Question

Use these data to draw a qualitative phase diagram for \(\mathrm{H}_{2}\). Does \(\mathrm{H}_{2}\) sublime at 0.05 atm? Explain. \(\mathrm{mp}\) at \(1 \mathrm{~atm}\): \(13.96 \mathrm{~K}\) bp at 1 atm: \(20.39 \mathrm{~K}\) Triple point: \(13.95 \mathrm{~K}\) and \(0.07 \mathrm{~atm}\) Critical point: \(33.2 \mathrm{~K}\) and \(13.0 \mathrm{~atm}\) Vapor pressure of solid at \(10 \mathrm{~K}: \quad 0.001\) atm

Short answer

Yes, \(\text{H}_2\) sublimes directly at 0.05 atm.

Step-by-step solution

Understand Key Points on Phase Diagram

Identify critical temperatures and pressures: Melting point (mp) at 1 atm: 13.96 K, Boiling point (bp) at 1 atm: 20.39 K, Triple point: 13.95 K at 0.07 atm, Critical point: 33.2 K at 13.0 atm, Vapor pressure of solid at 10 K: 0.001 atm.

Draw Axes for the Phase Diagram

On graph paper, draw the temperature (K) on the x-axis and pressure (atm) on the y-axis, ensuring a suitable range for given data points.

Plot Key Points

Mark the triple point at (13.95 K, 0.07 atm), the mp at (13.96 K, 1 atm), the bp at (20.39 K, 1 atm), the critical point at (33.2 K, 13 atm), and the vapor pressure of solid at 10 K and 0.001 atm.

Draw Phase Boundaries and Regions

Connect the data points to show phase boundaries: solid-liquid boundary from triple point to mp, liquid-gas boundary from mp to bp and from bp to critical point with a curve, solid-gas boundary from triple point through vapor pressure point and extending towards lower pressures and temperatures.

Analyze Sublimation at 0.05 atm

Check the phase diagram at 0.05 atm. Find the temperature region where hydrogen is a solid, and follow the solid-gas boundary to determine if hydrogen sublimes directly from solid to gas.

Key concepts

Sublimation

Sublimation is the process in which a substance transitions directly from a solid to a gas without going through the liquid phase. This usually happens under specific temperature and pressure conditions.
In the phase diagram of hydrogen \((\mathrm{H}_{2})\), you can observe sublimation occurring when the vapor pressure of the solid intersects with the external pressure below the triple point's pressure.
Given the vapor pressure of solid hydrogen at 10 K is 0.001 atm, we can determine if hydrogen sublimes at various pressures.
For instance, at 0.05 atm, you need to follow the solid-gas boundary on the diagram. If the line leads you to temperatures where hydrogen remains in the solid phase and directly transitions to gas, it signifies sublimation.

Triple Point

The triple point is a unique condition on the phase diagram where three phases—solid, liquid, and gas—exist in equilibrium.
For hydrogen, the triple point is at 13.95 K and 0.07 atm.
This means at this exact temperature and pressure, hydrogen can simultaneously be in a solid, liquid, and gaseous state.
Any shift in this balance, either by changing the temperature or pressure, will disrupt this equilibrium, converting the substance into one or two different phases.
Understanding the triple point is crucial for precisely predicting phase transitions under varying conditions.

Critical Point

The critical point marks the end of the liquid-gas boundary on the phase diagram.
At this point, the liquid and gas phases become indistinguishable from each other. For hydrogen, the critical point is defined at 33.2 K and 13.0 atm.
Above this temperature and pressure, hydrogen forms a supercritical fluid, which is neither purely a liquid nor a gas, but rather has properties of both.
Knowing the critical point helps to understand behavior under high-temperature and high-pressure environments.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its solid or liquid phases.
In the phase diagram, vapor pressure lines show how the vapor pressure changes with temperature.
For hydrogen, at 10 K, the vapor pressure of the solid phase is 0.001 atm.
This is a very low pressure, meaning that hydrogen hardly vaporizes at this temperature.
By tracing the vapor pressure curve of the solid on the phase diagram, you can predict the behavior of hydrogen under various temperatures and pressures, including where sublimation might occur.