Question

In your own words, explain why (a) \(\Delta S^{\circ}\) is negative for a reaction in which the number of moles of gas decreases. (b) we take \(\Delta S^{\circ}\) to be independent of \(T,\) even though entropy increases with \(T\). (c) a solid has lower entropy than its corresponding liquid.

Short answer

Question: Explain the relationship between the change in the number of moles of gas and the change in entropy (∆S°), why we consider ∆S° to be independent of temperature (T), and how the entropy of a solid compares to its corresponding liquid. Answer: The change in entropy (∆S°) is negative when the number of moles of gas decreases because the system becomes more ordered. ∆S° is considered independent of temperature because we assume that the temperature dependency of the reactants and products is roughly equal, simplifying the analysis. The entropy of a solid is lower than its corresponding liquid because the particles in a solid are more ordered and have less freedom of movement compared to those in a liquid.

Step-by-step solution

Part (a) - Explaining why ∆S° is negative when the number of moles of gas decreases

Entropy is often associated with the degree of randomness or disorder in a system. In a gaseous state, molecules are distributed more randomly and have a higher entropy compared to a state with fewer gas molecules. A reaction that decreases the number of moles of gas involves a reduction in the level of disorder, leading to a decrease in entropy. Therefore, when the number of moles of gas decreases in a reaction, the change in entropy (∆S°) is negative because the system has become more ordered.

Part (b) - Explaining why we consider ∆S° to be independent of T

It is true that entropy increases with temperature, as the molecules in a system have higher energy and are more randomly distributed at higher temperatures. However, when we study the change in entropy for a reaction (∆S°), we usually consider it to be independent of T because we are looking at a difference between two states, not the absolute entropy value at a specific temperature. When calculating the change in entropy for a reaction, we generally assume that the temperature dependency of the entropy of the reactants and products is roughly equal, which allows us to neglect any temperature dependence for ∆S°. This simplification makes it easier to understand and analyze the entropy changes involved in chemical reactions.

Part (c) - Explaining why a solid has lower entropy than its corresponding liquid

Entropy is a measure of the randomness or disorder in a system. In a solid state, the particles (atoms, molecules, or ions) are arranged in a regular pattern and are relatively fixed in position. The movement of these particles is restricted to small vibrations around their fixed positions. As a result, a solid has a lower entropy compared to a liquid. In a liquid state, the particles can move more freely and flow around each other. These particles are not confined to fixed positions and can interact with a greater number of neighboring particles. This higher degree of motion and more disordered arrangement of particles in a liquid leads to a higher entropy compared to its corresponding solid.

Key concepts

Change in Entropy

Entropy is a fundamental concept describing the disorder or randomness of particles in a system. A remarkable aspect of chemical reactions is the change in entropy, denoted as \( \Delta S \). Whenever a chemical reaction occurs, the entropy of the system can either increase or decrease. If the products of a reaction have a higher degree of randomness compared to the reactants, \( \Delta S \) will be positive, indicating an increase in disorder. Conversely, when a reaction results in products that are more ordered than the reactants — for instance, when the number of moles of gas decreases — \( \Delta S \) becomes negative, reflecting a decrease in entropy.

Understanding the changes in entropy is critical for predicting whether a reaction will occur spontaneously. Reactions tend to favor conditions that increase the overall disorder, thus a positive change in entropy often supports spontaneity under certain conditions. However, it's important to remember that entropy is only one component of a larger thermodynamic picture, which also includes enthalpy and temperature factors encapsulated by the Gibbs free energy equation.

Entropy in States of Matter

The concept of entropy extends to the different states of matter: solid, liquid, and gas. By comparing entropy across these states, we can grasp how particle arrangement and movement contribute to disorder. In solids, particles are tightly packed in a fixed, orderly structure, limiting their ability to move and thus resulting in lower entropy.

Liquids: A Step Up in Disorder

When a solid transforms into a liquid, the particles gain freedom to move around, sliding past one another, which increases the system's randomness or entropy. Hence, liquids have higher entropy than solids because their particles are less orderly and can interact with a broader set of neighboring particles.

Gases: The Epitome of Randomness

As we further transition to a gaseous state, particles spread out and move more independently, reaching the pinnacle of disorder. This dispersion represents a significant increase in entropy, explaining why gases have the highest entropy among the three states of matter. This hierarchy of entropy is crucial in predicting the direction of phase changes and the behavior of substances under varying conditions.

Temperature Dependence of Entropy

Temperature has a direct impact on entropy. As temperature increases, so does the kinetic energy of particles, resulting in greater motion and higher entropy. This relationship is grounded in the idea that heat energy 'excites' particles, causing them to move more vigorously and assume a wider variety of positions and speeds. This heightened movement creates more configurations and states for the particles, which is synonymous with increased randomness or entropy.

Despite the clear relationship between temperature and entropy, when evaluating the entropy change for a reaction \( \Delta S^\circ \), it is often treated as independent of temperature. The rationale behind this approach is that it simplifies comparisons between the entropy of reactants and products. The assumption is that over a small temperature range, the temperature dependency of the system's entropy will not significantly alter the outcome, especially for standard conditions. While this assumption isn't always perfect, it is a practical approach for many calculations, allowing chemists to focus on the inherent entropy differences due to molecular structure and phase rather than the influence of temperature fluctuations.