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Chapter 11: Problem 112

Sketch the phase diagram for a substance that has a triple point at \(-57.0^{\circ} \mathrm{C}\) and \(5.10 \mathrm{~atm} .\) It has a vapor pressure of 1 atm at \(-78{ }^{\circ} \mathrm{C}\) and has a critical point at \(31^{\circ} \mathrm{C}\) and \(73 \mathrm{~atm} .\) Its solid expands as temperature increases.

Short Answer

Expert verified
The phase diagram consists of a triple point at (-57.0°C, 5.10 atm), a critical point at (31°C, 73 atm), and a vapor pressure of 1 atm at -78°C. Solid-liquid boundary line slopes positively from the triple point, the solid-gas line is a curve reaching down to the vapor pressure point, and the liquid-gas line ends at the critical point with a steep slope.

Step by step solution

01

Understanding the Phase Diagram

A phase diagram displays the states of matter of a substance at various temperatures and pressures. It typically shows regions for solid, liquid, and gas states, as well as distinct lines representing phase transitions such as melting, boiling, and sublimation. The triple point is where three phases coexist in equilibrium, and the critical point is where the liquid and gas phases become indistinguishable.
02

Plotting the Triple Point

Begin the phase diagram by plotting the triple point at ewline(-57.0^{ewlinedeg} ewlineC, 5.10ewline atmewline). This is the condition where the solid, liquid, and gas phases coexist in equilibrium. Place a dot on the graph representing this point.
03

Plotting the Critical Point

Next, plot the critical point at ewline(31^{ewlinedeg} ewlineC, 73 ewlineatmewline). The critical point is the end of the liquid-gas equilibrium curve and indicates the temperature and pressure beyond which liquid and gas phases cannot be distinguished. Place a dot at the appropriate position to indicate the critical point.
04

Drawing the Phase Boundaries

Draw the phase boundaries starting from the triple point. Since the vapor pressure is 1 atm at -78°C, you can plot this condition for the solid-gas boundary and draw a curve reaching down to it from the triple point. For the solid-liquid boundary, considering that the solid expands with the increase in temperature, this line should have a positive slope. Draw the liquid-gas boundary, which should terminate at the critical point, with a steep positive slope indicating that higher pressures are needed to maintain the liquid phase as temperature increases.
05

Labeling the Phases

Label the regions on the diagram corresponding to solid, liquid, and gas. Solid should be to the left of the solid-liquid boundary and below the solid-gas boundary. Liquid will be above and to the right of the solid-liquid boundary, and below the liquid-gas boundary. Gas will be to the right of both the solid-gas boundary and the liquid-gas boundary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triple Point
The triple point of a substance is a unique stage where three phases of matter—solid, liquid, and gas—coexist in equilibrium. Imagine a dance floor where all three states get along perfectly; they're all present at the same time. This occurs only under specific conditions of temperature and pressure. Our exercise mentions a triple point at ewline(-57.0^{ewlinedeg} ewlineC, 5.10ewline atmewline), marking the fascinating scenario where, for this particular substance, you could find ice, water, and vapor all at once if you were to chill and pressurize a sample to exactly those surreal specifications.
Critical Point
Consider the critical point as the ultimate blending zone in the world of physics. Here, at a certain temperature and pressure, the difference between the liquid and gas phases of a substance becomes indiscernible—they essentially become one supercritical fluid. For our exercise, this critical merge happens at ewline(31^{ewlinedeg} ewlineC, 73 ewlineatmewline). It's like reaching the top of a mountain where, above this point, the usual rules separating liquids and gases no longer apply. Steaming water and its vapor look and act the same under these extreme conditions.
States of Matter
The states or phases of matter we typically talk about are solid, liquid, and gas. In the chilling realm of solid, molecules are tightly packed in a fixed structure. Warm up to the flowy state of liquid, and molecules slip and slide around—a little freedom, but still in touch. Escalate to the gaseous state, and it's total freedom; molecules move rapidly and are far apart. Some substances can also exhibit plasma and Bose-Einstein condensate states under more exotic conditions. Ordinary phase diagrams plot the three main states, exemplified in our exercise by showing the boundaries and transitions between solid, liquid, and gas.
Phase Transitions
Transitioning between phases is like a matter makeover. As you adjust temperature or pressure, a substance can switch between being a solid, liquid, or gas. Common transitions include melting (solid to liquid), freezing (liquid to solid), boiling (liquid to gas), condensing (gas to liquid), sublimation (solid to gas), and deposition (gas to solid). Each transition has a specific pathway on a phase diagram, dictating the conditions for these transformations. Intriguingly, our substance expands as a solid when heated, hinting that its molecules become more spread out, causing the solid-liquid boundary to slope upward on the diagram.
Vapor Pressure
Vapor pressure is the force that a substance's vapor exerts on its liquid or solid phase at a given temperature, in a closed system. It's a bit like the gas's way of saying 'I'd rather be a cloud.' When the vapor pressure equals the external pressure, boiling occurs. Every temperature change can shift the vapor pressure, influencing how a substance transitions between phases. For our exercise, at ewline(-78^{ewlinedeg} ewlineC, 1 ewlineatmewline), the vapor pressure is the hero setting the scene for the solid to skip the liquid phase dance and go straight to gas, which is a sublime move known as sublimation.

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Most popular questions from this chapter

The molar heat of vaporization of acetone, \(\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O},\) is \(30.3 \mathrm{~kJ} / \mathrm{mol}\) at its boiling point. How many kilojoules of heat would be liberated by the condensation of \(5.00 \mathrm{~g}\) of acetone?

What is diffusion? Give an example of diffusion in everyday life.

When warm, moist air sweeps in from the ocean and rises over a mountain range, it expands and cools. Explain how this cooling is related to the attractive forces between gas molecules. Why does this cause rain to form? When the air drops down the far side of the range, its pressure rises as it is compressed. Explain why this causes the air temperature to rise. How does the humidity of this air compare with the air that originally came in off the ocean? Now explain why the coast of California is lush farmland, whereas valleys (such as Death Valley) that lie to the east of the tall Sierra Nevada mountains are arid and dry.

Arrange the following substances in order of their increasing values of \(\Delta H_{\text {vaporization }}: \mathrm{HF}, \mathrm{CH}_{4}, \mathrm{CF}_{4}, \mathrm{HCl}\).

For a given substance, how do the intermolecular attractive forces compare in its gaseous, liquid, and solid states?
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