### Explanation

The concept of the **coexistence of phases** in thermodynamics refers to the condition under which different phases (solid, liquid, gas) of a substance exist together at equilibrium. This typically occurs at specific combinations of pressure and temperature, known as *phase boundaries*.

### Phase Diagrams

A phase diagram is a graphical way to depict the conditions under which distinct phases occur and coexist at equilibrium. These diagrams usually have pressure on the y-axis and temperature on the x-axis. Points or lines on the diagram where two or more phases coexist are called **coexistence curves**.

### Gibbs Free Energy

The equilibrium between phases is determined by the minimization of the Gibbs free energy $G$. For two phases to coexist, their Gibbs free energies must be equal:

$G_{\text{phase 1}} = G_{\text{phase 2}}$
### Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes the phase boundary between two phases (such as solid and liquid or liquid and gas). It is given by:

$\frac{dP}{dT} = \frac{\Delta S}{\Delta V}$
Here:

- $\frac{dP}{dT}$ is the slope of the coexistence curve.
- $\Delta S$ is the change in entropy during the phase transition.
- $\Delta V$ is the change in volume during the phase transition.

### Triple Point

A special point on a phase diagram is the **triple point**, where all three phases coexist in equilibrium. For water, the triple point is at a precise temperature of 273.16 K and a pressure of 611.657 Pa.

### Key Insights

**Equilibrium Conditions**: Coexistence implies that the thermodynamic potentials are equal for the phases in equilibrium.
**Phase Boundaries**: The lines or curves separating single-phase regions are where phases can coexist.
**Triple Point**: A unique point delineating conditions for the coexistence of three phases.

Understanding the coexistence of phases helps in various scientific and industrial processes, including the design of refrigeration systems, the study of material properties, and the synthesis of new compounds.