Entropy and the Second Law of Thermodynamics

A new concept: entropy $S$ is a state function introduced by Rudolf Clausius in 1865 to quantify the "direction" of thermodynamic processes. While energy is conserved (First Law), entropy tends to increase (Second Law), determining the thermodynamic arrow of time.

Definition: for a reversible process, $dS = \delta Q_{rev}/T$. Entropy is the heat exchanged divided by the absolute temperature. Integrating: $\Delta S = \int_1^2 \delta Q_{rev}/T$.

Fundamental characteristics:

• $S$ is a state function: $\Delta S$ depends only on the initial and final states, not on the path. This means we can compute $\Delta S$ for any process (even irreversible) using a fictitious reversible path connecting the same states.
• $\Delta S$ can be positive (increase in disorder), negative (decrease in disorder), or zero (reversible process).
• $S$ has units J/K.

Clausius inequality: $\oint \delta Q/T \leq 0$. For a thermodynamic cycle, the algebraic sum of heats exchanged divided by their respective temperatures is ≤ 0. Equality holds only for reversible cycles.

Second Law in entropic form: $\Delta S_{universe} \geq 0$. The entropy of the universe cannot decrease — it increases for irreversible processes (all real ones) and stays constant for reversible processes (ideal). This is the deepest formulation of the Second Law.

Example: when an ice cube melts in a glass of water at room temperature, the entropy of the system (ice + water) increases. The process is irreversible — we will never see the ice spontaneously reform from the water.

Reversible process: an ideal process in which the system passes through a succession of equilibrium states. It can be reversed without leaving any trace on the surroundings. Condition: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} = 0$.

• Example: quasi-static isothermal expansion of a gas against a frictionless piston.

Irreversible process: all real processes are irreversible (friction, turbulence, finite temperature gradients). $\Delta S_{univ} > 0$.

• Example: free expansion of a gas into a vacuum, mixing of two fluids, heat flow across a finite temperature difference.
• Microscopic cause: irreversible processes drive the system toward states with greater statistical probability (more microstates).

Computing entropy for an ideal gas: since $S$ is a state function, we can compute $\Delta S$ using a combination of reversible processes. For an ideal gas: $$\Delta S = nC_v\ln(T_2/T_1) + nR\ln(V_2/V_1)$$ In alternative form (using $P$ and $T$): $$\Delta S = nC_p\ln(T_2/T_1) - nR\ln(P_2/P_1)$$
Typical examples:

• Free expansion ($T$ const): $\Delta S = nR\ln(V_2/V_1)$.
• Isobaric heating: $\Delta S = nC_p\ln(T_2/T_1)$.
• Reversible adiabatic compression: $\Delta S = 0$.

Boltzmann's equation: $S = k_B\ln\Omega$, engraved on Ludwig Boltzmann's tombstone in Vienna. $\Omega$ is the number of microstates (microscopic configurations of particles) corresponding to a given macrostate (P, V, T).

Meaning:

• Microstate: a specific configuration of positions and velocities of all molecules.
• Macrostate: the thermodynamic state described by macroscopic variables (P, V, T). Many different microstates correspond to the same macrostate.
• $\Omega$ is the number of microstates accessible to the system for a given macrostate.
• $S$ measures "disorder" or, more precisely, the number of ways the system can realise a given macrostate.

Why entropy increases: a system evolves spontaneously toward the state with the largest number of microstates (maximum statistical probability). This is the Second Law in statistical form: $\Delta S > 0$ is equivalent to $\Omega_{final} > \Omega_{initial}$.

Classic example: if we release a drop of ink in a glass of water, the ink spreads until it is uniformly distributed. The initial state (concentrated ink) has few microstates ($\Omega$ small). The final state (dispersed ink) has a huge number of microstates ($\Omega$ large). Entropy increases. The reverse process (ink spontaneously gathering) is statistically impossible because $\Omega_{final} \ll \Omega_{initial}$.

Third Law (Nernst, 1906): $S \to 0$ as $T \to 0$ K. As absolute zero is approached, the system reaches its ground state, with a single microstate ($\Omega = 1$), so $S = k_B\ln 1 = 0$. This principle allows us to define absolute entropy (not just changes).

Deep connection: statistical entropy is the foundation of statistical mechanics, which unifies macroscopic thermodynamics with microscopic physics. The arrow of time (why we remember the past but not the future) is linked to the increase of the universe's entropy.