Statistical Thermodynamics: Master the Microscopic to Macroscopic Connection

Question 1

In statistical thermodynamics, the microcanonical ensemble is characterized by which fixed properties? Provide an example of a system that can be accurately described using this ensemble.

Accepted Answer

Fixed number of particles, volume, and energy

Answer

Fixed number of particles and temperature

Answer

Fixed number of particles and pressure

Question 2

The Maxwell-Boltzmann distribution, which describes the distribution of molecular velocities in a gas at equilibrium, depends on which of the following factors?

Accepted Answer

Both temperature and volume

Answer

Temperature only

Answer

Volume only

Answer

Mass, temperature, and volume

Question 3

According to statistical thermodynamics, which of the following is the most probable energy state of a system at absolute zero?

Accepted Answer

The ground state

Answer

The excited state

Answer

Any state is equally probable

Answer

The state with the highest energy

Question 4

Which of the following techniques can be employed to calculate the partition function?

Accepted Answer

Any of the above

Answer

Monte Carlo simulation

Answer

Molecular dynamics simulation

Answer

Density functional theory

Question 5

What is the Boltzmann factor, which represents the probability of a system occupying a specific microstate?

Accepted Answer

P = e^(-E/kT)

Answer

P = kT/E

Answer

P = 1/E

Answer

P = E/kT

Question 6

Define the Boltzmann distribution.

Accepted Answer

A probability distribution outlining the distribution of energy among a system's constituent particles.

Answer

A statistical measure quantifying the level of disorder within a system.

Answer

An equation that establishes a relationship between a system's entropy and its temperature.

Answer

A graphical representation depicting the spectrum of possible microstates within a system.

Question 7

Which of the following systems possesses the highest entropy?

Accepted Answer

A gas that has reached its critical point.

Answer

A solid existing at 0 K.

Answer

A liquid at its freezing point.

Answer

A gas at room temperature.

Question 8

How can statistical thermodynamics be employed to predict the equilibrium constant of a chemical reaction?

Accepted Answer

By calculating the difference in Gibbs free energy between the reactants and products.

Answer

By calculating the Boltzmann factor for the reaction.

Answer

By solving the Schrödinger equation for the system.

Answer

By utilizing the law of mass action.

Question 9

Which fundamental assumption underpins statistical thermodynamics?

Accepted Answer

The system comprises an extensive number of identical particles.

Answer

The system is in a state of thermal equilibrium.

Answer

The system is isolated from its surroundings.

Answer

The system's particles solely interact via a harmonic force.

Question 10

Define the Boltzmann distribution.

Accepted Answer

A probability distribution describing the energy distribution among particles within a system.

Answer

A thermodynamic equation relating entropy change to temperature variation.

Answer

An equation correlating a gas's pressure to its volume and temperature.

Answer

A model representing the behavior of ideal gases.

Question 11

Which of the following quantifies a system's disorder?

Accepted Answer

Entropy

Answer

Free energy

Answer

Internal energy

Answer

Enthalpy

Question 12

State a consequence of the second law of thermodynamics.

Accepted Answer

The entropy of an isolated system invariably increases over time.

Answer

The entropy of a system can never decrease.

Answer

A system's entropy can only decrease if work is done on it.

Answer

A system's entropy remains constant.

Question 13

Define the Gibbs free energy.

Accepted Answer

The maximum work extractable from a system at constant temperature and pressure.

Answer

The minimum work required to bring a system to a given state.

Answer

The change in internal energy at constant temperature and volume.

Answer

The change in entropy at constant temperature and pressure.

Question 14

What is the partition function?

Accepted Answer

A function describing the energy distribution among particles in a system.

Answer

A function relating a gas's pressure to its volume and temperature.

Answer

A function describing the change in entropy with temperature.

Answer

A function relating a system's free energy to its temperature.

Question 15

Explain the concept of 'ergodicity' in statistical thermodynamics.

Accepted Answer

The time average of a system's properties equals the ensemble average.

Answer

A system's properties do not change over time.

Answer

A system's properties are determined by its initial conditions.

Answer

A system's properties are random.

Question 16

In statistical thermodynamics, which postulate suggests that all microstates of a given macrostate have an equal probability of occurrence?

Accepted Answer

Equal a priori probability

Answer

Boltzmann's constant

Answer

Maxwell-Boltzmann distribution

Answer

Gibbs free energy

Question 17

Which mathematical expression represents the probability of finding a particle occupying a specific energy state at temperature T?

Accepted Answer

e^(-E/kT)

Answer

e^(E/kT)

Answer

kT/E

Answer

E/kT

Question 18

In statistical thermodynamics, which formula accurately defines entropy?

Accepted Answer

k * ln(W)

Answer

k * ln(E)

Answer

E * k * ln(T)

Answer

T * k * ln(E)

Question 19

What is the defining criterion for a system to be considered in thermal equilibrium?

Accepted Answer

Zero net energy transfer between the system and its environment.

