Definition and Characteristics of Second-Order Phase Transitions (Discontinuous Specific Heat, No Latent Heat)

In the study of thermodynamics, a phase transition describes the transformation of a substance from one physical state to another. These transitions are not merely changes in appearance; they represent fundamental reorganizations of the system's microscopic structure. To categorize these phenomena, physicist Paul Ehrenfest proposed a classification scheme based on the behavior of the Gibbs Free Energy ($G$) and its derivatives at the transition point.

The equilibrium state of a system under constant temperature ($T$) and pressure ($P$) is determined by the minimization of the Gibbs Free Energy, $G(T, P)$. The nature of a phase transition is defined by how the derivatives of this function behave as the system crosses the transition boundary.

First-Order vs. Second-Order Transitions

The distinction between the two primary types of transitions lies in which derivative of $G$ exhibits a discontinuity:

1. First-Order Phase Transitions: These occur when the first-order derivatives of the Gibbs Free Energy are discontinuous at the transition point. Since entropy ($S$) and volume ($V$) are defined as:
   $$S = -\left( \frac{\partial G}{\partial T} \right)_P, \quad V = \left( \frac{\partial G}{\partial P} \right)_T$$
   a first-order transition is characterized by a sudden "jump" in entropy and volume. Common examples include the melting of ice or the boiling of water.

2. Second-Order Phase Transitions: In these transitions, the first-order derivatives ($S$ and $V$) remain continuous across the transition point. However, the second-order derivatives of $G$ undergo a discontinuity or a divergence. These second-order derivatives represent the system's response functions, such as:

   • Isobaric Heat Capacity ($C_p$): $C_p = -T \left( \frac{\partial^2 G}{\partial T^2} \right)_P$
   • Isothermal Compressibility ($\kappa_T$): $\kappa_T = -\frac{1}{V} \left( \frac{\partial^2 G}{\partial P^2} \right)_T$
   • Thermal Expansion Coefficient ($\alpha$): $\alpha = \frac{1}{V} \left( \frac{\partial^2 G}{\partial P \partial T} \right)$

Key Characteristic I: The Absence of Latent Heat

One of the most defining physical differences between first-order and second-order transitions is the presence or absence of latent heat.

The Thermodynamic Mechanism

Latent heat ($L$) is the energy absorbed or released by a substance during a phase change that occurs without a change in temperature. Mathematically, it is expressed as:
$$L = T \Delta S$$
where $\Delta S = S_{\text{new}} - S_{\text{old}}$ represents the change in entropy during the transition.

• In a First-Order Transition: Because there is a discrete jump in entropy ($\Delta S \neq 0$), the system must exchange a specific amount of energy with its surroundings to facilitate the change in state. This is why boiling water stays at $100^\circ\text{C}$ even as heat is continuously added; the energy is being used to overcome intermolecular forces rather than increasing kinetic energy.
• In a Second-Order Transition: By definition, the entropy $S$ is continuous at the transition point, meaning $\Delta S = 0$. Consequently:
  $$L = T \cdot 0 = 0$$

This implies that second-order transitions are continuous processes. There is no sudden "burst" of energy required; instead, the system evolves smoothly from one phase to another through a continuous reorganization of its internal degrees of freedom.

Key Characteristic II: Discontinuity and Divergence in Heat Capacity

While second-order transitions lack latent heat, they are far from "quiet" thermodynamically. They are marked by dramatic fluctuations in the system's ability to absorb heat, known as the heat capacity jump or divergence.

Mathematical Origin

Since the heat capacity $C_p$ is proportional to the second derivative of the Gibbs Free Energy with respect to temperature, any discontinuity in that second derivative manifests as a sudden change in $C_p$. As a system approaches the critical temperature ($T_c$), the internal structure (such as magnetic moments or electron pairings) undergoes intense reorganization. This makes the system extremely sensitive to temperature fluctuations.

From Discontinuity to Divergence

In the classical Ehrenfest view, $C_p$ was thought to simply "jump" from one value to another. However, modern statistical mechanics and the theory of critical phenomena have refined this. For many real-world second-order transitions, the heat capacity does not just jump; it diverges (tends toward infinity) as the temperature approaches $T_c$. This behavior typically follows a power law:
$$C_p \propto |T - T_c|^{-\alpha}$$
where $\alpha$ is a critical exponent. This divergence is a hallmark of the complex, long-range correlations that emerge near a critical point, a concept central to the study of universality classes.

Real-World Examples

To ground these theoretical concepts, we can examine two classic examples found in condensed matter physics.

1. The Superconducting Transition

When certain materials (like lead or tin) are cooled below a critical temperature $T_c$, they transition from a normal metallic state to a superconducting state.

• Behavior: At $T_c$, electrical resistance drops to zero and the Meissner effect (expulsion of magnetic fields) occurs.
• Classification: This is a second-order transition because there is no latent heat involved (entropy is continuous), but the heat capacity shows a distinct, sharp jump at the transition temperature.

2. Ferromagnetic to Paramagnetic Transition

A ferromagnetic material (like iron) loses its spontaneous magnetization when heated above a specific temperature known as the Curie temperature ($T_C$).

• Behavior: Below $T_C$, the material possesses a permanent magnetic moment. Above $T_C$, thermal agitation destroys the alignment, resulting in paramagnetism.
• Classification: The order parameter (magnetization, $M$) decreases continuously to zero as $T$ approaches $T_C$. Because $M$ changes continuously, the entropy change is also continuous, meaning no latent heat is released. However, the magnetic susceptibility and heat capacity exhibit massive fluctuations or divergences near $T_C$.

Summary Comparison

The following table summarizes the fundamental differences between the two types of transitions:

Understanding these transitions is vital for modern physics, providing the framework necessary to study symmetry breaking, critical phenomena, and the development of advanced materials like high-temperature superconductors.