Question

For each of the following pairs, predict which substance possesses the larger entropy per mole: (a) 1 1 mol of \(\mathrm{O}_{2}(g)\) at \(300^{\circ} \mathrm{C}, 0.01\) atm, or 1 \(\mathrm{mol}\) of \(\mathrm{O}_{3}(g)\) at \(300^{\circ} \mathrm{C}, 0.01\) 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 \(\mathrm{H}_{2} \mathrm{O}(l)\) at \(100^{\circ} \mathrm{C}, 1\) atm; \((\mathbf{c}) 0.5 \mathrm{mol}\) of \(\mathrm{N}_{2}(g)\) at \(298 \mathrm{K}, 20 \mathrm{-L}\) volume, or 0.5 \(\mathrm{mol} \mathrm{CH}_{4}(g)\) at \(298 \mathrm{K}, 20-\mathrm{volume} ;(\mathbf{d}) 100 \mathrm{g} \mathrm{Na}_{2} \mathrm{SO}_{4}(s)\) at \(30^{\circ} \mathrm{C}\) or 100 \(\mathrm{g} \mathrm{Na}_{2} \mathrm{SO}_{4}(a q)\) at \(30^{\circ} \mathrm{C} .\)

Short answer

Pair (a): O3(g) has a larger entropy per mole compared to O2(g). Pair (b): H2O(g) has a larger entropy per mole compared to H2O(l). Pair (c): CH4(g) has a larger entropy per mole compared to N2(g). Pair (d): Na2SO4(aq) has a larger entropy compared to Na2SO4(s).

Step-by-step solution

Pair (a) - Comparing O2(g) and O3(g)

For both substances, the temperatures and pressure are the same, and both are gases. The difference between these two substances is in their molecular complexity. O3 has a more complex structure, which would increase its entropy. Therefore, O3(g) has a larger entropy per mole compared to O2(g).

Pair (b) - Comparing H2O(g) and H2O(l)

For these two substances, the temperature and pressure are the same, and both are made of the same molecules. The primary difference between them is their phase – one is a gas, and the other is a liquid. Typically, gases have higher entropy than liquids because their molecules have more freedom of motion. Therefore, the 1 mol of H2O(g) has a larger entropy per mole compared to 1 mol of H2O(l) at the same conditions.

Pair (c) - Comparing N2(g) and CH4(g)

For both substances, the temperature and volume are the same, and both are gases. They also have the same number of moles (0.5 mol). The difference between these substances is found in their molecular complexity. CH4 has a more complex structure and contains more atoms than N2, which would lead to higher entropy. Therefore, CH4(g) has a larger entropy per mole compared to N2(g).

Pair (d) - Comparing Na2SO4(s) and Na2SO4(aq)

For these two substances, the temperature and the amount (100 g) are the same, but one is a solid, and the other is an aqueous solution. The entropy of ions in a solution tends to be higher than the entropy of a solid. In the solid state, the ions are ordered in a crystal lattice, while in the solution state, the ions are dispersed and have more freedom of motion. Therefore, 100 g of Na2SO4(aq) has a larger entropy compared to 100 g of Na2SO4(s) at the same conditions.

Key concepts

Molecular Complexity and Entropy

Molecular complexity is a term that refers to the number of atoms in a molecule and the intricacy of its structure. In terms of entropy, a measure of disorder or randomness in a system, molecular complexity plays a key role.

When comparing molecules, those with more complex structures often have higher entropy. This is because such molecules can exhibit a larger number of arrangements and motions at any given temperature and pressure, leading to a state of greater disorder. For example, ozone \(\mathrm{O}_{3}(g)\) has a larger entropy per mole than oxygen \(\mathrm{O}_{2}(g)\) due to its more complex molecular structure, despite both being gases under the same conditions.

Phase Changes and Their Effect on Entropy

Phase changes are transformations that occur when a substance shifts from one state of matter to another—such as from a solid to a liquid, or a liquid to a gas. These changes significantly impact entropy.

For instance, when water vapor condenses into liquid water, its entropy decreases because the gas molecules are confined to a more ordered liquid state. Conversely, when water boils, its entropy increases as it transitions from liquid to the less ordered gas phase. Thus, comparing water in its gaseous form (vapor) to its liquid form at the same temperature and pressure, \(\mathrm{H}_{2} \mathrm{O}(g)\) would possess higher entropy than \(\mathrm{H}_{2}\mathrm{O}(l)\) because gases inherently have more freedom of movement, translating to higher disorder.

Entropy and Temperature

Entropy is not only influenced by molecular complexity and phase but also by temperature. Generally speaking, as temperature increases, so does entropy. Higher temperatures provide energy to particles, increasing their movement and the number of possible arrangements, leading to greater disorder.

In any system, if you raise the temperature while keeping other factors constant, the entropy will rise. This means at higher temperatures, substances have a greater potential for randomness and chaos in particle arrangements and motions.

Entropy in Gases vs Liquids vs Solids

Entropy varies greatly among the three primary phases of matter: gases, liquids, and solids. Gases have the highest entropy because their particles can move freely and occupy a larger volume, which corresponds to a high number of microstates and therefore a high level of disorder.

Liquids have lower entropy than gases as their particles are more restricted in movement but still have some freedom to move past one another. Solids possess the lowest entropy; their particles are fixed in place and can only vibrate about their fixed positions, leading to the fewest microstates and the highest level of order among the three phases.

For example, comparing \(\mathrm{Na}_{2}\mathrm{SO}_{4}(s)\), a solid, to \(\mathrm{Na}_{2}\mathrm{SO}_{4}(aq)\), its aqueous solution form, the latter will have higher entropy because the ions in solution are free to move around more than the ions fixed in the crystal lattice of the solid form.