Answer

Zero net momentum transfer between the system and its environment.

Answer

Constant internal energy within the system.

Answer

Equal temperatures between the system and its environment.

Question 20

Which of the following accurately describes a microstate?

Accepted Answer

It delineates the precise position and momentum of each particle within a given system.

Answer

It characterizes the macroscopic state of a system.

Answer

It represents a hypothetical state that may not align with reality.

Answer

It signifies the most probable state of a system.

Question 21

In statistical thermodynamics, what is the relationship between the number of microstates and the entropy of a system?

Accepted Answer

Entropy is directly proportional to the number of microstates.

Answer

Entropy is inversely proportional to the number of microstates.

Answer

Entropy is equal to the number of microstates.

Answer

Entropy is independent of the number of microstates.

Question 22

What is the key concept of ergodicity in statistical thermodynamics?

Accepted Answer

The time-averaged value of a property over an extended period equals the ensemble average.

Answer

The microstate of a system cannot be predicted with certainty.

Answer

Entropy of a system perpetually increases.

Answer

Free energy of a system is always negative.

Question 23

In phase transitions, what is the significance of critical points? How do they determine the conditions for different phases to coexist in equilibrium?

Accepted Answer

Critical points define the specific conditions where different phases can coexist in equilibrium, acting as boundaries between phases.

Answer

Critical points mark the start of phase separation, indicating the transition from one phase to another.

Answer

Critical points are unrelated to phase transitions, representing only the maximum or minimum of a thermodynamic property.

Answer

Critical points apply only to pure substances and not to mixtures.

Question 24

The Boltzmann distribution law describes the:

Accepted Answer

Probability of a system occupying a particular microstate

Answer

Entropy of a system

Answer

Total energy of a system

Question 25

Which of the following is a consequence of the second law of thermodynamics?

Accepted Answer

The entropy of an isolated system can never decrease

Answer

The entropy of an isolated system can always increase

Answer

The entropy of an isolated system can fluctuate randomly

Question 26

Identify the assumption that is not part of the ideal gas model:

Accepted Answer

The gas particles have finite volume.

Answer

Collisions between gas particles are perfectly elastic.

Answer

The gas particles behave randomly.

Answer

The average kinetic energy of the gas particles is directly proportional to the absolute temperature.

Question 27

Which of the following correctly defines the measure of a system's disorder?

Accepted Answer

Entropy

Answer

Enthalpy

Answer

Gibbs free energy

Answer

Internal energy

Question 28

Identify a direct consequence of the third law of thermodynamics:

Accepted Answer

At absolute zero, the entropy of a perfect crystal is zero.

Answer

The Gibbs free energy of a system is always negative.

Answer

As temperature approaches absolute zero, the heat capacity of a substance becomes infinite.

Question 29

Describe the Maxwell-Boltzmann distribution:

Accepted Answer

A probability distribution describing the distribution of molecular speeds in a gas.

Answer

A probability distribution describing the distribution of molecular energies in a gas.

Answer

A distribution of the number of molecules with a given energy.

Question 30

Define the Fermi-Dirac distribution:

Accepted Answer

A probability distribution describing the distribution of energies of electrons in a metal.

Answer

A probability distribution describing the distribution of energies of molecules in a gas.

Answer

A distribution of the number of electrons with a given energy.

Question 31

What is the fundamental relationship between entropy (S) and the number of microstates (W) of a system in statistical thermodynamics?

Accepted Answer

S = k * ln(W), where k is the Boltzmann constant.

Answer

S = W * k

Answer

S = ln(W) / k

Question 32

Which thermodynamic process is characterized by no change in entropy (ΔS = 0) and involves a closed system?

Accepted Answer

Reversible adiabatic process

Answer

Isothermal process

Answer

Irreversible adiabatic process

Question 33

During a phase transition from a solid to a liquid state, what is the typical change in entropy of the system?

Accepted Answer

Positive

Answer

Zero

Answer

Negative

Answer

Depends on the specific substance

Question 34

What is the mathematical definition of the partition function (Z) in statistical thermodynamics?

Accepted Answer

Z = ∑ᵢ exp(-Eᵢ / kT), where Eᵢ represents the energy of microstate i, k is the Boltzmann constant, and T is the temperature.

Answer

Z = ∑ᵢ Eᵢ * exp(-Eᵢ / kT)

Answer

Z = kT * ∑ᵢ exp(-Eᵢ)

Question 35

Which statement accurately describes the Gibbs free energy (G)?

Accepted Answer

It represents the maximum amount of non-expansion work that can be obtained from a system at constant temperature and pressure.

Answer

It is equivalent to the enthalpy change (ΔH) of the system.

Answer

It is always negative for a closed system.

Question 36

State the second law of thermodynamics.

Accepted Answer

The total entropy of an isolated system can never decrease over time.

Answer

Heat always flows from hot to cold objects.

Answer

No process can be 100% efficient.

Answer

The total energy of the universe is constant.