Statistical Mechanics: Distribution of Energy, Entropy, and Boltzmann Distribution

Question 1

Which concept refers to the measure of randomness or disorder within a system?

Accepted Answer

Entropy

Answer

Energy

Answer

Temperature

Answer

Heat capacity

Question 2

The Third Law of Thermodynamics states that:

Accepted Answer

The entropy of a perfect crystal at absolute zero is zero.

Answer

The entropy of a system always increases with increasing temperature.

Answer

The entropy of a closed system can never decrease.

Answer

The entropy of the universe remains constant.

Question 3

The microstate of a system refers to:

Accepted Answer

A specific arrangement and momentum of its particles

Answer

The total energy of the system

Answer

The probability of the system being in a particular state

Answer

The entropy of the system

Question 4

The Boltzmann constant (k) is equal to:

Accepted Answer

1.38 × 10^-23 J/K

Answer

8.62 × 10^-5 eV/K

Answer

6.63 × 10^-34 J/Hz

Answer

1.60 × 10^-19 C

Question 5

In an irreversible process, the entropy of the system:

Accepted Answer

Increases

Answer

Decreases

Answer

Remains constant

Answer

Can either increase or decrease

Question 6

Which distribution describes the most probable distribution of energy among particles?

Accepted Answer

Boltzmann distribution

Answer

Planck distribution

Answer

Maxwell distribution

Answer

Fermi distribution

Question 7

Entropy is a measure of what property of a system?

Accepted Answer

Disorder or randomness

Answer

Energy

Answer

Temperature

Answer

Pressure

Question 8

What is the approximate value of the Boltzmann constant?

Accepted Answer

1.38064852 × 10^-23 J/K

Answer

6.62607015 × 10^-34 J/s

Answer

6.02214076 × 10^23 mol^-1

Answer

1.602176634 × 10^-19 C

Question 9

According to the statistical interpretation, entropy is related to which property?

Accepted Answer

Number of possible microstates

Answer

Amount of disorder

Answer

Average energy

Answer

Absolute temperature

Question 10

Which application of statistical mechanics is used to determine the properties of gases?

Accepted Answer

Kinetic theory of gases

Answer

Thermodynamics

Answer

Quantum mechanics

Answer

Electromagnetism

Question 11

How does temperature affect the distribution of energy and entropy in a thermodynamic system?

Accepted Answer

Increasing temperature raises both average energy and entropy.

Answer

Higher temperature shifts energy distribution to higher levels, leaving entropy unchanged.

Answer

Rising temperature narrows energy distribution, reducing entropy.

Answer

Temperature has no impact on energy distribution or entropy.

Question 12

In statistical mechanics, how do statistical fluctuations help us understand the behavior of a system?

Accepted Answer

They measure the uncertainty of a system's microscopic state.

Answer

They are insignificant when a system has many particles.

Answer

They are the only cause of phase transitions and similar phenomena.

Answer

They only matter for systems in equilibrium.

Question 13

How is entropy mathematically related to the count of microstates?

Accepted Answer

Entropy is proportional to the logarithm of the number of microstates

Answer

Entropy is inversely proportional to the number of microstates

Answer

Entropy is independent of the number of microstates

Question 14

Of the following, which system possesses the highest entropy?

Accepted Answer

Gas contained in a large container

Answer

Gas at low pressure

Answer

Solid at low temperature

Answer

Liquid confined in a small container

Question 15

What is the significance of the partition function in the context of statistical mechanics?

Accepted Answer

It represents the probability distribution for all potential energy states

Answer

It determines the system's average energy

Answer

It calculates the system's entropy

Question 16

The ideal gas model employed in statistical mechanics assumes that:

Accepted Answer

Collisions between gas particles are exclusively elastic

Answer

Gas particles exhibit non-random motion

Answer

Gas particles have finite size

Question 17

How is temperature related to the mean kinetic energy of a particle system?

Accepted Answer

Mean kinetic energy is proportional to temperature

Answer

Mean kinetic energy is inversely proportional to temperature

Answer

Mean kinetic energy is independent of temperature

Question 18

What is the physical interpretation of chemical potential?

Accepted Answer

Change in free energy when a single molecule is added to the system

Answer

Energy required to remove a molecule from the system

Answer

Entropy of the system per molecule

Question 19

The Boltzmann distribution describes the probability of a molecule occupying a specific energy level. Which of the following expressions accurately represents this probability?

Accepted Answer

P(E) = (1/Q) * exp(-E/kT)

Answer

P(E) = (1/N) * exp(-E/kT)

Answer

P(E) = (1/Z) * exp(E/kT)

Answer

P(E) = exp(-E/kT)

Question 20

How is the entropy (S) of a system related to the number of microstates (W) it can occupy?

Accepted Answer

S = k ln(W)

Answer

S = k/W

Answer

S = W/k

Answer

S = W

Question 21

A gas undergoes an isothermal expansion from an initial volume V1 to a final volume V2. What is the change in entropy (ΔS) of the gas?

Accepted Answer

ΔS = nR ln(V2/V1)

Answer

ΔS = nCv ln(V2/V1)

Answer

ΔS = nCp ln(V2/V1)

Answer

ΔS = 0

Question 22

The partition function (Q) in statistical mechanics is a crucial concept. What does it represent?

Accepted Answer

The sum of Boltzmann factors over all possible energy states of a system.

Answer

The probability of a system being in a specific energy state.

Answer

The number of microstates accessible to a system.

Answer

The average energy of a system.

Question 23

What is the relationship between the partition function (Q) and the Helmholtz free energy (A) of a system?

Accepted Answer

A = -kT ln(Q)

Answer

A = kT ln(Q)

Answer

A = Q/kT

Answer

A = -Q/kT

Question 24

A system at maximum entropy is characterized by:

Accepted Answer

An equal probability of occupying any of its accessible microstates.

Answer

A state of minimum energy.

Answer

A state of maximum energy.

Answer

A single, specific microstate.

Question 25

The probability of a molecule occupying a specific energy state is primarily determined by:

Accepted Answer

The Boltzmann factor, which depends on the energy of the state and the temperature.

Answer

The volume of the system.

Answer

The number of molecules in the system.

Answer

The pressure of the system.