Problem 87 - Chemistry: The Central Science

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 \(\Delta H_{\mathrm{vap}}^{\circ}\) 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 web site.

Step-by-Step Solution

Answer

The estimated normal boiling point of bromine (Br₂) using Trouton's rule is approximately 338.64 K, which is close to the actual value of 332.00 K found in chemistry handbooks or the WebElements web site. To find this estimate, we first determined the standard enthalpy of vaporization for Br₂ (ΔHᵥ° = 29,800 J/mol) and then used Trouton's rule, which states that the molar entropy of vaporization (ΔSᵥ°) is approximately 88 J/mol-K. We applied the Gibbs free energy equation to find the boiling point temperature (T) by solving for T in the equation ΔGᵥ° = ΔHᵥ° - T * ΔSᵥ°.

Step 1: (Step 1: Find the standard enthalpy of vaporization for Br₂)

For this, we will refer to Appendix C and look for the enthalpy of vaporization for Br₂, which is given as ΔHᵥ° = 29.8 kJ/mol.

Step 2: (Step 2: Convert the enthalpy of vaporization to J/mol)

We need to express the ΔHᵥ° in J/mol for further calculations. We know that 1 kJ = 1000 J, so we can convert it using the following equation: ΔHᵥ° = 29.8 kJ/mol * (1000 J/1 kJ) = 29,800 J/mol

Step 3: (Step 3: Apply Trouton's rule to find the molar entropy of vaporization ΔSᵥ°)

Trouton's rule states that for many liquids at their normal boiling points, the molar entropy of vaporization is approximately 88 J/mol-K. So we can write: ΔSᵥ° ≈ 88 J/mol-K

Step 4: (Step 4: Use the Gibbs free energy equation to find the boiling point of Br₂)

The relationship between enthalpy, entropy, and Gibbs free energy is given by the equation: ΔGᵥ° = ΔHᵥ° - T * ΔSᵥ° At the boiling point, the phase transition is in equilibrium, which results in a Gibbs free energy of zero: ΔGᵥ° = 0 = ΔHᵥ° - T * ΔSᵥ° Now, we will rearrange the equation to find the boiling point temperature (T): T = ΔHᵥ° / ΔSᵥ° Substitute the values for ΔHᵥ° and ΔSᵥ°: T = (29,800 J/mol) / (88 J/mol-K) = 338.64 K Hence, the estimated normal boiling point of bromine, Br₂, using Trouton's rule is approximately 338.64 K.

Step 5: (Step 5: Compare the calculated boiling point with reference values)

According to chemistry handbooks or the WebElements web site, the actual normal boiling point of Br₂ is 332.00 K. The estimated boiling point of Br₂ using Trouton's rule is quite close to the actual value, indicating that Trouton's rule is a useful method for estimating boiling points when data is not readily available.

Key Concepts

Standard Molar Entropy of Vaporization

The standard molar entropy of vaporization ( ்ப MiKTeX) is a key thermodynamic function that measures the entropy change when one mole of a substance changes phase from liquid to gas at a constant temperature, typically at the substance's boiling point. It serves as an indicator of how dispersed the molecules of a substance become when they transition to a gaseous state. According to Trouton's rule, this entropy change for many substances falls around 88 J/mol•K at their normal boiling points.

Using Trouton's rule as a tool for estimation, we can predict the boiling points of substances by establishing a relationship between entropy and enthalpy changes during vaporization. This is particularly helpful in situations where precise experimental data might not be available.

Enthalpy of Vaporization

Enthalpy of vaporization, commonly represented as ்ஜ MiKTeX) here. This is due to the fact that the addition of heat can increase the kinetic energy of the molecules to a point where intermolecular forces are overcome, permitting the transition from liquid to gas. The value of the enthalpy of vaporization is used in various calculations, such as using Trouton's rule to estimate the normal boiling point of substances. It's a prime example of how interconnected thermodynamic properties are in understanding and predicting the physical behavior of substances in chemical processes.

Gibbs Free Energy Equation

The Gibbs free energy equation is a fundamental cornerstone of thermodynamics

்ஜ MiKTeX), represents the maximum amount of work that can be extracted from a thermodynamic process at constant temperature and pressure. It is defined by the relationship ்ஜ MiKTeX), where ்ஜ (Gibb's free energy), ்ஜ (enthalpy), and T ்ஜ MiKTeX) represents the absolute temperature.

When a substance reaches its boiling point and vaporizes, the Gibbs free energy change ( ்ஜ) for the phase transition is zero, because the substance is in equilibrium. This concept helps to calculate the normal boiling point of a substance by setting the ்ஜ MiKTeX) within the Gibbs energy equation to zero.

Normal Boiling Point Estimation

Estimating the normal boiling point of a substance is a practical application of thermodynamic concepts, particularly when experimental data may be lacking. The normal boiling point is the temperature at which a liquid boils under standard atmospheric pressure (1 atm). Using Trouton's rule provides a rough estimate of this boiling point.

To apply Trouton's rule for boiling point estimation, we calculate the enthalpy of vaporization and equate it to the product of the normal molar entropy of vaporization and the estimated boiling temperature ( ்ஜ). Rearranging this relationship allows us to solve for the temperature. This method has proven reliable for many substances, as demonstrated by the reasonably accurate estimate for the boiling point of bromine compared to the actual value. However, it's important to note that there are exceptions to Trouton's rule, and it should be used with a measure of caution for substances with highly associative or structurally complex molecules.