Question

Rank each set of substances in order of increasing standard molar entropy \(\left(S^{\circ}\right) .\) Explain your reasoning. \begin{equation} \begin{array}{l}{\text { a. } \mathrm{I}_{2}(s) ; \mathrm{F}_{2}(g) ; \mathrm{Br}_{2}(g) ; \mathrm{Cl}_{2}(g)} \\ {\text { b. } \mathrm{H}_{2} \mathrm{O}(g) ; \mathrm{H}_{2} \mathrm{O}_{2}(g) ; \mathrm{H}_{2} \mathrm{S}(g)}\end{array} \end{equation} c. C(s, graphite); C(s, diamond); C(s, amorphous)

Short answer

\(\rm{a.} I_{2}(s) < F_{2}(g) < Cl_{2}(g) < Br_{2}(g)\); \(\rm{b.} H_{2}O(g) < H_{2}S(g) < H_{2}O_{2}(g)\); \(\rm{c.} C(s, diamond) < C(s, amorphous) < C(s, graphite)\)

Step-by-step solution

Understanding Entropy

Understand that standard molar entropy (\(S^{\text{o}}\)) is a measure of the amount of disorder or randomness in a substance. Generally, gases have higher entropies than liquids, which in turn have higher entropies than solids. For substances in the same state of matter, complexity and molecular mass can affect their entropy.

Ranking Group a

For set a, which includes \(\mathrm{I}_{2}(s), \mathrm{F}_{2}(g), \mathrm{Br}_{2}(g), \mathrm{Cl}_{2}(g)\), consider their states and molecular complexity. Iodine (\(\mathrm{I}_{2}\)) is a solid and has the lowest entropy. Among the gases, fluorine (\(\mathrm{F}_{2}\)) has the lowest molecular weight, followed by chlorine (\(\mathrm{Cl}_{2}\)), and bromine (\(\mathrm{Br}_{2}\)) has the highest molecular weight and thus the highest entropy. The order of increasing entropy is \(\mathrm{I}_{2}(s) < \mathrm{F}_{2}(g) < \mathrm{Cl}_{2}(g) < \mathrm{Br}_{2}(g)\).

Ranking Group b

For set b, which includes \(\mathrm{H}_{2}\mathrm{O}(g), \mathrm{H}_{2}\mathrm{O}_{2}(g), \mathrm{H}_{2}\mathrm{S}(g)\), all substances are in the gaseous state. Water (\(\mathrm{H}_{2}\mathrm{O}\)) has the simplest molecule and the lowest entropy. Hydrogen sulfide (\(\mathrm{H}_{2}\mathrm{S}\)) has a slightly more complex molecule and higher molecular weight than water, giving it a higher entropy. Hydrogen peroxide (\(\mathrm{H}_{2}\mathrm{O}_{2}\)) is more complex than both water and hydrogen sulfide, so it has the highest entropy. The order of increasing entropy is \(\mathrm{H}_{2}\mathrm{O}(g) < \mathrm{H}_{2}\mathrm{S}(g) < \mathrm{H}_{2}\mathrm{O}_{2}(g)\).

Ranking Group c

For set c, we have three allotropes of carbon: graphite, diamond, and amorphous carbon. Graphite is the most stable and has the highest entropy. Amorphous carbon is less ordered than graphite and diamond but more ordered than graphite; therefore, it has a moderate entropy. Diamond, being the most ordered and compact allotrope of carbon, has the lowest entropy. The order of increasing entropy is \(\text{C(s, diamond)} < \text{C(s, amorphous)} < \text{C(s, graphite)}\).

Key concepts

Entropy Ranking

Entropy is a fundamental concept in thermodynamics and chemistry that describes the level of disorder or randomness in a system. When ranking substances based on their standard molar entropy \(S^{\circ}\), we must consider their physical state (gas, liquid, or solid) and molecular structure.

As a rule of thumb, gases possess higher entropy than liquids, which in turn have higher entropy than solids. This is due to the increased freedom of movement and the larger volume occupied by gas molecules, which create more possible arrangements, or microstates, contributing to higher disorder.

The rankings within a group of substances that are in the same state of matter can also be influenced by other factors, such as molecular complexity and molecular weight. More complex molecules with larger molecular weights typically have a greater number of atoms and bonds, leading to a larger number of vibrational and rotational modes that contribute to overall entropy. For instance, when comparing \(\mathrm{H}_{2}\mathrm{O}(g)\) with \(\mathrm{H}_{2}\mathrm{S}(g)\), we acknowledge the slightly higher molecular weight and complexity of hydrogen sulfide, leading to higher entropy.

Molecular Complexity and Entropy

Molecular complexity significantly influences the entropy of a substance. This complexity can mean the number of atoms within a molecule, the variety of elements present, and the intricacy of the molecule's structure. Each additional atom or bond allows for more ways in which a molecule can rotate, vibrate, and move, increasing the number of possible microstates.

Take the example of water \(\mathrm{H}_{2}\mathrm{O}\) and hydrogen peroxide \(\mathrm{H}_{2}\mathrm{O}_{2}\). Although both contain hydrogen and oxygen, hydrogen peroxide contains an additional oxygen atom, making the molecule more complex. This leads to a greater number of microstates and, therefore, a higher entropy compared to water. In essence, the greater the complexity, the more arrangements the atoms can adopt and the more entropy the substance will have.

This understanding is pivotal when it comes to explaining variations in entropies among molecules that might otherwise seem quite similar. By keeping this in mind, students can better interpret entropy rankings beyond mere memorization of facts, enhancing their grasp of the underlying principles.

Allotropes of Carbon

Carbon is an element that exists in multiple forms—known as allotropes—each with different physical properties. Allotropes of carbon include graphite, diamond, and amorphous carbon, amongst others. These differences in structure have a direct impact on their entropy.

Graphite has a layered structure with weak forces between the layers, allowing them to slide over one another. This feature grants it a high degree of disorder and, consequently, a higher entropy. Diamond, in contrast, has a rigid, three-dimensional lattice structure that is extremely organized, yielding the lowest entropy among the allotropes. Amorphous carbon lacks a clear, regular structure and is somewhere between the organized diamond and the more disordered graphite in terms of entropy.

Understanding the relationship between the arrangement of carbon atoms in these allotropes and their associated entropy helps students appreciate why certain forms of carbon are more thermodynamically stable than others. As entropy is a measure of disorder, the more ordered the structure of the allotrope, the lower its entropy will be.