Question

In each of the following pairs, choose the substance with a lower entropy. (a) One mole of \(\mathrm{O}_{2}(g)\) with \(758 \mathrm{~mm} \mathrm{Hg}\) pressure, one mole of \(\mathrm{O}_{2}(\mathrm{~g})\) with \(493 \mathrm{~mm} \mathrm{Hg}\) pressure, both at room temperature (b) glucose \((s)\), glucose \((a q)\) (c) \(\mathrm{Hg}(l), \mathrm{Hg}(\mathrm{g})\), both at room temperature

Short answer

Question: Compare the entropy of the following substances or conditions: a) One mole of O2(g) at pressures 758 mm Hg and 493 mm Hg, both at room temperature. b) Glucose in its solid and aqueous states. c) Mercury in its liquid and gaseous states, both at room temperature. Answer: a) The one mole of O2(g) at 758 mm Hg pressure has a higher entropy than the one mole of O2(g) at 493 mm Hg pressure. b) Glucose in its aqueous state has a higher entropy than in its solid state. c) Mercury in its gaseous state at room temperature has higher entropy than in its liquid state at the same temperature.

Step-by-step solution

(a) Comparing the entropy of one mole of O2(g) at different pressures

We are given two samples of one mole of oxygen gas \((O_2(g))\), both at room temperature, but with different pressures: \(758 \,\mathrm{mm} \,\mathrm{Hg}\) and \(493\, \mathrm{mm} \, \mathrm{Hg}\). The same substance at a higher pressure generally has a higher entropy, so since \(758\, \mathrm{mm}\, \mathrm{Hg}\) is greater than \(493\, \mathrm{mm}\, \mathrm{Hg}\), the oxygen gas with a pressure of \(758\, \mathrm{mm}\, \mathrm{Hg}\) has a higher entropy. The oxygen gas with a pressure of \(493\, \mathrm{mm}\, \mathrm{Hg}\) has a lower entropy.

(b) Comparing the entropy of glucose in solid and aqueous states

We are given two samples of glucose, one in solid state \((s)\), and the other in aqueous state \((aq)\). Solid substances generally have a lower entropy than substances in solution. Therefore, glucose \((s)\) has a lower entropy compared to glucose \((aq)\).

(c) Comparing the entropy of Hg(l) and Hg(g) both at room temperature

We are given two samples of mercury, both at room temperature, one in the liquid state \((l)\), and the other in gaseous state \((g)\). At any given temperature, the gas phase of a substance generally has a higher entropy than the other phases. Therefore, Hg(l) at room temperature has a lower entropy compared to Hg(g) at room temperature.

Key concepts

Entropy and Pressure

Understanding how pressure affects entropy is fundamental when studying chemical thermodynamics. Entropy, a measure of the molecular disorder within a system, can change with varying pressure. When the pressure on a gas increases, the molecules are forced closer together, thus reducing the space available for them to move around. This reduction in volume leads to increased interactions and a decrease in randomness of motion, which in turn lowers entropy.

For example, if we take a mole of a gas at room temperature and compress it from a lower pressure to a higher pressure, we are decreasing the entropy because the gas molecules have less freedom of motion. The same concept applies to the exercise where oxygen gas at 493 mm Hg has more freedom to move than at 758 mm Hg, leading to higher entropy at the lower pressure.

In everyday scenarios, this is observed when you pump air into a bicycle tire. As the pressure inside the tire increases, the entropy of the air molecules decreases due to the confined space.

Entropy of Substances in Different States

Let's explore the concept of entropy as it relates to the physical states of substances. Entropy is associated with the amount of disorder or randomness in a system, and it varies significantly among solids, liquids, and gases. Solids, which have a tightly ordered structure, present low levels of entropy. As a substance transitions to the liquid state, its particles have more freedom to move, thus increasing entropy. Gases have the most disordered state with the highest entropy since the particles are freely moving and widely spaced apart.

In the context of the provided exercise, a solid form of glucose has ordered molecular arrangements; therefore, it has lower entropy compared to glucose in an aqueous solution, where it is dispersed with water molecules, increasing its disorder. Analogously, we experience the difference in entropy in the kitchen when ice (solid) melts into water (liquid).

Higher entropy in the aqueous and gaseous states reflects the increased energy levels and movement freedom of the particles as opposed to the more rigid structure of solids.

Entropy Comparison in Chemical Systems

When comparing entropy among different chemical systems, it's essential to consider both the physical state of the substances and the molecular complexity. Entropy generally increases with temperature, state changes (solid to liquid to gas), and with the complexity of the molecules because more complex molecules have more ways to distribute energy internally.

In the exercise's case of mercury, we observe the entropy of its liquid form compared to its gaseous form. Mercury as a gas has more entropy than in its liquid state due to the increased disorder and freedom of motion of its particles as a gas. This entropy difference accompanies the energy absorbed during the phase change from liquid to gas.

Understanding these entropy changes allows us to predict the direction of spontaneous processes, which is invaluable for fields like materials science, predicting weather patterns, or even during complex reactions in biochemistry, where the concept of entropy helps us to understand the structure and behavior of proteins and enzymes.