Question

Which substance in each of the following pairs would you expect to have the higher standard molar entropy? Explain. (a) \(\mathrm{NO}(g)\) or \(\mathrm{NO}_{2}(g)\) (b) \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}(l)\) or \(\mathrm{HCO}_{2} \mathrm{H}(l)\) (c) \(\mathrm{Br}_{2}(l)\) or \(\mathrm{Br}_{2}(s)\) (d) \(\mathrm{S}(s)\) or \(\mathrm{SO}_{3}(g)\)

Short answer

(a) NO₂(g); (b) CH₃CO₂H(l); (c) Br₂(l); (d) SO₃(g) have higher entropies.

Step-by-step solution

Analyze Pair (a) - NO(g) vs NO₂(g)

For gases, entropy generally increases with molecular complexity and molar mass. Here, NO₂(g) has more atoms than NO(g). Because it is a more complex molecule, it also has more ways to store energy (degrees of freedom), resulting in a higher molar entropy.

Analyze Pair (b) - CH₃CO₂H(l) vs HCO₂H(l)

Both compounds are liquids, but CH₃CO₂H (acetic acid) is a larger molecule than HCO₂H (formic acid). Larger molecules tend to have higher entropy because they have more atoms and more vibrational modes, which can increase the overall entropy of the substance.

Analyze Pair (c) - Br₂(l) vs Br₂(s)

Entropy of a substance is higher in its liquid state than in the solid state, due to increased molecular motion and disorder in the liquid phase. Thus, Br₂(l), being in liquid form, will have a higher molar entropy than Br₂(s).

Analyze Pair (d) - S(s) vs SO₃(g)

According to the states of matter and complexity of molecules, SO₃(g) will have a higher molar entropy than S(s). SO₃ is a gas and inherently has more entropy than a solid due to increased molecular motion and larger number of microstates it can access.

Key concepts

Molecular Complexity

Molecular complexity is a key factor influencing the standard molar entropy of a substance. A more complex molecule, generally one with more atoms, will have a higher entropy. This is because it has more ways to store energy—through rotation, vibration, and other movements.

• More atoms mean more possible vibrational modes, increasing the complexity of the molecule.
• Larger molecules can have intricate shapes, providing more arrangements or microstates for the distribution of energy.

When you compare substances like \( \text{NO(g)} \) and \( \text{NO}_2(g) \), the latter, \( \text{NO}_2(g) \), is more complex due to the additional oxygen atom. This increase in complexity directly contributes to a higher standard molar entropy.

Degrees of Freedom

Degrees of freedom in a molecule refer to the independent ways a molecule can move in space, which include translational, rotational, and vibrational movements. A molecule with more degrees of freedom tends to have a higher entropy because it can distribute energy over more movements.

• Translational freedom: involves the movement of the entire molecule in space.
• Rotational freedom: involves spinning around various axes.
• Vibrational freedom: involves periodic bond stretches and angle bends.

For instance, a larger molecule like \( \text{CH}_3\text{CO}_2\text{H}(l) \) has more vibrational degrees of freedom compared to \( \text{HCO}_2\text{H}(l) \), contributing to its higher entropy.

States of Matter

The state of matter significantly affects the standard molar entropy. The order of entropy generally increases from solid to liquid to gas because gases have the highest molecular motion and disorder.

• Solids: Molecules are tightly packed, leading to limited motion and lower entropy.
• Liquids: More freedom of movement compared to solids, with higher entropy due to increased disorder.
• Gases: Molecules are widely spaced and move freely, resulting in high entropy values.

Comparing \( \text{Br}_2(l) \) and \( \text{Br}_2(s) \), the liquid form, \( \text{Br}_2(l) \), has a higher entropy due to the increased molecular movement.

Molar Mass

Molar mass refers to the mass of one mole of a substance. While it is typically secondary to molecular complexity and degrees of freedom, a higher molar mass can influence entropy. Generally, heavier molecules have more atoms and thus more ways energy can be stored and distributed.Molecular weight can impact how a molecule vibrates and rotates, slightly enhancing its entropy. For gases, as in the comparison between sulfur \( \text{S}(s) \) and sulfur trioxide \( \text{SO}_3(g) \), the latter not only benefits from being in the gaseous state but also from the greater mass and complexity, leading to higher standard molar entropy.