Problem 26 - Chemistry: An Atoms-Focused Approach

Question

Rank the compounds in each of the following groups in order of increasing standard molar entropy \(\left(S^{\circ}\right):\) a. \(\mathrm{CH}_{4}(g), \mathrm{CH}_{3} \mathrm{CH}_{3}(g),\) and \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{3}(g)\) b. \(\mathrm{CCl}_{4}(\ell), \mathrm{CHCl}_{3}(\ell),\) and \(\mathrm{CH}_{2} \mathrm{Cl}_{2}(\ell)\) c. \(\mathrm{CO}_{2}(\ell), \mathrm{CO}_{2}(g),\) and \(\mathrm{CS}_{2}(g)\)

Step-by-Step Solution

Answer

Question: Rank the following compounds in increasing order of standard molar entropy: a. CH4(g), CH3CH3(g), CH3CH2CH3(g); b. CCl4(l), CHCl3(l), CH2Cl2(l); c. CO2(l), CO2(g), CS2(g). Answer: a. CH4(g) < CH3CH3(g) < CH3CH2CH3(g) b. CCl4(l) < CHCl3(l) < CH2Cl2(l) c. CO2(l) < CO2(g) < CS2(g)

Step 1: Evaluate the molecular complexity

Among the three compounds, CH4 has the smallest and simplest molecular structure, followed by CH3CH3 and CH3CH2CH3, respectively. As molecules become more complex, the number of possible microstates and, consequently, the entropy increases.

Step 2: Rank the compounds

Since all three molecules are in the gas phase, we only need to rank them based on their molecular complexity. The order of increasing standard molar entropy is as follows: \(\mathrm{CH}_{4}(g) < \mathrm{CH}_{3}\mathrm{CH}_{3}(g) < \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{3}(g)\) #b. CCl4(l), CHCl3(l), and CH2Cl2(l)#

Step 1: Evaluate the molecular complexity

Here, we have three molecules in the liquid phase. Among them, CCl4 has the simplest molecular structure, followed by CHCl3 and CH2Cl2. Although it might seem counterintuitive, the entropy increases with the number of hydrogen atoms.

Step 2: Rank the compounds

Since all three molecules are in the liquid phase, we just need to rank them based on their molecular complexity. The order of increasing standard molar entropy is as follows: \(\mathrm{CCl}_{4}(l) < \mathrm{CHCl}_{3}(l) < \mathrm{CH}_{2} \mathrm{Cl}_{2}(l)\) #c. CO2(l), CO2(g), and CS2(g)#

Step 1: Evaluate the molecular complexity and phases

For this group, we have two different factors that impact entropy: molecular complexity and phase of matter. CO2 has the same molecular structure in both liquid and gas phase, while CS2 has a similar structure but with a sulfur atom instead of oxygen atoms. Additionally, gases typically have higher entropy than liquids.

Step 2: Rank the compounds

Taking into account both molecular complexity and phase of matter, we can rank the compounds as follows: \(\mathrm{CO}_{2}(l) < \mathrm{CO}_{2}(g) < \mathrm{CS}_{2}(g)\)

Key Concepts

Molecular Complexity

Understanding molecular complexity is key to predicting why some substances have a higher standard molar entropy than others. Simply put, molecular complexity refers to the structure of a molecule, including how many atoms it contains and how these atoms are arranged. - **Simpler Molecule Examples:** A molecule like methane (\(\mathrm{CH}_{4}\) is simpler because it consists of fewer atoms. - **More Complex Molecule Examples:** Larger molecules, like ethane (\(\mathrm{CH}_{3}\mathrm{CH}_{3}\) or propane (\(\mathrm{CH}_{3}\mathrm{CH}_{2}\mathrm{CH}_{3}\) have more atoms, more complex structures, and consequently, more possible ways of arranging these atoms (more microstates). The more complex a molecule, the more microstates it can have. This means higher molecular complexity leads to higher entropy. That's because there are more possible arrangements or motions for molecules to explore in complex species, contributing to a higher disorder.

Gas Phase Entropy

Entropy largely depends on the phase of matter. Gases typically have the highest entropy among different states (solid, liquid, gas) because gas molecules have greater freedom of movement. - **Why Gases Have High Entropy:** Gas molecules are not confined to a specific volume and can move freely, translating into a higher number of microstates. These free molecules distribute themselves in space, allowing maximum disorder and randomness. - **Examples:** For instance, when comparing \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}_{2}(\ell)\), the gaseous form demonstrates much higher entropy because the molecules can occupy many more positions than in the liquid phase. This is why gases like \(\mathrm{CS}_{2}(g)\) will intuitively have a higher entropy compared to their liquid counterparts or simpler gas species with the same molecular structure.

Liquid Phase Entropy

Liquidity comes with less molecular freedom than gases, but more than solids. This intermediate phase impacts the standard molar entropy, where liquid molecules can move past each other but are still bonded somewhat closely. - **Differences with Gas Phase:** While gas phase molecules are entirely free to move without restriction, liquid molecules slide and tumble around each other, constrained by intermolecular forces but not stuck in a fixed position as in a solid. - **Impact on Entropy:** Liquid phase entropy sits between solid and gas phases. The molecules are free to a moderate degree, leading to more microstates than a solid, but far fewer than a gas. An interesting aspect of liquid phase entropy can be seen when examining \(\mathrm{CCl}_{4}(\ell)\), \(\mathrm{CHCl}_{3}(\ell),\) and \(\mathrm{CH}_{2}\mathrm{Cl}_{2}(\ell).\) These substances illustrate how even within liquids, differing molecular complexities can influence entropy, with the number of possible hydrogen bonding sites adding more microstates and thus increasing entropy as molecular complexity increases.