One gram each of ice, water, and water vapor are in...

One gram each of ice, water, and water vapor are in equilibrium together in a closed container. The pressure is 4.58 mm of Hg , the temperature is $0.01^{\circ} \mathrm{C}$. Sixty calories of heat are added to the system. The total volume is kept constant. Calculate to within $2 \%$ the masses of ice, water, and water vapor now present in the container. Justify your answers. (Hint: For water at $0.01^{\circ} \mathrm{C}$, the latent heat of fusion is $80 \mathrm{cal} / \mathrm{g}$, the latent heat of vaporization is $596 \mathrm{cal} / \mathrm{g}$, and the latent heat of sublimation is $676 \mathrm{cal} / \mathrm{g}$. Also note that the volume of the vapor is much larger than the volume of the water or the volume of the ice.)

One gram each of ice, water, and water vapor are in equilibrium together in a closed container. The pressure is 4.58 mm of Hg , the temperature is $0.01^{\circ} \mathrm{C}$. Sixty calories of heat are added to the system. The total volume is kept constant. Calculate to within $2 \%$ the masses of ice, water, and water vapor now present in the container. Justify your answers.
(Hint: For water at $0.01^{\circ} \mathrm{C}$, the latent heat of fusion is $80 \mathrm{cal} / \mathrm{g}$, the latent heat of vaporization is $596 \mathrm{cal} / \mathrm{g}$, and the latent heat of sublimation is $676 \mathrm{cal} / \mathrm{g}$. Also note that the volume of the vapor is much larger than the volume of the water or the volume of the ice.)

Key Concepts

Energy Conservation in Phase Transitions

The principle of energy conservation, formulated in the first law of thermodynamics, plays a critical role in phase transitions. When heat is added to a system at equilibrium, that energy is partitioned to overcome the latent heats of various phase changes rather than to raise the temperature. This allocation of energy determines the amounts of material in each phase after the change.

Volume Disparities Among Phases

In many systems, especially for substances like water, the volume occupied by the vapor phase is significantly larger than that of the liquid or solid phases. This large disparity allows for simplifying assumptions in calculations: the contributions of the condensed phases to the overall volume can often be neglected, which is important when determining changes in pressure and phase distribution after the addition of heat.

Constant Volume Processes

A constant volume process occurs when a system is confined to a fixed volume, meaning that any added heat or other form of energy can change the pressure rather than the volume. This constraint is crucial for analyzing systems in closed containers, as it influences the equilibrium conditions and the resulting phase composition, particularly for the gaseous phase.

Latent Heat

Latent heat is the amount of energy per unit mass required to change the phase of a substance without changing its temperature. Each phase change—whether it’s fusion (melting), vaporization, or sublimation—has its specific latent heat, which quantifies the energy needed to overcome the intermolecular forces holding the molecules in a particular phase.

Phase Equilibrium

Phase equilibrium refers to the state in which different phases of a substance (e.g., solid, liquid, and gas) coexist under conditions where the chemical potentials of each phase are equal. This concept is central to understanding systems at the triple point, where small amounts of energy added or removed can drive phase transitions without an overall change in temperature when equilibrium is maintained.

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