Problem 93 - Chemistry: The Central Science in SI Units, Global Edition

Question

Trouton's rule states that for many liquids at their normal boiling points, the standard molar entropy of vaporization is about \(88 \mathrm{~J} / \mathrm{mol}-\mathrm{K} .(\) a) Estimate the normal boiling point of bromine, \(\mathrm{Br}_{2}\), by determining \(\Delta H_{\text {vap }}^{\circ}\) for \(\mathrm{Br}_{2}\) using data from Appendix \(C\). Assume that \(\Delta H_{\text {vap }}^{\circ}\) remains constant with temperature and that Trouton's rule holds. (b) Look up the normal boiling point of \(\mathrm{Br}_{2}\) in a chemistry handbook or at the WebElements website (www.webelements.com) and compare it to your calculation. What are the possible sources of error, or incorrect assumptions, in the calculation?

Step-by-Step Solution

Answer

Using Trouton's rule and the given data, we estimated the normal boiling point of Br₂ to be around 340.9 K. The actual value is 332 K. Errors in the calculation could arise from Trouton's rule being an approximation, the assumption that ∆Hₐₚ° remains constant with temperature, and using rounded values for ∆Hₐₚ° and ∆Sₐₚₙ. Despite these possible sources of errors, the estimated boiling point is still reasonably close to the actual value.

Step 1: Find ∆Hₐₚ° for Br₂

First, we need to find the standard molar enthalpy of vaporization (∆Hₐₚ°) for Br₂. You can find this value in Appendix C of your textbook or other chemistry references. The value for ∆Hₐₚ° for Br₂ is approximately 30 kJ/mol.

Step 2: Calculate the standard molar entropy of vaporization for Br₂

According to Trouton's rule, the standard molar entropy of vaporization for many liquids at their normal boiling points is about 88 J/mol·K. Therefore, for Br₂, we can assume: \[\Delta S_{\text{vap}}^{\circ} = 88\, \mathrm{J/mol} \cdot \mathrm{K}\]

Step 3: Estimate the normal boiling point of Br₂ using Trouton's rule

To estimate the normal boiling point of Br₂, we can use the following relation between ∆Hₐₚ° and ∆Sₐₚ: \[\Delta H_{\text{vap}}^{\circ} = T_{\text{bp}} \cdot \Delta S_{\text{vap}}^{\circ}\] Now, we can rearrange the equation to solve for the normal boiling point (Tₒ) of Br₂: \[T_{\text{bp}} = \frac{\Delta H_{\text{vap}}^{\circ}}{\Delta S_{\text{vap}}^{\circ}}\] Plug in the values for ∆Hₐₚ° and ∆ Sₐₚ: \[T_{\text{bp}} = \frac{30,000\, \mathrm{J/mol}}{88\, \mathrm{J/mol} \cdot \mathrm{K}}\] \[T_{\text{bp}} \approx 340.9\, \mathrm{K}\] So, using Trouton's rule, the estimated normal boiling point of Br₂ is around 340.9 K.

Step 4: Compare with the actual value and discuss possible sources of errors

Now, we need to compare our calculated value with the actual value for the normal boiling point of Br₂, which can be found in a chemistry handbook or on the WebElements website. The actual value is 332 K. There is a difference between our estimated value and the actual value. Possible sources of error and incorrect assumptions include: 1. Trouton's rule is an approximation, and it might not hold true for all substances, including halogens such as bromine. 2. We assumed that ∆Hₐₚ° remains constant with temperature, but it can vary for some substances. 3. Errors could also be introduced by using rounded values for ∆Hₐₚ° and ∆Sₐₚₙ. Despite these possible sources of errors, the estimated boiling point is still reasonably close to the actual value, demonstrating the usefulness of Trouton's rule as an estimation tool.

Key Concepts

Entropy of Vaporization

Entropy is a measure of disorder or randomness in a system, and the entropy of vaporization represents the change in disorder when a liquid becomes a gas. At the boiling point, molecules gain enough energy to overcome intermolecular forces and transition from a liquid to a vapor.

When a substance vaporizes, its entropy increases significantly because gas molecules have more freedom to move compared to their liquid state. According to Trouton's Rule, the standard molar entropy of vaporization (\( \Delta S_{\text{vap}}^{\circ} \)) for many liquids is approximately 88 J/mol·K.

This rule suggests a constant value due to the similarly random distribution of gas molecules at boiling points, across many different liquids. However, this estimation may vary slightly based on the specific characteristics and molecular interactions of a given substance.

Enthalpy of Vaporization

Enthalpy of vaporization (\( \Delta H_{\text{vap}}^{\circ} \)) is the amount of energy required to convert one mole of a liquid into a gas at its boiling point, under standard conditions. It reflects the strength of intermolecular forces in the liquid.

Substances with stronger intermolecular forces require more energy to vaporize, resulting in higher enthalpy of vaporization values. For bromine, the standard molar enthalpy of vaporization was found to be approximately 30 kJ/mol.

• The enthalpy of vaporization can provide insights on the amount of heat needed for phase changes.
• Knowing this value is crucial for estimating boiling points using Trouton's Rule.

Though this energy requirement can change slightly with temperature, for practical estimations, it is often assumed constant over small temperature ranges like those involved in calculating boiling points.

Boiling Point Estimation

Estimating the boiling point of a substance can be approached using the relationship between enthalpy of vaporization and entropy of vaporization. According to Trouton's Rule, this estimation involves dividing the enthalpy of vaporization by the entropy of vaporization:

\[ T_{\text{bp}} = \frac{\Delta H_{\text{vap}}^{\circ}}{\Delta S_{\text{vap}}^{\circ}} \]

In the case of bromine (\( \text{Br}_2 \)), plugging in the values:

• \( \Delta H_{\text{vap}}^{\circ} = 30,000 \, \text{J/mol} \)
• \( \Delta S_{\text{vap}}^{\circ} = 88 \, \text{J/mol} \cdot \text{K} \)

provides an estimated boiling point of approximately 340.9 K.

This predicted temperature is slightly higher than the actual boiling point of 332 K for bromine, indicating possible deviations due to simplifications in the calculation, such as:

• Natural variance in Trouton's Rule applicability across different substances.
• Assuming enthalpy remains constant with temperature, which might not hold true over larger ranges.
• Use of average or rounded data values in calculation affecting precision.

Despite these potential errors, this method provides a useful estimation, especially when detailed data or alternative methods are not available.