Overview#

• Entropy \(S\) is a state function—it depends only on the state of the system, not on how that state was reached.

• \(\Delta S = \int dQ/T\) for a reversible process; \(T\) must be in kelvins.

• Second law: In a closed system, \(\Delta S \ge 0\)—entropy increases for irreversible processes and stays constant for reversible processes; it never decreases.

Entropy change#

For a reversible process from state \(i\) to state \(f\):

• \(Q\): heat transferred to the system. \(T\): temperature (kelvins).

• Units: J/K.

• For an irreversible process, \(\Delta S\) is computed using any reversible path between the same initial and final states.

Reversible isothermal (\(T\) constant):

Small \(\Delta T\): \(\Delta S \approx Q/T_{\text{avg}}\).

Ideal gas#

For an ideal gas undergoing a reversible process from \((T_i, V_i)\) to \((T_f, V_f)\):

Free expansion (irreversible): \(Q = 0\) along the actual path, but \(T\) and \(Q\) are not well defined. Use a reversible isothermal expansion between the same \(V_i\) and \(V_f\):

Second law of thermodynamics#

Entropy postulate / Second law: If a process occurs in a closed system, the entropy of the system:

• Increases for irreversible processes (\(\Delta S > 0\)).

• Remains constant for reversible processes (\(\Delta S = 0\)).

• Never decreases (\(\Delta S \ge 0\)).

The direction of irreversible processes is set by entropy increase—the “arrow of time.”

Summary#

• \(\Delta S = \int dQ/T\) for reversible process; \(\Delta S = Q/T\) for isothermal.

• \(\Delta S = nR\ln(V_f/V_i) + nC_V\ln(T_f/T_i)\) for ideal gas.

• Irreversible: use a reversible path between same states to compute \(\Delta S\).

• Second law: \(\Delta S \ge 0\) in a closed system.