Question

For each of the following pairs, indicate which substance possesses the larger standard entropy: (a) \(1 \mathrm{~mol}\) of \(\mathrm{P}_{4}(\mathrm{~g})\) at \(300{ }^{\circ} \mathrm{C}, 0.01 \mathrm{~atm}\), or \(1 \mathrm{~mol}\) of \(\mathrm{As}_{4}(\mathrm{~g})\) at \(300{ }^{\circ} \mathrm{C}, 0.01 \mathrm{~atm}\); (b) \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(g)\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm}\), or \(1 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{l})\) at \(100^{\circ} \mathrm{C}, 1 \mathrm{~atm} ;\) (c) \(0.5 \mathrm{~mol}\) of \(\mathrm{N}_{2}(g)\) at \(298 \mathrm{~K}, 20\) - \(\mathrm{L}\) volume, or \(0.5 \mathrm{~mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{~K}, 20-\mathrm{L}\) volume; (d) \(100 \mathrm{~g}\) \(\mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or \(100 \mathrm{~g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C} .\)

Short answer

In summary, for each pair of substances, the one with the larger standard entropy is: (a) \(\mathrm{As}_{4}(\mathrm{~g})\), (b) \(\mathrm{H}_{2} \mathrm{O}(g)\), (c) \(\mathrm{CH}_4(g)\), and (d) \(\mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\).

Step-by-step solution

(Step 1: Compare substances in pair (a))

We will compare the standard entropies of \(\mathrm{P}_{4}(\mathrm{~g})\) and \(\mathrm{As}_{4}(\mathrm{~g})\). Both substances are in gaseous state and have the same number of moles and similar conditions. However, the atomic weight of arsenic (As) is larger than that of phosphorus (P). Since heavier molecules generally have more atoms and vibrational modes than lighter molecules, the substance with the heavier molecules will have a higher standard entropy. Thus, \(\mathrm{As}_{4}(\mathrm{~g})\) has a larger standard entropy than \(\mathrm{P}_{4}(\mathrm{~g})\).

(Step 2: Compare substances in pair (b))

We will compare the standard entropies of \(\mathrm{H}_{2} \mathrm{O}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\). Both substances have the same molecular structure. However, they differ in their state of matter (gas vs. liquid). In general, gas particles move more freely and randomly compared to particles in a liquid. Therefore, the molecular disorder and entropy are higher in a gas than in a liquid. Based on this, \(\mathrm{H}_{2} \mathrm{O}(g)\) has a larger standard entropy than \(\mathrm{H}_{2} \mathrm{O}(l)\).

(Step 3: Compare substances in pair (c))

The substances in question are \(\mathrm{N}_{2}(g)\) and \(\mathrm{CH}_{4}(g)\). Both substances are in gaseous state and have similar conditions. We need to compare their molecular structures to determine which one has a higher standard entropy. Nitrogen (N) molecules are diatomic, while methane (CH4) molecules are more complex, consisting of carbon and hydrogen atoms. As a result, methane molecules have more vibrational modes than nitrogen molecules. Therefore, \(\mathrm{CH}_4(g)\) has a larger standard entropy than \(\mathrm{N}_2(g)\).

(Step 4: Compare substances in pair (d))

The substances in question are \(\mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) and \(\mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\). They have the same molecular structure but differ in their state of matter (solid vs. aqueous). In an aqueous solution, the ions present separate and move more freely than they can in a solid. Thus, the molecular disorder and entropy are higher in the aqueous solution than in the solid. Based on these factors, \(\mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) has a larger standard entropy than \(\mathrm{Na}_{2} \mathrm{SO}_{4}(s)\).

Key concepts

Gaseous State

When discussing entropy, it's essential to understand the gaseous state. Gases have particles that move freely and occupy the entire volume of their container.
This freedom of movement contributes significantly to a higher degree of disorder, or entropy.
Unlike solids and liquids, gases are less structured, as their atoms or molecules are far apart with minimal interactions.

• Increased particle energy levels.
• Higher range of random motion.
• Less inter-particle forces compared to solids and liquids.

In scenarios where substances in different states of matter are compared, the gaseous form almost always has the greatest entropy. This is because gases have significantly more microstates compared to the more orderly arrangement of particles in liquids or solids. So, when comparing the entropy of water in its gaseous state (H₂O(g)) vs. liquid state (H₂O(l)), H₂O(g) will possess a higher standard entropy.

Molecular Structure

Molecular structure plays a critical role in determining a substance's standard entropy. This is because the way atoms are arranged within a molecule affects its vibrational and rotational capacities.
Simplest molecules such as diatomic gases can only vibrate back and forth, yet more complex molecules can have a variety of vibrational and rotational movements.

• Number of atoms in a molecule.
• The overall 3D configuration.
• Interatomic bonds and angle strength.

For example, in comparing nitrogen (N₂, a diatomic molecule) with methane (CH₄, a tetrahedral molecule), methane's complex 3D shape allows it more freedom in vibrational and rotational modes. This results in methane having a higher standard entropy than nitrogen, as it can explore more configurations and internal vibrations.

Vibrational Modes

Every molecule has specific vibrational modes that it can access based on its structure and atomic composition. These vibrational motions contribute significantly to a molecule's entropy.
In simpler terms, vibrational modes refer to the different ways the atoms within a molecule can oscillate.

• Motion involves stretching, bending, or twisting of bonds.
• More complex molecules have more vibrational modes, leading to higher entropy.

Consider two gases, like phosphorus gas (P₄) and arsenic gas (As₄). Due to arsenic's higher atomic mass, molecules of As₄ will have more available vibrational modes than those in P₄. This is due to each larger mass being able to store more energy in vibrational forms even with similar molecular structures. Therefore, As₄ will have a higher standard entropy.

Atomic Weight

Atomic weight, or atomic mass, significantly influences standard entropy. Heavier atoms mean larger molecules, often translating to an increased number of possible vibrational, translational, and rotational states.
When you compare two substances where the atomic composition and conditions are similar, the one with heavier atomic weight typically exhibits higher entropy.

• More energy levels accessible.
• Increased moments of inertia allowing more rotations and vibrations.

For example, in comparing phosphorus gas (P₄) with arsenic gas (As₄), arsenic's greater atomic weight leads to more vibrational modes, which results in higher disorder and thus more entropy. Similarly, when substances transition from solid or liquid to aqueous solutions, like Na₂SO₄ going from solid to aqueous, the ions have more freedom, enhancing entropy.