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.

Short answer

In conclusion, \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher standard molar entropy than \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) due to its higher molecular complexity, and \(\mathrm{CO}_{2}(g)\) has a higher standard molar entropy than \(\mathrm{CO}(g)\) due to its greater number of atoms and bonds.

Step-by-step solution

Understanding standard molar entropy

Standard molar entropy is a measure of the randomness or disorder of a substance in its standard state (1 atm pressure and 25 degrees Celsius). It is represented by the symbol \(S^{\ominus}\). The higher the standard molar entropy of a substance, the more randomness or disorder it has. There are several factors affecting the standard molar entropy of a compound: 1. Molecular complexity: More complex molecules with more atoms tend to have higher standard molar entropy as there are more ways in which the atoms can arrange themselves. 2. Phase: Gases generally have higher standard molar entropy than liquids and solids, as the particles in a gas are more randomly distributed. 3. Molecular size: Larger molecules with more atoms and bonds tend to have a greater standard molar entropy. Now, let's compare the compounds in both pairs and determine which one has a higher standard molar entropy.

Comparing the standard molar entropy of \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) and \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\)

In the given pair, both compounds are in the gaseous phase but their molecular complexity is different. \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) has four atoms while \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has eight atoms. Since \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) is more complex with more atoms, it has more ways in which the atoms can arrange themselves, and thus, higher standard molar entropy. So, out of \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) and \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\), \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has the higher standard molar entropy.

Comparing the standard molar entropy of \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\)

In this pair, both compounds are also in the gaseous phase, but their molecular complexity is different. \(\mathrm{CO}_{2}(g)\) has three atoms, and \(\mathrm{CO}(g)\) has two atoms. Since \(\mathrm{CO}_{2}(g)\) has more atoms and bonds, it has a greater standard molar entropy than \(\mathrm{CO}(g)\). So, out of \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\), \(\mathrm{CO}_{2}(g)\) has the higher standard molar entropy. In conclusion, \(\mathrm{C}_{2} \mathrm{H}_{6}(g)\) has a higher standard molar entropy than \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\), and \(\mathrm{CO}_{2}(g)\) has a higher standard molar entropy than \(\mathrm{CO}(g)\).

Key concepts

Molecular Complexity and Entropy

In chemistry, molecular complexity refers to the number and types of atoms in a molecule, as well as how these atoms are arranged and bonded together. More complex molecules, which have a greater number and variety of atoms and bonds, generally possess higher standard molar entropy. This is because there are more possible ways for the energy within the molecule to be distributed. Each atom and bond can vibrate, rotate, and move in different ways, which leads to greater randomness or disorder.

Consider the example of ethylene (\(\mathrm{C}_{2} \mathrm{H}_{4}(g)\)) versus ethane (\(\mathrm{C}_{2} \mathrm{H}_{6}(g)\)). Both are gaseous hydrocarbons, but ethane is more complex because it has more atoms. These additional atoms create more vibrational modes and increase the molecule's rotational freedom, adding to the total number of microstates (specific ways of arranging energy) the molecule can occupy.

Therefore, ethane with six hydrogen atoms has higher entropy compared to ethylene, which simplifies the understanding of how molecular complexity affects molecular entropy.

Phase Effects on Entropy

The phase of a substance – whether it is solid, liquid, or gas – can significantly impact its standard molar entropy. In general, gases have higher entropy than liquids, and liquids have higher entropy than solids. This trend is due to the differences in particle arrangements and movement freedom across different phases.

In a gas, particles are far apart and move freely, leading to a high degree of randomness and disorder. This results in higher entropy. In comparison, liquids have particles that are closer together but can still move past each other, offering moderate entropy. Solids, however, have particles packed tightly in fixed positions, allowing minimal movement, thus exhibiting the least entropy.

This is why even if two substances have the same number of atoms and molecular complexity, the gaseous one will typically have a greater entropy. It's a general rule of thumb that highlights the typical segment of phase effects on entropy.

Molecular Size and Entropy

Molecular size, referring to the number of atoms and the mass of the molecule, is another factor that influences the standard molar entropy of substances. Larger molecules generally have higher entropy than smaller ones due to several factors.

Firstly, larger molecules have more atoms and potentially more types of bonds, leading to a greater number of vibrational and rotational modes. This increases the number of possible microstates the molecule can achieve, thus increasing its entropy.

For instance, comparing carbon monoxide (\(\mathrm{CO}(g)\)) with carbon dioxide (\(\mathrm{CO}_2(g)\)), the latter has a higher molecular size due to the additional oxygen atom. This added complexity and mass in carbon dioxide increases its number of molecular motions compared to carbon monoxide and thus results in a higher entropy value.

So, when comparing molecules, considering their size along with complexity and phase helps provide a comprehensive understanding of their entrpotic properties.