Problem 47 - Chemistry The Central Science

Question

In each of the following pairs, which compound would you expect to have the higher standard molar entropy: (a) \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) or \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) (b) \(\mathrm{CO}_{2}(g)\) or \(\mathrm{CO}(g) ?\) Explain.

Step-by-Step Solution

Answer

In pair (a), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher standard molar entropy due to its greater number of atoms and possible arrangements in space. In pair (b), \(\mathrm{CO}_{2}(g)\) has a higher standard molar entropy for the same reason.

Step 1: Identify factors that affect molar entropy

For each pair of compounds, we will consider factors such as the complexity of the molecules (e.g., size, number of atoms) and the phase (solid, liquid, or gas) to determine which compound has a higher molar entropy.

Step 2: Compare the complexity of molecules in pair (a)

In pair (a), we are comparing \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) and \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\). Both compounds are in the gas phase, so we don't need to consider the phase. Let's compare the complexity of these molecules. \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) contains 2 carbon atoms and 2 hydrogen atoms, for a total of 4 atoms. \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) contains 2 carbon atoms and 6 hydrogen atoms, for a total of 8 atoms. Since \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has more atoms, it has more possible arrangements in space, which means it has a higher entropy.

Step 3: Determine the compound with higher molar entropy in pair (a)

In pair (a), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher molar entropy because it has more atoms and therefore more possible arrangements in space. So, the answer for pair (a) is \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\).

Step 4: Compare the complexity of molecules in pair (b)

In pair (b), we are comparing \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\). Both compounds are in the gas phase, so we don't need to consider the phase. Let's compare the complexity of these molecules. \(\mathrm{CO}_{2}(g)\) contains 1 carbon atom and 2 oxygen atoms, for a total of 3 atoms. \(\mathrm{CO}(g)\) contains 1 carbon atom and 1 oxygen atom, for a total of 2 atoms. Since \(\mathrm{CO}_2(g)\) has more atoms, it has more possible arrangements in space, which means it has a higher entropy.

Step 5: Determine the compound with higher molar entropy in pair (b)

In pair (b), \(\mathrm{CO}_{2}(g)\) has a higher molar entropy because it has more atoms and therefore more possible arrangements in space. So, the answer for pair (b) is \(\mathrm{CO}_{2}(g)\). In summary, - In pair (a), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher standard molar entropy. - In pair (b), \(\mathrm{CO}_{2}(g)\) has a higher standard molar entropy.

Key Concepts

Molecular Complexity

Molecular complexity plays a vital role in determining the molar entropy of a substance. Entropy, in simple terms, measures the disorder or randomness in a system. The more complex a molecule, the more ways its atoms can be arranged. For instance, consider the molecules

• \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\): consisting of 2 carbon and 2 hydrogen atoms.
• \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\): consisting of 2 carbon and 6 hydrogen atoms.

Since \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has more atoms, it has greater molecular complexity
and hence more possible configurations compared to \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\). This leads to higher molar entropy.

The same principle applies to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\). With 3 atoms, \(\mathrm{CO}_{2}(g)\) has more ways to organize itself than \(\mathrm{CO}(g)\), which has only 2 atoms.
Hence, \(\mathrm{CO}_{2}(g)\) has higher molar entropy.

Entropy Comparisons

When comparing the entropy of different substances, several factors come into play. One key factor is the number and type
of atoms in a molecule. More atoms typically mean more potential arrangements, increasing the entropy. More complex molecules
offer more vibrational and rotational modes, contributing to higher entropy.

For example: comparing \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) with \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\), the latter has more hydrogen atoms, allowing more flexibility
and possible movements within the molecule. This increases its entropy. Similarly, \(\mathrm{CO}_{2}(g)\) has one more oxygen atom
than \(\mathrm{CO}(g)\), enabling more arrangements and thus a higher entropy.

By understanding these factors, you can predict which substances will have higher entropies.

Gas Phase Properties

In the gas phase, particles are more spread out and move freely, contributing to higher entropy compared to liquids and solids.
This freedom allows for countless possible arrangements and movements of atoms within a molecule.

All the molecules in our example (\(\mathrm{C}_{2} \mathrm{H}_{2}(g)\), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\), \(\mathrm{CO}_{2}(g)\), and \(\mathrm{CO}(g)\))
are gases. Therefore, we only need to focus on their molecular complexity when comparing their entropies, as phase influences
are constant. Being in the gas phase naturally assigns them a high baseline entropy.

It's important to remember that the gas phase inherently provides a higher level of entropy due to increased molecular freedom.