Question

For each pair of substances, choose the one that you expect to have the higher standard molar entropy \(\left(S^{\circ}\right)\) at \(25^{\circ} \mathrm{C}\) . Explain the reasons for your choices. \begin{equation} \begin{array}{l}{\text { a. } \mathrm{NaNO}_{3}(s) ; \mathrm{NaNO}_{3}(a q)} \\\ {\text { b. } \mathrm{CH}_{4}(\mathrm{g}) ; \mathrm{CH}_{3} \mathrm{CH}_{3}(\mathrm{g})} \\ {\text { c. } \mathrm{Br}_{2}(l) ; \mathrm{Br}_{2}(g)}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { d. } \operatorname{Br}_{2}(g) ; \mathrm{F}_{2}(g)} \\\ {\text { e. } \mathrm{PCl}_{3}(g) ; \mathrm{PCl}_{5}(g)} \\ {\mathrm{f} . \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{3}(\mathrm{g}) ; \mathrm{SO}_{2}(g)}\end{array} \end{equation}

Short answer

The substances with higher standard molar entropy are: a. NaNO3(aq), b. CH3CH3(g), c. Br2(g), d. Br2(g), e. PCl5(g), f. CH3CH2CH2CH3(g).

Step-by-step solution

Comparing Entropy in Different States

Substances in an aqueous state generally have higher standard molar entropy than the same substances in a solid state due to higher degrees of freedom in movement and the presence of solvation effects. Therefore, between NaNO3(s) and NaNO3(aq), NaNO3(aq) will have a higher standard molar entropy at 25°C.

Comparing Entropy of Different Gas Molecules

When comparing gases, the molecule with greater complexity, often indicated by a higher number of atoms or a larger mass, will have a higher standard molar entropy. Thus, between CH4(g) and CH3CH3(g), CH3CH3(g) will have a higher entropy due to its larger number of atoms and complexity.

Comparing Entropy in Different Phases

The gaseous state of a substance has a higher standard molar entropy than the same substance in a liquid state because of the significant increase in molecular motion and distance between molecules. Thus, between Br2(l) and Br2(g), Br2(g) will have a higher standard molar entropy.

Comparing Entropy of Different Gaseous Elements

For diatomic molecules in the gas phase, standard molar entropy increases with increased atomic size and mass. As bromine atoms are larger and have greater mass than fluorine atoms, Br2(g) should have a higher standard molar entropy than F2(g).

Comparing Entropy of Gases with Different Numbers of Atoms

A molecule with a greater number of atoms will have a higher standard molar entropy because of the increased number of possible microstates. PCl5(g) contains more atoms than PCl3(g) and will have a higher standard molar entropy.

Comparing Entropy of Organic Molecules vs. Inorganic Molecules

Comparing an organic molecule to an inorganic molecule requires considering the structure and molecular complexity. CH3CH2CH2CH3(g), being a larger molecule with more complex internal rotations, will likely have a higher standard molar entropy than SO2(g), which is smaller and has more limited internal modes of rotation.

Key concepts

Entropy Comparison

When discussing entropy comparison, it's vital to understand entropy as a measure of the system's disorder or randomness. A useful analogy is to think of entropy as how messy a room can be; the messier the room, the higher its entropy. In the realm of chemistry, substances that possess more complexity or freedom in their particles' movement exhibit higher entropy.

For example, when comparing two substances, one solid and one in aqueous solution, such as NaNO\(_3\)(s) and NaNO\(_3\)(aq), the latter has higher entropy due to increased molecular movement in solution. This is analogous to having more ways to arrange the furniture (molecules) in a larger room (solution) compared to a smaller one (solid).

Another comparison can be made between different gases, like CH\(_4\)(g) and CH\(_3\)CH\(_3\)(g). The molecule with greater molecular complexity – indicated by a larger number of atoms – has higher entropy. This is similar to having more objects in a room, increasing the number of ways the room can be disorganized.

Entropy in Different States

The concept of entropy in different states highlights how entropy varies as a substance changes from solid to liquid to gas. Solid-state molecules are tightly packed with limited movement, like books neatly arranged on a shelf. In contrast, gases have molecules far apart and moving freely, like papers scattered across a room.

Between Br\(_2\)(l) and Br\(_2\)(g), Br\(_2\)(g) will have higher standard molar entropy as gas molecules have greater freedom of movement and spacing, leading to more randomness. This change in entropy is significant during phase transitions, particularly from liquid to gas, where a substance experiences a remarkable increase in entropy due to the drastic change in particle arrangement and motion.

Entropy of Gas Molecules

Exploring the entropy of gas molecules takes us to the microscopic world of atoms and molecules. Gas molecules are akin to children running freely in a playground, with each child (molecule) having numerous paths (microstates) they can take.

The more complex the molecule (the more children or paths), the higher the entropy. For diatomic molecules like Br\(_2\)(g) and F\(_2\)(g), the larger and heavier bromine atoms make Br\(_2\)(g) more disorganized than F\(_2\)(g), similar to how heavier children might take up more space on the playground.

Phase Change Impact on Entropy

The impact of a phase change on entropy is one of the most vivid illustrations of how entropy changes with the state of matter. Phase changes, especially from liquid to gas (evaporation) or solid to gas (sublimation), represent a leap in the level of disorder.

As the phase changes to a gaseous state, molecules spread out and move vigorously, just as a room becomes messier when items are thrown about haphazardly. This results in a significant increase in standard molar entropy, reflecting a more disarrayed system. It's worth noting that this transition illustrates one of the most fundamental concepts in thermodynamics: systems tend to move towards greater disorder, which means higher entropy.