Problem 45 - Chemistry: The Central Science in SI Units, Global Edition

Question

In each of the following pairs, which compound would you expect to have the higher standard molar entropy: (a) \(\mathrm{C}_{3} \mathrm{H}_{\mathrm{s}}(g)\) or \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\), (b) \(\mathrm{C}_{4} \mathrm{H}_{10}(l)\) or \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\)

Step-by-Step Solution

Answer

In conclusion: - For pair (a): Butane, \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\), has a higher standard molar entropy due to its higher molecular complexity compared to propane, \(\mathrm{C}_{3} \mathrm{H}_{8}(g)\). - For pair (b): The gaseous phase of butane, \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\), has a higher standard molar entropy compared to the liquid phase, \(\mathrm{C}_{4} \mathrm{H}_{10}(l)\), due to its increased freedom of movement and spacing between particles.

Step 1: Identify the compounds

We have two compounds: \(\mathrm{C}_{3} \mathrm{H}_{8}(g)\), which is propane, and \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\), which is butane.

Step 2: Analyze the phase and molecular complexity

Both are in the gaseous phase, so there is no difference in entropy due to phase. Molecular complexity is higher in butane since it has one extra carbon atom and two extra hydrogen atoms compared to propane.

Step 3: Determine the higher entropy compound

Since both compounds are in the same phase, the only factor determining their entropy is the molecular complexity. In this case, butane (\(\mathrm{C}_{4} \mathrm{H}_{10}(g)\)) has a higher molecular complexity, resulting in a higher standard molar entropy. For pair (b):

Step 4: Identify the compounds and phases

We have the same compound (\(\mathrm{C}_{4} \mathrm{H}_{10}\)) in two different phases: liquid and gas.

Step 5: Analyze the influence of phase on entropy

A gaseous phase compound has more freedom of movement and spacing between particles, which leads to a higher entropy compared to a liquid phase compound.

Step 6: Determine the higher entropy compound

In this case, since the substance is the same in both cases, the difference in entropy is determined by the phase. The gaseous phase of butane (\(\mathrm{C}_{4} \mathrm{H}_{10}(g)\)) has a higher standard molar entropy compared to the liquid phase (\(\mathrm{C}_{4} \mathrm{H}_{10}(l)\)). In conclusion: - For pair (a): \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\) has a higher standard molar entropy; - For pair (b): \(\mathrm{C}_{4} \mathrm{H}_{10}(g)\) has a higher standard molar entropy.

Key Concepts

Molecular Complexity

Molecular complexity is a major factor in determining the standard molar entropy of a substance. Entropy measures the randomness or disorder within a system. The more complex a molecule is, the higher its entropy tends to be. This is because a complex molecule has more atoms, which means more ways to store energy and more possible arrangements.

In the case of **propane** \(\mathrm{C}_{3}\mathrm{H}_{8}(g)\) vs. **butane** \(\mathrm{C}_{4}\mathrm{H}_{10}(g)\):

• Propane consists of 3 carbon atoms and 8 hydrogen atoms.
• Butane has an additional carbon and two more hydrogen atoms, totaling 4 carbon atoms and 10 hydrogen atoms.

This additional complexity gives butane more ways to disperse energy, leading to higher entropy. Thus, butane will have a higher standard molar entropy than propane.

Gaseous Phase

The gaseous phase of a compound significantly affects its standard molar entropy. In gases, particles are not in a fixed position. Instead, they have considerable freedom of motion in three dimensions. This freedom increases the disorder significantly when compared to liquids or solids.

When considering butane in **gaseous** phase \(\mathrm{C}_{4}\mathrm{H}_{10}(g)\), we observe:

• Particles can move freely and are widely spaced.
• This causes more randomness and leads to higher entropy.

Because gas molecules are free to move around randomly, the entropy is naturally higher compared to the same substance in a liquid or solid state.

Liquid Phase

In the liquid phase, compounds experience less entropy compared to their gaseous counterparts. Liquids have particles that are still able to move about, but they are more restricted compared to gases. They slide past each other but remain closely packed due to the intermolecular forces keeping them together.

For butane in the **liquid** phase \(\mathrm{C}_{4}\mathrm{H}_{10}(l)\):

• There's less freedom of movement, leading to decreased randomness.
• Entropy in a liquid is lower as particles are more ordered than in a gas.

Therefore, in the same substance, a liquid will always have a lower standard molar entropy compared to its gaseous phase. In our example, \(\mathrm{C}_{4}\mathrm{H}_{10}(g)\) has a higher entropy than \(\mathrm{C}_{4}\mathrm{H}_{10}(l)\), due to phase differences alone.