
Answer: d) Temperature
20. What does the Helmholtz free energy
𝐹
F represent in thermodynamics?
a) The energy available for work
b) The total internal energy
c) The work done at constant pressure
d) The maximum work obtainable at constant volume
Answer: d) The maximum work obtainable at constant volume
21. Which principle states that the entropy of a system increases with the number of accessible microstates?
a) Second Law of Thermodynamics
b) Third Law of Thermodynamics
c) Boltzmann Principle
d) Maxwell’s Demon
Answer: c) Boltzmann Principle
22. In the Bose-Einstein distribution, what characterizes the particles involved?
a) Fermions with indistinguishable states
b) Bosons with indistinguishable states
c) Classical particles with distinguishable states
d) Ideal gas particles
Answer: b) Bosons with indistinguishable states
23. Which quantity is minimized in the grand canonical ensemble to find the equilibrium state?
a) Helmholtz free energy
b) Gibbs free energy
c) Grand potential
d) Entropy
Answer: c) Grand potential