Question

The London dispersion forces in the three different isomers of \(\mathrm{C}_{5} \mathrm{H}_{12}\) lead to different boiling points for pentane, 2-methylbutanc, and 2,2 -dimethylbutanc \((309 \mathrm{K}, 301 \mathrm{K}\) and \(282 \mathrm{K}\), respectively, Figure 6.3 ). a. What are the root-mean-square speeds of the three isomers of pentane at \(325 \mathrm{K} ?\) b. Using the KMT model, describe why calculated root-mean-square speeds are not influenced by the inter-molecular forces that the molecules of these isomers apparently experience.

Short answer

To summarize: a. The root-mean-square speeds for the three isomers of C5H12 at 325 K are approximately 205.85 m/s. b. According to the Kinetic Molecular Theory model, there are no inter-molecular forces between the gas particles. As a result, the calculated root-mean-square speeds do not take these forces into account, and they are not influenced by the inter-molecular forces experienced by the molecules of these isomers.

Step-by-step solution

Calculate the molar mass of C5H12 isomers

The three isomers of C5H12 have the same molecular formula and thus have the same molar mass. To calculate the molar mass of C5H12, we will add the individual molar masses of Carbon (C) and Hydrogen (H) atoms in the compound. Molar mass of C5H12 = 5 * Molar mass of C + 12 * Molar mass of H = 5 * 12.01 g/mol + 12 * 1.01 g/mol = 72.15 g/mol

Calculate the root-mean-square speeds for the isomers

With the molar mass of C5H12 (72.15 g/mol), we can calculate the root-mean-square speeds for the isomers using the following formula: \(v_{rms} = \sqrt{\frac{3RT}{M}}\) where: - \(v_{rms}\) is the root-mean-square speed - R is the ideal gas constant (8.314 J/mol.K) - T is the temperature (325 K) - M is the molar mass of the substance in kg/mol (0.07215 kg/mol, converting the molar mass of C5H12 from g/mol to kg/mol) Plug in the values: \(v_{rms} = \sqrt{\frac{3 \times 8.314 \text{ J/mol.K} \times 325 \text{ K}}{0.07215 \text{ kg/mol}}}\) \(v_{rms} \approx 205.85 \text{ m/s}\) So, the root-mean-square speeds for the three isomers of pentane at 325 K are approximately 205.85 m/s.

Explain the KMT model and its assumptions

The Kinetic Molecular Theory (KMT) model is a microscopic model describing the behavior of gases. It makes the following assumptions: 1. Gases consist of a large number of small particles (atoms or molecules) that are in constant random motion. 2. The volume occupied by the individual particles is negligible compared to the volume of the gas. 3. The particles' collisions with each other and the container walls are perfectly elastic (no energy is lost in the collisions). 4. There are no inter-molecular forces between the particles. 5. The average kinetic energy of the gas particles is directly proportional to the temperature in Kelvin. Since the KMT model assumes no inter-molecular forces between particles, the calculated root-mean-square speeds do not take them into account. This explains why the root-mean-square speeds are not influenced by the inter-molecular forces experienced by the molecules of these isomers, such as London dispersion forces.

Key concepts

Isomers of C5H12

Isomers are molecules that have the same molecular formula but differ in the arrangement of their atoms. For the hydrocarbon \(\text{C}_5\text{H}_{12}\), these different structures are what we call isomers.
In the case of \(\text{C}_5\text{H}_{12}\), there are three main isomers:

• Pentane - a straight chain of five carbon atoms.
• 2-Methylbutane - a four-carbon chain with a methyl group attached to the second carbon.
• 2,2-Dimethylpropane - a three-carbon chain with two methyl groups attached to the middle carbon.

The diversity in structure affects their physical properties like boiling points. Despite having the same number of atoms, the geometry and branching play critical roles. For instance, more compact (branched) isomers tend to have lower boiling points because they have smaller surface areas compared to linear isomers. This leads to weaker London dispersion forces.

Root-Mean-Square Speed

The root-mean-square speed (\(v_{\text{rms}}\)) is an average speed which provides a meaningful measure of the speed of gas molecules. In essence, it is the square root of the mean of the squares of the speeds of all molecules in the gas.
It's calculated with the formula: \[v_{\text{rms}} = \sqrt{\frac{3RT}{M}}\]

• \(R\) is the universal gas constant, valued at approximately 8.314 J/mol·K.
• \(T\) is the absolute temperature measured in Kelvin.
• \(M\) is the molar mass of the gas, in kg/mol.

For the isomers of \(\text{C}_5\text{H}_{12}\), given the same temperature and molecular mass, they will have the same \(v_{\text{rms}}\). Though isomers have different structural configurations, \(v_{\text{rms}}\) depends solely on mass and temperature and remains unaffected by molecular geometry.

Kinetic Molecular Theory (KMT)

The Kinetic Molecular Theory (KMT) provides insights into the behavior of gases by depicting them as a large number of tiny particles in constant motion. It supports the understanding of gas properties and behaviors as explained here:

• Gas molecules move randomly and rapidly.
• The volume of the molecules themselves is negligibly small compared to the gas volume.
• Collisions among molecules, and between molecules and container walls, are elastic, conserving total energy.
• No forces of attraction or repulsion exist between gas molecules.
• The average kinetic energy is directly linked to temperature.

In calculating \(v_{\text{rms}}\), KMT's assumption that no intermolecular forces act means the result is purely based on kinetic aspects. Therefore, despite the presence of London dispersion forces in these isomers, they do not affect their calculated speeds.

Boiling Points of Isomers

When considering the boiling points of isomers of a molecule like \(\text{C}_5\text{H}_{12}\), London dispersion forces play a significant role. These temporary attractive forces occur due to momentary shifts in electron density within molecules.
London dispersion forces get stronger with:

• Larger surface areas, as more area allows more interactions among molecules.
• Less branching, as linear molecules can pack together more effectively, providing more contact for these forces.

Therefore, the boiling points vary among the isomers:

• Pentane, being linear, experiences stronger dispersion forces than its more compact counterparts, leading to the highest boiling point.
• 2-Methylbutane has more branching, resulting in a lower boiling point than pentane.
• 2,2-Dimethylpropane, being the most branched, has the lowest boiling point.

Understanding these differences is key to understanding the relationship between molecular structure and physical properties.