Worked example 1: One mole of ideal gas expands freely from V to 2V at the same temperature. What is the sign of the system entropy change?

Delta S = nR ln(V2/V1) for the equivalent reversible isothermal path
Even though the real free expansion is irreversible, the same endpoints can be connected reversibly for the system entropy calculation.
Delta S = nR ln 2 > 0
Doubling the accessible volume increases the number of positional microstates.
check: the process can have Q = 0 and still have Delta S > 0
Entropy is not merely a heat tally; multiplicity can rise even when no heat enters.

Worked example 2: Two different ideal gases, A and B, begin separated with 1 mol of each on opposite sides of a partition at the same temperature. What is the sign of the entropy change when they mix?

Delta S_mix = -R(nA ln xA + nB ln xB)
Ideal-gas mixing entropy is computed from the mole fractions in the mixed state.
xA = xB = 0.5, so Delta S_mix = -R(ln 0.5 + ln 0.5) = 2R ln 2 > 0
Each species can now occupy the full container, so the number of distinguishable arrangements rises.
check: positive Delta S comes from more accessible arrangements, not from a claim that the gases were heated
The entropy sign follows from the new state count for the mixed system.

Worked example 3: Why is melting ice at 273 K associated with positive system entropy change while freezing is negative?

Delta S_system = q_rev/T at the phase boundary
Phase change at equilibrium temperature lets the latent heat be treated reversibly for the system.
melting: q_rev > 0 so Delta S_system > 0
The liquid phase offers more accessible molecular arrangements than the ordered crystal.
freezing: q_rev < 0 so Delta S_system < 0
The system becomes more constrained, even though the surroundings can gain enough entropy for the whole process to proceed spontaneously.

Worked example 4: A hot metal block cools while a colder water bath warms by the same energy transfer. Why does total entropy rise?

Delta S_hot = -Q/Th
The hot block loses heat, so its entropy change is negative.
Delta S_cold = +Q/Tc with Tc < Th
The colder bath gains the same energy at a lower temperature, so the positive term is larger in magnitude.
Delta S_total > 0
Heat flows from hot to cold because that direction raises the total entropy of the isolated pair.

Entropy & free energy