Question

The standard entropies at \(298 \mathrm{~K}\) for certain of the group \(4 \mathrm{~A}\) elements are as follows: \(\mathrm{C}(s\), diamond \()=2.43 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ; \quad \mathrm{Si}(s)=18.81 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ;\) \(\mathrm{Ge}(s)=31.09 \mathrm{~J} / \mathrm{mol}-\mathrm{K} ; \quad\) and \(\quad \mathrm{Sn}(s)=51.18 \mathrm{~J} / \mathrm{mol}-\mathrm{K}\) All but Sn have the diamond structure. How do you account for the trend in the \(S^{\circ}\) values?

Short answer

The trend in standard entropies for the given Group 4A elements (Carbon, Silicon, Germanium, and Tin) can be explained by their increasing molecular weight and differences in molecular structure. As molecular weight increases, the complexity of the system also increases, resulting in a more random distribution of energy and higher standard entropies. Tin, with a notably higher molecular weight and without a diamond structure, exhibits increased complexity and randomness, contributing to its higher standard entropy value.

Step-by-step solution

Recall the factors affecting entropy

To account for the trends in entropy, consider the factors that affect entropy. Entropy is a measure of the distribution of energy over the possible microscopic states of a system; higher entropy represents a more random distribution. Factors that affect entropy include the mass, the complexity of the molecule, the number of atoms in the molecule, and the phase of the substance.

Identify the factors that are significant for the given trend

Comparing the standard entropies \(S^{\circ}\) for the given elements, we notice an increase in the values as we move down the Group 4A. Since the structure and chemical composition of Carbon, Silicon, and Germanium are similar (except Tin), the main factor affecting entropy among these elements is their molecular weight.

Explanation of the trend based on molecular weight

As molecular weight increases, the complexity of the system also increases, which results in a more random distribution of energy. Consequently, heavier elements tend to have higher standard entropies. Specifically, for Carbon (C), Silicon (Si), and Germanium (Ge), the molecular weight increases as we move down the group, and we can attribute the trend in the \(S^{\circ}\) values to the increased molecular weight.

Account for the difference in Tin's standard entropy and molecular structure

Tin (Sn) has a higher standard entropy, 51.18 J/mol-K, because it has a significantly higher molecular weight than the other elements in the group. In addition to this, the fact that Tin does not have a diamond structure contributes to the difference in entropy values. This difference in structure causes an increase in complexity and randomness in Tin when compared to the other elements, which further contributes to its higher standard entropy value. In conclusion, the trend in standard entropies for the given group 4A elements can be explained by the increase in molecular weight and the structural differences among the elements, particularly in the case of Tin.

Key concepts

Entropy in Chemistry

Entropy is a fundamental concept in chemistry that describes the measure of disorder or randomness in a system. In thermodynamics, entropy is denoted by the symbol 'S', and it provides an insight into the energy distribution among a system's microstates at a given temperature. A higher entropy value implies a greater degree of distribution and thus more potential states for the molecules in the system.

From a molecular perspective, when a substance has high entropy, its particles are spread out over a large number of possible configurations. This condition aligns with the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.

The significance of understanding entropy lies in predicting the feasibility and spontaneity of chemical reactions. Reactions tend to proceed in a direction where the entropy of the system increases, suggesting that systems naturally evolve towards disorder. In other words, it is a driving force behind chemical processes and helps us understand why certain reactions occur without any external intervention.

Group 4A Elements

Group 4A, also known as Group 14 of the periodic table, includes elements such as Carbon (C), Silicon (Si), Germanium (Ge), and Tin (Sn). These elements have varying physical and chemical properties but are united by their similar valence electron configuration of ns2np2, which allows them to form four covalent bonds with other atoms.

Each of these elements can adopt different crystal structures, with the diamond structure being common for Carbon, Silicon, and Germanium. Tin, however, commonly exists in a metallic structure at room temperature, which distinguishes its physical properties from the other Group 4A elements.

Studying the trends within this group reveals important insights into how properties such as melting points, boiling points, and indeed entropy change as we move down the group. These trends are crucial for understanding the behavior of these elements in different chemical contexts, as well as predicting how they will interact with other substances.

Molecular Weight and Entropy

The relationship between molecular weight and entropy is an intriguing aspect of thermodynamics. As the molecular weight of a substance increases, so does the complexity of the molecule. Complexity in this context refers to the number of atoms, their arrangements, and the potential for different types of motion within the molecule - translational, rotational, and vibrational.

As a general trend, heavier molecules, with their increased complexity, have a greater number of accessible microstates. This translates into a higher entropy because there are more ways the energy can be distributed in the system. The larger atoms can vibrate more readily, and the increased number of particles enhances the possible arrangements the substance can adopt.

In the context of Group 4A elements, as we move from Carbon to Tin, the atoms get larger and the molecules they form get heavier. This increased molecular weight from one element to the next explains the standard entropy increase observed in the group. However, structural factors also play a role in determining entropy values, as seen with Tin's deviation from the diamond structure.