Problem 93 - Chemistry The Central Science In Si Units

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

The estimated boiling point of bromine is about 336 K, close to the actual 332 K. Errors may stem from assumption errors in Trouton's rule or specific \( \Delta H_{\text{vap}}^{\circ} \) values.

Step 1: Understanding Trouton's Rule

Trouton's rule suggests that the standard molar entropy of vaporization \( \Delta S_{\text{vap}}^{\circ} \) is about \( 88 \, \text{J/mol⋅K} \) for many substances. This implies a relationship between the enthalpy of vaporization \( \Delta H_{\text{vap}}^{\circ} \), the boiling point \( T_b \), and the entropy of vaporization such that \( \Delta H_{\text{vap}}^{\circ} = T_b \times \Delta S_{\text{vap}}^{\circ} \).

Step 2: Finding \( \Delta H_{\text{vap}}^{\circ} \) for \( \mathrm{Br}_2 \)

Using Appendix C, find the standard enthalpy of vaporization for bromine, \( \Delta H_{\text{vap}}^{\circ} \). Suppose it is given as \( 29.6 \, \text{kJ/mol} \) (this may need checking from the actual appendix). Convert this to \( \text{J/mol} \) since we need consistent units: \( 29.6 \, \text{kJ/mol} = 29600 \, \text{J/mol} \).

Step 3: Applying Trouton's Rule

According to Trouton's rule, \( \Delta H_{\text{vap}}^{\circ} = T_b \times 88 \, \text{J/mol⋅K} \). By rearranging, \( T_b = \frac{\Delta H_{\text{vap}}^{\circ}}{88} \). Substitute \( \Delta H_{\text{vap}}^{\circ} = 29600 \, \text{J/mol} \) to find \( T_b \).

Step 4: Calculating the Boiling Point

Calculate \( T_b = \frac{29600}{88} \approx 336.36 \, \text{K} \).

Step 5: Comparing with Actual Data

Look up the normal boiling point of \( \mathrm{Br}_2 \) from a reliable source. According to the data, it is approximately \( 59 \, ^\circ \text{C} \) or \( 332 \, \text{K} \). Compare this with the calculated result.

Step 6: Evaluating Assumptions and Errors

Consider that any deviation may be caused by errors in \( \Delta H_{\text{vap}}^{\circ} \), non-ideal behavior of bromine, or the assumption that \( \Delta H_{\text{vap}}^{\circ} \) remains constant. Trouton's rule is an approximation and might not completely hold for bromine.

Key Concepts

Standard Molar Entropy of Vaporization

The concept of standard molar entropy of vaporization, denoted as \( \Delta S_{\text{vap}}^{\circ} \), is a measure of the change in entropy when one mole of a liquid evaporates into a gas state at its boiling point under standard conditions. Trouton's Rule simplifies this by suggesting a common value of approximately \( 88 \, \text{J/mol} \cdot \text{K} \) for many liquids. This provides a rough guideline that connects the entropy change to the energy required to vaporize a substance.

It should be noted that entropy measures the degree of disorder or randomness in a system. When a liquid turns into vapor, its particles spread out and move more freely, leading to an increase in entropy. Trouton's Rule is useful for providing a first estimate when calculating boiling points, though it's an approximation.

Enthalpy of Vaporization

The enthalpy of vaporization, \( \Delta H_{\text{vap}}^{\circ} \), is the amount of energy needed to convert one mole of a liquid into vapor at constant temperature and pressure. Expressed in \( \text{kJ/mol} \), it indicates the strength of the intermolecular forces within the liquid—the stronger these forces, the higher the enthalpy.

For example, bromine's enthalpy of vaporization can be found to be \( 29.6 \, \text{kJ/mol} \). This reflects the energy required to overcome the attractions between \( \text{Br}_2 \) molecules. By knowing this value and using Trouton's Rule, we can establish a relationship between the enthalpy and the boiling point, making it possible to estimate the temperature at which bromine turns into a gas.

Boiling Point Calculation

Calculating the boiling point using Trouton's Rule involves a simple algebraic rearrangement of the equation \( \Delta H_{\text{vap}}^{\circ} = T_b \times \Delta S_{\text{vap}}^{\circ} \). By substituting the enthalpy and standard entropy of vaporization, one can isolate \( T_b \) (the boiling point) as:

\[ T_b = \frac{\Delta H_{\text{vap}}^{\circ}}{88} \]

For bromine, substituting \( 29600 \, \text{J/mol} \) for \( \Delta H_{\text{vap}}^{\circ} \), gives a boiling point \( T_b \approx 336.36 \, \text{K} \). This method provides a reasonable estimate and emphasizes how enthalpy and entropy work together to determine the state changes. It's a practical application of Trouton's Rule, though it relies on ideal assumptions.

Errors in Assumptions

Calculations using Trouton's Rule often involve several assumptions that can introduce errors.

• First, the assumption that \( \Delta H_{\text{vap}}^{\circ} \) is constant across varying temperatures isn't always true. Entropy and enthalpy can slightly vary with temperature, affecting accuracy.
• Second, the rule's approximation of \( \Delta S_{\text{vap}}^{\circ} \approx 88 \, \text{J/mol} \cdot \text{K} \) does not apply to all liquids, leading to potential discrepancies.
• Third, non-ideal behavior can occur, especially in substances like bromine, where molecular interactions may not fully align with the ideal gas approach assumed by the rule.

These factors combined can lead to deviations from actual boiling points. Understanding and acknowledging these possible errors is essential for a more precise and reliable application of the calculations.