Problem 45 - Chemistry The Central Science In Si Units

Question

In each of the following pairs, which compound would you expect to have the higher standard molar entropy: (a) \(\mathrm{C}_{3} \mathrm{H}_{8}(g)\) or \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\) (b) \(\mathrm{C}_{4} \mathrm{H}_{10}(l)\) or \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\)

Step-by-Step Solution

Answer

(a) \(C_{4}H_{10}(g)\) has higher entropy; (b) \(C_{4}H_{10}(g)\) has higher entropy.

Step 1: Understanding Entropy

Entropy is a measure of disorder or randomness in a system. In general, gases have higher entropy than liquids or solids due to the greater freedom of movement of the molecules. Moreover, larger molecules tend to have higher entropy because they have more possible configurations.

Step 2: Analyzing Part (a)

For part (a), we compare \( C_{3}H_{8}(g) \) and \( C_{4}H_{10}(g) \). \( C_{4}H_{10}(g) \) is a larger molecule with more atoms, offering more possible positional and vibrational states. Thus, \( C_{4}H_{10}(g) \) should have a higher entropy than \( C_{3}H_{8}(g) \).

Step 3: Analyzing Part (b)

For part (b), we examine \( C_{4}H_{10}(l) \) versus \( C_{4}H_{10}(g) \). As a liquid, the molecules are more closely packed and have fewer ways to move compared to a gas. Therefore, \( C_{4}H_{10}(g) \) has a higher entropy due to the greater freedom of motion in the gaseous state.

Key Concepts

Entropy and States of Matter

Understanding the concept of entropy involves recognizing how different states of matter contribute to randomness and disorder in a system. Entropy is often described as the measure of disorder, with gases typically having the highest entropy compared to liquids and solids. Why? Because gas molecules possess a greater range of movement.

Gases have broad freedom to move in any direction, leading to increased randomness and, thus, greater entropy. In contrast, liquid molecules are more densely packed than gases, reducing their freedom to move but still allowing more flexibility than in solids. That said, solids have the lowest entropy among the states of matter because their particles are tightly locked into place, resulting in a more ordered structure. Given this behavior, the order of entropy from highest to lowest is: gas > liquid > solid. This hierarchy helps us predict which states of matter will have higher entropy when comparing similar substances.

Entropy and Molecular Complexity

When considering entropy, the complexity of a molecule also plays a significant role in determining its entropy. Generally, the more complex a molecule, the higher its entropy. This can be attributed to several reasons:

• Larger molecules have more atoms, which increases the number of possible arrangements or configurations.
• More complex molecules can exhibit more vibrational modes, where atoms within a molecule vibrate in various ways, adding to the system's overall entropy.
• Complex molecules often possess more surface area, allowing for greater interaction potential.

Take, for example, two gaseous molecular compounds: \(C_3H_8\) and \(C_4H_{10}\). \(C_4H_{10}\) is larger and more complex, featuring more atoms and potential configurations than \(C_3H_8\). This increased complexity results directly in higher entropy. Thus, understanding such molecular intricacies aids in discerning why more complex molecules hold greater entropy.

Comparing Entropy in Different Phases

An essential aspect of understanding entropy is recognizing how it varies across different phases. Consider a specific compound, like \(C_4H_{10}\). In its gaseous state, the molecules possess significant freedom, moving independently throughout space, resulting in high entropy. As the same compound transitions to a liquid state, the molecules are restricted, with reduced freedom of movement, which decreases its entropy.

Here's a simple way to compare:

• In gases, molecules can spread and interact randomly, maximizing entropy.
• In liquids, molecules have less freedom, with slightly restricted movement compared to gases, resulting in moderate entropy.
• In solids, molecules are tightly packed, with very limited movement, resulting in low entropy.

Whenever we wish to determine which phase a substance might exhibit higher entropy in, we consider the degree of molecular freedom in each phase. This comparison is quintessential when analyzing phase transitions and how these shifts affect entropy, leading us to a clearer understanding of the thermodynamic equilibrium in different environments.