Problem 52 - Chemistry A Molecular Approach

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 your choices. a. \(\mathrm{NaNO}_{3}(s) ; \mathrm{NaNO}_{3}(a q)\) b. \(\mathrm{CH}_{4}(g) ; \mathrm{CH}_{3} \mathrm{CH}_{3}(g)\) c. \(\operatorname{Br}_{2}(l) ; \mathrm{Br}_{2}(g)\) d. \(\mathrm{Br}_{2}(g) ; \mathrm{F}_{2}(g)\) e. \(\mathrm{PCl}_{3}(g) ; \mathrm{PCl}_{5}(g)\) f. \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{3}(g) ; \mathrm{SO}_{2}(g)\)

Step-by-Step Solution

Answer

\(\mathrm{NaNO}_{3}(aq)\), \(\mathrm{CH}_{3} \mathrm{CH}_{3}(g)\), \(\mathrm{Br}_{2}(g)\), \(\mathrm{F}_{2}(g)\), \(\mathrm{PCl}_{5}(g)\), \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{3}(g)\) have higher standard molar entropy.

Step 1: Understanding Standard Molar Entropy

Standard molar entropy, denoted as \(S^\circ\), is a measure of the amount of disorder or randomness in a substance. Typically, gases have higher standard molar entropy than liquids, which in turn have higher entropy than solids. Solutions have higher entropy than pure substances in the solid state because of increased randomness. Furthermore, the complexity of the molecular structure and number of atoms in a molecule can also increase entropy.

Step 2: Comparing \(\mathrm{NaNO}_{3}(s)\) and \(\mathrm{NaNO}_{3}(aq)\)

Between sodium nitrate in the solid state and in an aqueous solution, the solution form \(\mathrm{NaNO}_{3}(aq)\) will have a higher standard molar entropy because the ions are more dispersed and have more freedom to move in solution compared to the ordered, rigid structure of the solid.

Step 3: Comparing \(\mathrm{CH}_{4}(g)\) and \(\mathrm{CH}_{3} \mathrm{CH}_{3}(g)\)

Methane (\(\mathrm{CH}_{4}\)) has a simpler molecular structure compared to ethane (\(\mathrm{CH}_{3}\mathrm{CH}_{3}\)). Ethane has more atoms and more possible configurations due to the additional carbon and hydrogen, leading to a higher standard molar entropy for \(\mathrm{CH}_{3} \mathrm{CH}_{3}(g)\).

Step 4: Comparing \(\operatorname{Br}_{2}(l)\) and \(\mathrm{Br}_{2}(g)\)

Bromine as a liquid (\(\operatorname{Br}_{2}(l)\)) will have a lower standard molar entropy than bromine as a gas (\(\mathrm{Br}_{2}(g)\)) because the gaseous state allows for more disorder and freedom of movement for the bromine molecules.

Step 5: Comparing \(\mathrm{Br}_{2}(g)\) and \(\mathrm{F}_{2}(g)\)

Bromine gas (\(\mathrm{Br}_{2}(g)\)) and fluorine gas (\(\mathrm{F}_{2}(g)\)) are both diatomic molecules; however, fluorine is lighter and has smaller atoms. Therefore, the intermolecular forces are weaker in fluorine gas, allowing greater freedom of movement. As a result, the standard molar entropy is expected to be higher in \(\mathrm{F}_{2}(g)\).

Step 6: Comparing \(\mathrm{PCl}_{3}(g)\) and \(\mathrm{PCl}_{5}(g)\)

Phosphorus trichloride (\(\mathrm{PCl}_{3}\)) and phosphorus pentachloride (\(\mathrm{PCl}_{5}\)) are both gaseous compounds containing phosphorus and chlorine. \(\mathrm{PCl}_{5}(g)\) has a higher standard molar entropy due to its larger size and the increased number of atoms, which allows for more molecular complexity and a greater number of microstates.

Step 7: Comparing \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{3}(g)\) and \(\mathrm{SO}_{2}(g)\)

Butane (\(\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{CH}_{2}\mathrm{CH}_{3}\)) is a larger molecule compared to sulfur dioxide (\(\mathrm{SO}_{2}\)), with a more complex structure and thus more possible microstates. This suggests a higher standard molar entropy for butane gas.

Key Concepts

Entropy and Molecular Complexity

The concept of entropy is intimately connected with the idea of molecular complexity. Entropy, often symbolized as \(S\), represents the degree of randomness or disorder within a system. As a system becomes more complex—having more atoms or a larger, more intricate structure—it inherently possesses a greater number of microstates, which are possible states that the system can occupy.

Take for instance the comparison between methane \(\mathrm{CH}_{4}\) and ethane \(\mathrm{CH}_{3} \mathrm{CH}_{3}\). Ethane has a more complex molecular structure due to an additional carbon and three extra hydrogen atoms. This complexity implies that there are more ways in which the atoms can be arranged and more possible rotations and vibrations within the molecule, leading to a higher standard molar entropy.

The same principle applies when comparing \(\mathrm{PCl}_{3}(g)\) and \(\mathrm{PCl}_{5}(g)\). The molecule \(\mathrm{PCl}_{5}\) has more atoms than \(\mathrm{PCl}_{3}\), indicating a higher level of molecular complexity. This equates to a greater number of potential microstates and, as a result, a higher standard molar entropy. This relationship between molecular structure and entropy is crucial for students to grasp, as it lays the foundation for understanding thermodynamic behavior in chemical reactions.

States of Matter and Entropy

Entropy is also affected by the state of matter, whether it be solid, liquid, or gas. A solid, with its tightly packed and ordered structure, has the least number of ways its particles can be arranged, resulting in a lower entropy. Conversely, a liquid has more randomness as its particles are not held in a rigid structure and can move about more freely, meaning it has a higher entropy than a solid.

For example, the entropy of bromine in its liquid state \(\operatorname{Br}_{2}(l)\) is lower than that of bromine in its gaseous form \(\mathrm{Br}_{2}(g)\). This is a direct consequence of the increased freedom of movement and greater number of orientations the molecules can have in the gas phase. Similarly, the entropy of a substance increases when it is dissolved to form a solution, such as with sodium nitrate \(\mathrm{NaNO}_{3}(s)\) compared to \(\mathrm{NaNO}_{3}(aq)\). The ions in the solution have more freedom to move and are less ordered than in the solid lattice.

Overall, the greater freedom for particles to move and interact in liquids and gases, as opposed to the rigidity of a solid, significantly contributes to the higher entropy seen in these states of matter.

Comparing Entropy of Substances

In order to compare the standard molar entropy of different substances, we must consider both their molecular complexity and their state of matter. This approach is essential, as it allows students to predict the relative entropy values between different substances under standard conditions.

For example, when comparing butane \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{CH}_{3}(g)\) and sulfur dioxide \(\mathrm{SO}_{2}(g)\), the larger size and greater complexity of butane's molecular structure suggest a higher entropy. Moreover, the comparison between bromine gas \(\mathrm{Br}_{2}(g)\) and fluorine gas \(\mathrm{F}_{2}(g)\) takes into account the weaker intermolecular forces in fluorine due to its smaller atoms, which translates into a higher standard molar entropy for fluorine gas, despite both substances being in the same state of matter.

Understanding these concepts thoroughly allows students to make informed predictions about the relative entropies of various substances. A comprehensive grasp of these fundamental ideas forms a cornerstone for further study in thermodynamics and physical chemistry.