Question

Propanol \(\left(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{OH}\right)\) melts at \(-126.5^{\circ} \mathrm{C}\) and boils at \(97.4^{\circ} \mathrm{C}\). Draw a qualitative sketch of how the entropy changes as propanol vapor at \(150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) is cooled to solid propanol at \(-150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

Short answer

The qualitative sketch of the entropy changes for propanol cooling from \(150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) to \(-150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) should show a continuous decrease in entropy, with more significant decreases during the vapor to liquid and liquid to solid phase transitions. This represents the cooling process and the increased order of the molecules as they change from vapor to liquid and finally to the solid state.

Step-by-step solution

Identify the starting point

The initial state of propanol is vapor at \(150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\). This is the starting point of our entropy sketch.

Cooling the vapor

As the vapor is cooled, the entropy will decrease because the molecules have less energy and are more ordered. The cooling process will continue until the boiling point, which is \(97.4^{\circ} \mathrm{C}\). The sketch should show a decreasing entropy.

Vapor to liquid phase transition

At the boiling point, propanol will transition from the vapor phase to the liquid phase. Entropy will decrease during the phase transition as the molecules become more ordered and are in closer proximity. At this point, the sketch should show a more significant decrease in entropy.

Cooling the liquid

After the phase transition, the liquid propanol will continue to cool down from \(97.4^{\circ} \mathrm{C}\) to its melting point at \(-126.5^{\circ} \mathrm{C}\). The entropy will keep decreasing as the temperature decreases. The sketch should show a gradual decrease in entropy during this part of the process.

Liquid to solid phase transition

At the melting point, propanol will transition from the liquid phase to the solid phase. Entropy will decrease again during this phase transition, as the molecules become even more ordered and are in a fixed lattice arrangement. The sketch should show another significant decrease in entropy at this point.

Cooling the solid

Finally, the solid propanol will continue cooling down from \(-126.5^{\circ} \mathrm{C}\) to the final temperature of \(-150^{\circ} \mathrm{C}\). The entropy will continue to decrease as the temperature drops. The sketch should show a slight decrease in entropy in the last part of the process. In conclusion, the qualitative sketch of the entropy changes for propanol cooling from \(150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) to \(-150^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) should show a continuous decrease in entropy, with more significant decreases during the vapor to liquid and liquid to solid phase transitions.

Key concepts

Entropy in Thermodynamics

Entropy, in thermodynamics, is a measure of the disorder or randomness in a system. It’s a fundamental concept that helps us understand how energy is distributed within a system and how systems evolve over time. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time; in other words, systems naturally tend toward disorder.

In the context of cooling propanol, the entropy decreases as the system loses heat and the molecules assume more structured arrangements. During a phase transition, where propanol changes from gas to liquid or liquid to solid, entropy sees a significant drop. This is because the molecular movement is restricted during the change of state, leading to a more ordered system. The importance of entropy in such processes is that it provides a numerical measure that can predict the direction in which processes can spontaneously occur.

Phase Changes

Phase changes refer to the transitions between different states of matter, typically solid, liquid, and gas. These changes occur at specific temperatures and pressures and are crucial in thermodynamic processes. When propanol transitions from a vapor to a liquid (condensation) or from a liquid to a solid (freezing), it undergoes a phase change.

Each phase change is accompanied by a change in entropy. During condensation, the gas molecules, which are far apart and move freely, come closer together and start moving less freely, indicating a decrease in entropy. Freezing, on the other hand, restricts the movement of the molecules even more as they form a lattice structure, indicating an even larger decrease in entropy. These significant changes in entropy during phase transitions are pivotal in understanding the behavior of substances as they cool.

States of Matter

The states of matter -- solid, liquid, and gas -- represent different phases that materials can exist in. Gases have high entropy because their molecules are spread out and move independently of one another. Liquids have less entropy than gases because their molecules are closer together but still move past each other. Solids exhibit the lowest entropy among the three states since their molecules are in a fixed, orderly arrangement.

As propanol moves from a gas to a liquid and finally to a solid during cooling, its entropy decreases due to the increasing order of the molecules. Understanding how the states of matter relate to entropy gives insight into the energy distribution of a substance and helps predict the nature of its transformations under various temperature and pressure conditions.