Question

Refer to Figure \(11.27(\mathrm{~b})\), and describe the phase changes (and the temperatures at which they occur) when \(\mathrm{CO}_{2}\) is heated from \(-80{ }^{\circ} \mathrm{C}\) to \(-20{ }^{\circ} \mathrm{C}\) at \((\mathrm{a})\) a constant pressure of \(3 \mathrm{~atm}\), (b) a constant pressure of \(6 \mathrm{~atm}\).

Short answer

When CO2 is heated from -80°C to -20°C: (a) At a constant pressure of 3 atm, it undergoes a phase change from solid to gas through sublimation at around -57°C. (b) At a constant pressure of 6 atm, it undergoes two phase changes: first from solid to liquid at around -55°C (melting point) and then from liquid to gas at around -40°C (boiling point).

Step-by-step solution

Identify phases on the phase diagram

To perform this task, we need to locate the corresponding points at the given pressures and temperatures on the phase diagram. For each pressure, draw a horizontal line, as the pressure remains constant. For both cases, we need to mark the starting temperature (-80°C) and the final temperature (-20°C) on the corresponding pressure lines.

Determine phase changes at 3 atm pressure

Locate the 3 atm pressure line on the phase diagram and mark the starting and final temperatures, -80°C and -20°C, respectively. At -80°C and 3 atm, CO2 is in the solid phase (also known as dry ice). As the temperature rises, we observe a transition from the solid phase to the gas phase at the sublimation point. There is no liquid phase observed for CO2 at this pressure during the heating process. The temperature at which this sublimation occurs can be read from the phase diagram. In this case, the sublimation happens at around -57°C. Therefore, when CO2 is heated from -80°C to -20°C at a constant pressure of 3 atm, it undergoes a phase change from solid to gas through sublimation at around -57°C.

Determine phase changes at 6 atm pressure

Now, locate the 6 atm pressure line on the phase diagram and mark the starting and final temperatures, -80°C and -20°C, respectively. At -80°C and 6 atm, CO2 is in the solid phase (again, dry ice). As the temperature rises, we observe a transition from the solid phase to the liquid phase at the melting point. The temperature at which this melting occurs can be read from the phase diagram, approximately at -55°C. As the temperature continues to rise, the liquid CO2 reaches a boiling point and changes to the gas phase. Locate this boiling point on the phase diagram; in this case, it happens around -40°C. Therefore, when CO2 is heated from -80°C to -20°C at a constant pressure of 6 atm, it undergoes two phase changes: first from solid to liquid at around -55°C (melting point) and then from liquid to gas at around -40°C (boiling point). In conclusion, when CO2 is heated from -80°C to -20°C: - At 3 atm, it undergoes a phase change from solid to gas through sublimation at around -57°C. - At 6 atm, it undergoes two phase changes: first from solid to liquid at around -55°C (melting point) and then from liquid to gas at around -40°C (boiling point).

Key concepts

Phase Diagram

A phase diagram is a graphical representation that shows the conditions of temperature and pressure at which distinct phases occur and coexist at equilibrium. It is divided into regions that indicate whether a substance will exist as a solid, liquid, or gas. For carbon dioxide ( CO_2 ), the phase diagram is unique because it typically doesn't have a liquid state at 1 atm pressure.

With a phase diagram, we can easily see the boundaries where phase changes occur. These boundaries represent the lines along which two phases exist simultaneously, like solid-gas or liquid-gas. By following these lines, we can understand the behavior of substances under different conditions.

• Solid Region: Found at low temperatures and high pressures.
• Liquid Region: Typically found at intermediate temperatures and pressures, although less common for CO_2 .
• Gas Region: Appears at higher temperatures and lower pressures.

The intersections of these regions are important points such as the triple point, where all three phases coexist, and the critical point, beyond which the liquid and gas phases become indistinguishable.

Sublimation

Sublimation is the transition of a substance directly from the solid to the gas phase, skipping the liquid phase altogether. This can occur under specific conditions of temperature and pressure, particularly when a substance's phase diagram does not support a liquid state within the given range. For carbon dioxide, sublimation is a common phenomenon, especially at pressures like 1 atm where liquid CO_2 cannot exist.

In the context of our exercise, when CO_2 is heated at a constant pressure of 3 atm from -80°C to -20°C, it undergoes sublimation. This happens around -57°C, evident from the phase diagram. Sublimation is unique because it involves only two phases: solid and gas.

• Occurs commonly for dry ice ( CO_2 ) at low atmospheric pressures.
• It bypasses the liquid phase entirely.
• Observable as solid particles turning into gas without first becoming liquid.

Understanding sublimation is crucial because it underscores the distinct behavior substances can exhibit under specific thermodynamic conditions.

Melting Point

The melting point of a substance is the temperature at which it transitions from a solid to a liquid state at a given pressure. It is where the forces holding the molecules in a solid state break apart, allowing them to flow past each other as a liquid. Each substance has a unique melting point that depends on its intermolecular forces and other kinetic factors.

For CO_2 at 6 atm pressure, heating from -80°C leads to the phase change from solid to liquid at approximately -55°C. This is its melting point under these conditions.

• Marks the conversion of a structured solid lattice to a fluid state.
• Highly dependent on ambient pressure.
• An essential characteristic for identifying pure substances.

Knowing the melting point helps in understanding how substances behave under pressure and temperature changes, as this knowledge is crucial for various practical applications.

Boiling Point

The boiling point of a substance is the temperature at which its liquid form transitions to gas. At this point, the vapor pressure of the liquid equals the external atmospheric pressure, allowing bubbles of vapor to form within the liquid. For CO_2 , the boiling point can change significantly with pressure.

In the exercise scenario, when CO_2 is at a pressure of 6 atm, heating it from -80°C eventually reaches a temperature around -40°C. Here, the liquid phase turns into gas, reflecting CO_2 's boiling point under these conditions.

The boiling point is critical for several reasons:

• It indicates the temperature needed for a substance to undergo a phase change from liquid to gas.
• Varies with the pressure, decreasing as the ambient pressure falls.
• Helps define boiling, evaporative, and phase change processes.

Understanding the boiling point is key for many scientific and industrial processes where phase changes are pertinent, such as distillation and refrigeration.