entropy 7

For an ideal monatomic gas, how does the number of microstates depend on macroscopic variables?

Ω(U,V,N) ∝ V^N · U^(3N/2)

Why does increasing volume increase entropy even if temperature stays constant?

Increasing volume increases positional microstates available to particles.

Why does increasing temperature increase entropy even if volume stays constant?

Higher energy allows more possible momentum microstates.

Why does entropy scale with particle number N?

Each particle adds degrees of freedom, multiplying possible microstates.

If volume doubles for one particle, how does Ω change?

It doubles.

If volume doubles for N particles, how does Ω change?

Ω ∝ V^N so microstates increase exponentially.

Example: if 4 particles double volume, how does Ω change?

Ω₂/Ω₁ = (V₂/V₁)^4 = 2^4 = 16

Why does entropy depend on the logarithm of microstates?

Because microstates multiply while entropy must scale linearly with system size.

Using Boltzmann equation, what is entropy change between two states?

ΔS = k_B ln(Ω₂/Ω₁)

Using Ω ∝ V^N, derive entropy change for volume change.

ΔS = k_B N ln(V₂/V₁)

Convert the particle entropy equation to the mole form.

k_B N = nR so ΔS = nR ln(V₂/V₁)

Why can ΔS be calculated using reversible heat even if the process is irreversible?

Because entropy is a state function.

Why does the entropy formula use q_rev instead of actual heat?

Entropy is defined using a reversible reference path.

If a process is irreversible, what heat must be used to calculate ΔS?

The reversible heat q_rev.

Does entropy depend on the path taken?

No, it depends only on initial and final states.

For isothermal expansion of an ideal gas, what is ΔU?

ΔU = 0

Why is ΔU = 0 during isothermal expansion?

Internal energy depends only on temperature.

If ΔU = 0 during isothermal expansion, what relationship must hold between q and w?

q = −w

What is the reversible work for isothermal expansion?

w = −nRT ln(V₂/V₁)

What is the heat absorbed during reversible isothermal expansion?

q = +nRT ln(V₂/V₁)

What is the entropy change for isothermal expansion?

ΔS = nR ln(V₂/V₁)

Why does heating at constant volume increase entropy?

Increasing temperature increases the number of accessible energy microstates.

What is the entropy change at constant volume when temperature changes?

ΔS = nC_V ln(T₂/T₁)

What is the heat equation at constant volume?

qv = nCV ΔT

Why is work zero at constant volume?

w = −PΔV and ΔV = 0.

What is the entropy change at constant pressure when temperature changes?

ΔS = nC_P ln(T₂/T₁)

At constant pressure, what thermodynamic quantity equals heat?

q_p = ΔH

What is the entropy formula involving pressure and temperature?

ΔS = nC_P ln(T₂/T₁) − nR ln(P₂/P₁)

What contributes to positional entropy?

An increase in volume or available space.

What contributes to thermal entropy?

An increase in temperature.

What is the full entropy equation for an ideal gas with both V and T changes?

ΔS = nR ln(V₂/V₁) + nC_V ln(T₂/T₁)

What is the entropy change of the surroundings?

ΔSsurr = −qsys / T_surr

Why do we use the surroundings temperature instead of the system temperature?

The surroundings act as a large constant-temperature reservoir.

Why does heat for surroundings have the opposite sign of system heat?

Energy conservation between system and surroundings.

What is the condition for a spontaneous process?

ΔS_universe > 0

What is the condition for equilibrium?

ΔS_universe = 0

What happens if ΔS_universe < 0?

The process is nonspontaneous.

What must be true for reversible expansion?

External pressure must be infinitesimally close to internal pressure.

What is the work equation for irreversible expansion against constant external pressure?

w = −P_ext ΔV

Why does reversible expansion produce more work than irreversible expansion?

The gas pushes against higher pressure throughout the expansion.

What is the sign of entropy change during expansion?

Positive.

What is the sign of entropy change during compression?

Negative.

What is the sign of entropy change during heating?

Positive.

What is the sign of entropy change during cooling?

Negative.

Why is ΔS_system identical for reversible and irreversible paths?

Because entropy depends only on initial and final states.

Which entropy term differs between reversible and irreversible processes?

The entropy change of the surroundings.

Why is ΔS_universe larger for irreversible processes?

Irreversible processes generate additional entropy.

What is ΔS_universe for a reversible process?

0

What is ΔS_universe for a spontaneous irreversible process?

Positive.

Why does adding heat at higher temperature produce a smaller entropy change?

Because ΔS = q/T and larger T reduces ΔS.