Entropy & The Third Law of Thermodynamics

1. Clausius Theorem for Cyclic Reversible Processes

In 1854, Rudolf Clausius discovered that the Second Law implies the existence of a fundamental thermodynamic state function. Consider an arbitrary reversible cyclic process operating on a state plane $(P, V)$.

Any arbitrary reversible cycle $\mathcal{C}$ can be decomposed into an infinite grid of infinitesimal Carnot cycles operating between adjacent adiabats. For each infinitesimal Carnot cycle operating between reservoirs at temperatures $T_{1, i}$ and $T_{2, i}$:

Summing over all sub-cycles, the internal adiabatic and isothermal paths cancel pairwise because each internal segment is traversed in opposite directions by adjacent cycles. The only uncompensated heat exchanges are along the outer closed boundary:

Taking the continuum limit as the number of sub-cycles $N \to \infty$:

This is Clausius's Theorem.

2. Definition of Entropy as a State Function

Because the cyclic integral of $\frac{\delta Q_{\text{rev}}}{T}$ vanishes for every conceivable reversible loop, the line integral between any two equilibrium states $A$ and $B$ is strictly independent of the reversible path:

Consequently, $\frac{\delta Q_{\text{rev}}}{T}$ is an exact differential of an extensive state function, which Clausius named Entropy ($S$):

The finite entropy difference between states $A$ and $B$ is:

The absolute value of entropy is defined relative to an arbitrary reference state ($S_0$), until anchored by the Third Law.

1. The Clausius Inequality

Now consider a real, irreversible cyclic process $\mathcal{C}$. Let the system absorb heat $\delta Q$ at temperature $T$ during an irreversible forward path $A \to B$, and return to initial state $A$ via an idealized reversible path $B \to A$. By Carnot's theorem, an irreversible engine is less efficient than a reversible engine:

Combining reversible ($= 0$) and irreversible ($< 0$) cases yields the universal Clausius Inequality:

where equality holds if and only if the entire cycle is strictly reversible.

2. The Second Law in Differential Form

Splitting the cyclic integral into the irreversible path $I$ ($A \to B$) and reversible path $R$ ($B \to A$):

Since the second path is reversible, $\int_{B, R}^A \frac{\delta Q_{\text{rev}}}{T} = S_A - S_B = -\Delta S$:

In differential form:

For an isolated system, which cannot exchange heat with its surroundings ($\delta Q = 0$):

This is the Principle of Increase of Entropy:

3. Entropy of the Universe

Any system interacting with its environment can be embedded within an overall isolated composite universe:

While the entropy of a localized open subsystem can decrease (e.g., crystal freezing, living organism biological growth), it does so only by ejecting an even greater quantity of entropy into its surrounding environment, ensuring that total universal entropy relentlessly increases.

1. Analytical Expressions for Ideal Gas Entropy

From the combined First and Second Laws for a reversible process in a simple fluid:

For an ideal gas, substituting $dU = n C_{v, m} dT$ and $P = n R T / V$:

Integrating from state $(T_1, V_1)$ to $(T_2, V_2)$:

Alternatively, using enthalpy $T dS = dH - V dP = n C_{p, m} dT - V dP$:

2. Free Adiabatic Expansion (Joule Expansion)

Consider $n$ moles of an ideal gas expanding freely into an evacuated volume from $V_1$ to $V_2 = 2 V_1$ inside an adiabatic container:

• $Q = 0, W = 0 \implies \Delta U = 0 \implies T_2 = T_1$.
• Because the expansion is irreversible, we cannot integrate $\delta Q / T$ along the actual path.
• But because $S$ is a state function, $\Delta S$ depends solely on initial and final states. We construct an imaginary reversible isothermal expansion connecting $(T_1, V_1)$ to $(T_1, V_2)$:

• Surroundings experience zero heat transfer: $\Delta S_{\text{surroundings}} = 0$.
• Total universal entropy generated:

This positive entropy change quantifies the intrinsic irreversibility of free expansion.

