For each of the following pairs, choose the substance with the higher entropy per mole at a given temperature: (a) \(\operatorname{Ar}(l)\) or \(\mathrm{Ar}(g),\) (b) \(\mathrm{He}(g)\) at 3 atm pressure or \(\mathrm{He}(g)\) at 1.5 atm pressure, (c) \(1 \mathrm{~mol}\) of \(\mathrm{Ne}(g)\) in \(15.0 \mathrm{~L}\) or \(1 \mathrm{~mol}\) of \(\mathrm{Ne}(g)\) in \(1.50 \mathrm{~L}\), (d) \(\mathrm{CO}_{2}(g)\) or \(\mathrm{CO}_{2}(s)\).

Entropy, often symbolized by the letter 'S', is a fundamental concept in chemistry that measures the level of disorder or randomness in a system. Understanding the relationship between entropy and temperature can give insight into the behavior of substances under different thermal conditions.

As temperature increases, the kinetic energy of the particles within a substance also increases, promoting greater movement and spreading out of particles. This enhanced movement results in a higher level of disorder, leading to an increase in entropy. For example, when a liquid like argon (Ar(l)) transitions to its gaseous form (Ar(g)), this change occurs at a temperature where the added thermal energy allows the argon atoms to move around more freely, hence increasing its entropy.

When comparing substances at the same temperature, one must consider their phase. Gases typically exhibit higher entropy than liquids and solids because their particles can move freely and spread out, creating a higher degree of randomness. This is why, at the same temperature, Ar(g) possesses a higher entropy than Ar(l).

The entropy of a gas is also greatly influenced by pressure. Since gases are compressible, changing the pressure can alter their volume and therefore their entropy. When a gas is compressed to a higher pressure, the molecules are forced closer together, reducing the randomness of their movement and hence decreasing the entropy.

Conversely, if the pressure is decreased, the gas expands, the molecules have more space to move randomly, and thus, the entropy increases. This explains why helium gas (He(g)) at a lower pressure of 1.5 atm has a higher entropy than at a higher pressure of 3 atm, at the same temperature. The molecules at lower pressure are less confined and have more available microstates, increasing the system's disorder.

The concept of molar entropy is deeply connected to the volume occupied by a substance, particularly gases. Molar entropy represents the entropy of one mole of a substance, and for gases, an increase in volume provides more accessible microstates for the particles to occupy, driving up the entropy.

For instance, when comparing neon gas (Ne(g)) in a 15.0 L volume to the same amount in a 1.50 L volume, the larger volume allows for more particle positions and random movements. The Neon gas has more space in which to distribute itself, which leads to a greater number of possible microstates and therefore a higher molar entropy in the 15.0 L volume.

The state of matter—solid, liquid, or gas—affects the entropy of a substance because it determines the degree of particle movement and arrangement. Solids, with their structured lattice and limited particle movement, exhibit the lowest entropy. Liquids, with some degree of fluidity, have higher entropy than solids. Gases, which are not constrained by a lattice and have the most freedom to move, have the highest entropy among the three states.

Thus, when comparing carbon dioxide as a solid (CO2(s)) and as a gas (CO2(g)), the gaseous form manifests significantly higher entropy due to the greater level of particle disorder. In a solid state, the structured arrangement of particles constrains movement and limits the number of accessible microstates, leading to lower entropy.