Question

Which state in each of the following pairs has the higher entropy per mole of substance? (a) Ice at \(-40^{\circ} \mathrm{C}\) or ice at \(0^{\circ} \mathrm{C}\) (b) \(\mathrm{N}_{2}\) at STP or \(\mathrm{N}_{2}\) at \(0{ }^{\circ} \mathrm{C}\) and \(10 \mathrm{~atm}\) (c) \(\mathrm{N}_{2}\) at STP or \(\mathrm{N}_{2}\) at \(0{ }^{\circ} \mathrm{C}\) in a volume of \(50 \mathrm{~L}\) (d) Water vapor at \(150^{\circ} \mathrm{C}\) and 1 atm or water vapor at \(100{ }^{\circ} \mathrm{C}\) and \(2 \mathrm{~atm}\)

Short answer

(a) Ice at 0°C, (b) N₂ at STP, (c) N₂ at 50 L, (d) Water vapor at 150°C and 1 atm.

Step-by-step solution

Understand the Concept of Entropy

Entropy is a measure of the disorder or randomness in a system. Generally, higher temperature and lower pressure increase entropy as they lead to increased molecular motion and volume, respectively.

Analyze Part (a)

Ice at higher temperatures ( 0^{ ext{∘}} ext{C} ) is closer to melting and thus has more disorder compared to ice at lower temperatures ( -40^{ ext{∘}} ext{C} ). Thus, ice at 0^{ ext{∘}} ext{C} has higher entropy.

Analyze Part (b)

At STP, nitrogen ( ext{N}_2 ) has a standard pressure of 1 ext{ atm} . Compared to 10 ext{ atm} , the STP condition has a lower pressure, allowing for more volume and thus higher entropy. Therefore, ext{N}_2 at STP has higher entropy than at ext{0}^{ ext{∘}} ext{C} and 10 ext{ atm} .

Analyze Part (c)

At STP (Standard Temperature and Pressure), nitrogen has a specific volume. If the volume is increased to 50 ext{ L} , and temperature and pressure remain the same, the molecules can explore more space, leading to higher entropy. Therefore, ext{N}_2 in a volume of 50 ext{ L} has higher entropy compared to at STP.

Analyze Part (d)

Water vapor at 150^{ ext{∘}} ext{C} and 1 ext{ atm} is at a higher temperature and lower pressure compared to 100^{ ext{∘}} ext{C} and 2 ext{ atm} . Both conditions suggest higher entropy for the water vapor at 150^{ ext{∘}} ext{C} and 1 ext{ atm} .

Key concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the principles governing how energy is transferred within and between different systems. It helps us understand how heat and work influence the energy and organization of particles in a system. One key aspect of thermodynamics is the study of entropy, which is a measure of the system's disorder or randomness.
When we talk about entropy in the context of thermodynamics, we're often discussing how energy spread or disperses within a system over time. Entropy tends to increase in an isolated system, reflecting the tendency towards balance and uniform distribution of energy.

• In thermodynamics, the second law states that the total entropy of an isolated system can never decrease over time.
• This principle explains why certain processes occur spontaneously, as they naturally progress towards a state of higher entropy where energy and materials spread more evenly.

Understanding thermodynamics requires grasping these basic laws and principles, which explain many natural phenomena and are foundational in fields such as chemistry and engineering.

Molecular Motion

Molecular motion refers to the movement of atoms and molecules within a substance. This motion is directly related to temperature, as higher temperatures provide more energy for molecules to move. Greater molecular motion typically contributes to higher entropy because molecules can distribute themselves in a more varied and less ordered manner.
When considering the connection between temperature and molecular motion:

• At higher temperatures, the kinetic energy of molecules increases, leading them to move faster and collide more frequently.
• In solids, molecules primarily vibrate in fixed positions. In liquids, they have more freedom to move around each other. In gases, molecules move freely at high speeds.

Thus, an increase in temperature generally leads to increased molecular motion and, hence, increased entropy. In practical terms, this is reflected in how gases fill containers. The increased motion allows them to occupy larger volumes and create more disorder.

Disorder and Randomness

Disorder and randomness are fundamental concepts in understanding entropy. In a mechanical or structured arrangement, systems have lower entropy due to organized order. As a system becomes more random and disordered, its entropy increases. Entropy, therefore, helps quantify the amount of disorder within a system.
Examples illustrating disorder include:

• A solid has lower entropy than a liquid because its particles are more fixed and ordered.
• A gas has higher entropy than both solids and liquids due to the randomness in particle distribution.

In terms of randomness, entropy can be thought of as a measurement of uncertainty or the number of different ways a system can be arranged. The greater the number of potential configurations, the higher the entropy.

Temperature and Pressure Influence on Entropy

Entropy is significantly influenced by temperature and pressure changes, as these factors alter the energy and volume of a system.
Temperature directly affects the kinetic energy and, consequently, entropy. At higher temperatures:

• Molecules move more energetically, increasing the system's disorder.
• This added motion leads to increased possibilities for how molecules are distributed, heightening entropy.

Conversely, pressure compresses molecules into a smaller volume, generally reducing entropy. However, if temperature is increased while pressure decreases, entropy usually rises due to both increased motion and available volume for molecules to spread out.

• Low-pressure systems typically have higher entropy as there's more space for molecules to move about.
• High-pressure systems constrain molecular movement, lowering entropy.

Understanding these relationships is crucial in predicting how a substance will behave under different physical conditions.