Question

The density of solid gallium at its melting point is \(5.9 \mathrm{~g} / \mathrm{cm}^{3}\), whereas that of liquid gallium is \(6.1 \mathrm{~g} / \mathrm{cm}^{3} .\) Is the temperature at the triple point higher or lower than the normal melting point? Is the slope of the solid-liquid line for gallium positive or negative?

Short answer

The temperature at the triple point is lower than the normal melting point, and the slope of the solid-liquid line for gallium is negative.

Step-by-step solution

- Understand the significance of densities

At the melting point, the densities of the solid and liquid phases give insight into the behavior of the substance. For gallium, the solid phase has a density of 5.9 g/cm³ and the liquid phase has a higher density of 6.1 g/cm³.

- Analyze the relationship between density and volume

Given that density is mass per unit volume, a substance with a higher density has a smaller volume for the same mass. Therefore, as gallium melts, it contracts because the liquid gallium occupies less volume than the same mass of solid gallium.

- Determine the effect on the triple point

When a substance contracts upon melting, the solid-liquid boundary in the phase diagram has a negative slope according to the Clausius-Clapeyron equation. This implies that an increase in pressure will favor the solid phase, and thus the triple point temperature is lower than the normal melting point when compared at standard atmospheric pressure.

- Conclude the slope of the solid-liquid line

Since gallium contracts upon melting, the solid-liquid equilibrium line in the phase diagram must have a negative slope. This indicates that for gallium, as pressure increases, the melting point decreases.

Key concepts

Density and Phase Transitions

Understanding the density of a substance at different phases is crucial in determining its phase behavior. Density is defined as mass per unit volume. When gallium transitions from solid to liquid, its density increases from 5.9 g/cm³ to 6.1 g/cm³. This means that liquid gallium occupies less volume than an equal mass of solid gallium. Generally, substances expand upon melting due to the looser arrangement of molecules in the liquid phase, but gallium is an exception, contracting instead. This unique density behavior leads to interesting implications in its phase transitions and can be observed in a phase diagram.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation describes the relationship between pressure and temperature during phase transitions. This equation is vital in understanding how the phase boundaries shift in a phase diagram. For gallium, the solid-liquid boundary has a negative slope, which suggests that higher pressure decreases the melting point. Mathematically, the equation is expressed as:
\[ \frac{dP}{dT} = \frac{\triangle S}{\triangle V} \] Where \( \frac{dP}{dT} \) is the slope of the phase boundary, \( \triangle S \) is the change in entropy, and \( \triangle V \) is the change in volume. For gallium, the negative slope indicates that the entropy and volume change lead to lower melting points with increased pressure.

Phase Diagram Analysis

A phase diagram visually demonstrates the stability of different phases of a substance under varying conditions of pressure and temperature. For gallium, the phase diagram shows a negative slope at the solid-liquid equilibrium line. This means that as pressure increases, the melting point decreases. Such phase diagrams help predict the conditions under which gallium will exist in solid, liquid, or even gaseous phases. Key features like the triple point and critical point provide comprehensive insights into the behavior of gallium under different environmental conditions.

Triple Point

The triple point of a substance is the unique combination of temperature and pressure where all three phases (solid, liquid, and gas) can coexist in equilibrium. For gallium, the nature of its density and phase behavior suggests that the triple point occurs at a lower temperature than the normal melting point. This is because gallium contracts upon melting, resulting in a negative slope in its phase diagram. The triple point is a critical parameter in understanding the phase stability and transitions of gallium under varying pressure and temperature.

Solid-Liquid Equilibrium

Solid-liquid equilibrium refers to the stable coexistence of solid and liquid phases under a given set of conditions. In gallium, this equilibrium is influenced by its abnormal density behavior. Since liquid gallium is denser than solid gallium, the equilibrium line shows a negative slope in the phase diagram. This implies that solid gallium will melt into liquid at a lower temperature if the pressure increases. This concept is essential in understanding the melting and freezing processes and predicting the behavior of gallium in different environments.