Question

For each of the following pairs of substances, which substance has the greater value of \(S^{\circ} ?\) a. \(C_{\text {graphite }}(s)\) or \(C_{\text {diamond }}(s)\) b. \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)\) or \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{g})\) c. \(\mathrm{CO}_{2}(s)\) or \(\mathrm{CO}_{2}(g)\)

Short answer

In summary: a. \(C_{\text {graphite }}(s)\) has greater entropy than \(C_{\text {diamond }}(s)\) b. \(C_2 H_5 OH(g)\) has greater entropy than \(C_2 H_5 OH(l)\) c. \(CO_2(g)\) has greater entropy than \(CO_2(s)\)

Step-by-step solution

Comparison of Entropy for a. Carbon in Graphite and Diamond Forms

In carbon exists in two different forms - graphite (a more disordered and less structured form) and diamond (a highly ordered and structured form). Since entropy is a measure of randomness, the more disordered form will have higher entropy. Therefore, \(C_{\text {graphite }}(s)\) has a greater value of \(S^{\circ}\) than \(C_{\text {diamond }}(s)\).

Comparison of Entropy for b. Ethanol in Liquid and Gas Forms

In this pair, we need to compare ethanol (\(C_2H_5OH\)) in its liquid and in its gas states. Gas molecules have more freedom to move around and are more disorganized compared to their liquid counterparts, which are held together by intermolecular forces. Therefore, the gaseous form (\(C_2 H_5 OH(g)\)) has a greater value of \(S^{\circ}\) than the liquid form (\(C_2 H_5 OH(l)\)).

Comparison of Entropy for c. Carbon Dioxide in Solid and Gas Forms

Here, we need to compare carbon dioxide (\(CO_2\)) in its solid and gaseous states. As a solid (dry ice), carbon dioxide molecules are held together in a relatively more ordered arrangement compared to its gaseous form, where the molecules have more freedom to move and are more disordered. Therefore, \(CO_2(g)\) has a greater value of \(S^{\circ}\) than \(CO_2(s)\). In summary: a. \(C_{\text {graphite }}(s)\) has greater entropy than \(C_{\text {diamond }}(s)\) b. \(C_2 H_5 OH(g)\) has greater entropy than \(C_2 H_5 OH(l)\) c. \(CO_2(g)\) has greater entropy than \(CO_2(s)\)

Key concepts

Standard Entropy

Standard entropy, denoted by the symbol \(S^{\text{\textdegree}}\), is a key concept in thermodynamics representing the absolute entropy of a substance at a standard state. Simply put, it's a measure of the energy dispersal within a system at a specific temperature and pressure, typically 1 bar (100 kPa) pressure and a temperature of 25°C (298 K).

Understanding standard entropy is crucial because it helps us compare the disorderliness or randomness of different substances under equal conditions. For example, when comparing graphite and diamond, both being forms of carbon, standard entropy tells us that graphite is more disordered due to its layered structure, leading to higher \(S^{\text{\textdegree}}\) compared to diamond, which has a rigid, tightly-bound crystalline lattice.

This concept of entropy not only applies to different forms of the same substance but also to different states of matter of the same compound. As a rule of thumb, gases generally have higher standard entropies than liquids, which in turn have higher entropies than solids, because the molecules in gases have the most freedom to move and spread out.

Entropy and Molecular Structure

The link between entropy and molecular structure is rooted in the ways atoms and molecules are arranged and how they interact with one another. Entropy is intimately connected to these arrangements - a more complex molecule with more atoms and bonds will often have higher entropy because there are more ways for the energy to be distributed within the molecule.

When contrasting molecules like ethanol in liquid and gas phases, the molecular structure guides the energy distribution. The liquid phase, with stronger intermolecular forces, keeps molecules in a more orderly state. However, when ethanol evaporates into the gas phase, these forces are overcome, allowing the molecules to spread out and adopt a vastly greater number of configurations, increasing the entropy.

To provide a concrete example, solid carbon dioxide or 'dry ice' is organized in a tight lattice structure, limiting the possible arrangements of its molecules. Upon sublimation, dry ice transforms into CO2 gas, which allows for free movement of molecules and a significantly higher entropy owing to the greater number of microstates accessible to the gas molecules.

States of Matter and Entropy

The states of matter—solid, liquid, and gas—are distinguished by different physical properties, but also by their level of entropy. Entropy's relationship with the states of matter can be thought of in terms of particle movement and arrangement.

In solids, particles are closely packed and have minimal movement, resulting in a lower entropy. Liquids have more space between particles and additional kinetic energy, giving particles more freedom to move, which translates to a moderate increase in entropy. Gases have the highest entropy among the three traditional states of matter because particles move freely and occupy the available space, leading to a high level of disorderliness.

This pattern can be highlighted through the exercise: dry ice \(CO_2(s)\) showcases a structured solid state with low entropy, whereas \(CO_2(g)\), as a gas, exhibits far more disorder with molecules whizzing freely, resulting in a higher standard entropy. The idea here is that as matter changes from solid to liquid to gas, the entropy increases with each transition, reflecting the greater freedom and number of positions that the particles can assume.