1. Representation of Thermodynamic Cycles on T-S Coordinates

The Temperature-Entropy ($T$-$S$) diagram provides deep geometric insight into thermodynamic transformations because the area under a reversible curve directly represents heat exchanged:

• An isothermal process ($T = \text{const}$) maps to a horizontal straight line.
• A reversible adiabatic process ($dS = 0$) maps to a vertical straight line (an isentropic process).

2. The Carnot Cycle on a T-S Diagram

On a $T$-$S$ plane, the Carnot cycle maps to an exact rectangle:

1. Stage $A \to B$ (Isothermal expansion at $T_H$): Horizontal line from $S_A$ to $S_B$. Heat absorbed:

1. Stage $B \to C$ (Adiabatic expansion): Vertical line downward at constant entropy $S_B$ from $T_H$ to $T_C$ ($Q = 0$).
2. Stage $C \to D$ (Isothermal compression at $T_C$): Horizontal line from $S_B$ back to $S_A$. Heat rejected:

1. Stage $D \to A$ (Adiabatic compression): Vertical line upward at constant entropy $S_A$ from $T_C$ back to $T_H$ ($Q = 0$).

3. Net Mechanical Work and Efficiency

The net mechanical work done per cycle is the enclosed area of the rectangle:

The thermal efficiency is the ratio of enclosed area to total area under the upper horizontal curve:

This offers an elegant geometric proof of Carnot efficiency.

1. Nernst Heat Theorem & Planck Formulation

In 1906, Walther Nernst observed that during chemical reactions between condensed phases, the entropy change $\Delta S$ approaches zero as temperature approaches absolute zero:

Max Planck (1911) generalized this into the Third Law of Thermodynamics:

2. Physical Consequences of the Third Law

1. Vanishing Heat Capacities: The entropy of a substance at temperature $T$ is $S(T) = \int_0^T \frac{C_p(T')}{T'} dT'$. For this integral to converge at $T = 0$, $C_p$ and $C_v$ must vanish:

In quantum physics, Debye's law confirms $C_v \propto T^3$ for non-metallic crystals and $C_v \propto T$ for conduction electrons as $T \to 0$.

2. Vanishing Thermal Expansion: Using Maxwell's relation $(\partial V/\partial T)_P = -(\partial S/\partial P)_T$, and since $S \to 0$ independent of pressure:

3. Principle of Unattainability of Absolute Zero

The Third Law is equivalent to the proposition: It is physically impossible to reduce the temperature of any macroscopic system to absolute zero ($T = 0\text{ K}$) in a finite number of thermodynamic operations.

To cool a system, one alternates between isothermal parameter changes (e.g., magnetization $M$) and adiabatic changes (e.g., demagnetization). Because all isentropic curves converge to $S = 0$ at $T = 0\text{ K}$, the temperature steps $\Delta T$ achieved per cycle diminish asymptotically to zero as $T \to 0$, requiring an infinite sequence of operations to reach $0\text{ K}$.

1. Ehrenfest Classification Scheme

Paul Ehrenfest (1933) classified phase transitions according to the lowest order derivative of the Gibbs Free Energy $G(T, P)$ that displays a mathematical discontinuity:

2. First-Order Phase Transitions

A transition is first-order if the first partial derivatives of $G$ exhibit finite discontinuities across the phase boundary:

• Entropy discontinuity:

where $L$ is the latent heat of transformation.

• Volume discontinuity:

Examples include solid-liquid melting, liquid-vapor boiling, and solid-vapor sublimation. Along the coexistence curve, equality of chemical potentials $G_1(T, P) = G_2(T, P)$ leads directly to the Clausius-Clapeyron equation:

3. Second-Order Phase Transitions

A transition is second-order (continuous) if the first derivatives of $G$ are continuous ($\Delta S = 0, \Delta V = 0, L = 0$), but the second partial derivatives exhibit finite jump discontinuities or power-law divergences:

1. Specific Heat Discontinuity:

2. Thermal Expansion Discontinuity:

3. Isothermal Compressibility Discontinuity:

Examples include:

• Ferromagnetic to paramagnetic transition at the Curie temperature $T_C$.
• Superconducting transition in zero external magnetic field.
• Order-disorder transitions in binary metal alloys (e.g., $\beta$-brass).
• Liquid Helium-4 normal to superfluid lambda transition at $T_\lambda = 2.17\text{ K}$.