Problem 54 - Chemistry: Principles and Reactions

Question

Given the following data for bromine, $$ \begin{aligned} &\mathrm{Br}_{2}(l): S^{\circ}=152.2 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ &\mathrm{Br}_{2}(g): S^{\circ}=245.4 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \quad \Delta H_{\mathrm{f}}^{\circ}=30.91 \mathrm{~kJ} / \mathrm{mol} \end{aligned} $$ estimate the normal boiling point of bromine. $$ \mathrm{Br}_{2}(l) \rightleftharpoons \mathrm{Br}_{2}(g) $$

Step-by-Step Solution

Answer

The estimated normal boiling point of liquid bromine is approximately 331.7 K.

Step 1: Understand the Clausius-Clapeyron equation

The Clausius-Clapeyron equation relates the vapor pressure of a substance to its temperature, enthalpy of vaporization, and the value of the ideal gas constant. The equation can be written as: $$ \ln \frac{P_2}{P_1}= \frac{\Delta H_v}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) $$ Where \(P_1\) and \(P_2\) are the vapor pressures at temperatures \(T_1\) and \(T_2\) respectively, \(\Delta H_v\) is the enthalpy of vaporization, and \(R\) is the ideal gas constant in appropriate units.

Step 2: Calculate the enthalpy of vaporization

To find the enthalpy of vaporization (\(\Delta H_v\)), we can use the given enthalpy of formation value for gaseous bromine (\(\Delta H_{f}^{\circ}=30.91 \mathrm{~kJ} / \mathrm{mol}\)) since liquid and gaseous bromine are the only components in the equilibrium equation: $$ \Delta H_v = \Delta H_{f}^{\circ} = 30.91 \mathrm{~kJ} / \mathrm{mol} = 30,910 \mathrm{~J} / \mathrm{mol} $$

Step 3: Make use of the entropy values

Now we need to relate the given entropy values of liquid and gaseous bromine to the boiling point. At the boiling point, the vapor pressure of liquid bromine equals the vapor pressure of gaseous bromine. Therefore, we can write the relation between \(\Delta S\) (change in entropy), \(T_{b}\) (boiling point), and \(\Delta H_v\) (enthalpy of vaporization) as: $$ \Delta S = \frac{\Delta H_v}{T_b} $$ Using the given entropy values for liquid (\(S^{\circ}_{l} = 152.2 \mathrm{~J} / \mathrm{mol}\cdot K\)) and gaseous bromine (\(S^{\circ}_{g} = 245.4 \mathrm{~J} / \mathrm{mol}\cdot K\)), we can calculate the change in entropy (\(\Delta S\)) as: $$ \Delta S = S^{\circ}_{g} - S^{\circ}_{l} = 245.4 - 152.2 = 93.2 \mathrm{~J} / \mathrm{mol}\cdot K $$

Step 4: Calculate the boiling point

Now we can use the relation between \(\Delta S\), \(T_b\), and \(\Delta H_v\) to find the boiling point: $$ T_b = \frac{\Delta H_v}{\Delta S} = \frac{30,910 \mathrm{~J} / \mathrm{mol}}{93.2 \mathrm{~J} / \mathrm{mol}\cdot K} = 331.7 \mathrm{~K} $$ Thus, the estimated normal boiling point of bromine is approximately 331.7 K.

Key Concepts

Clausius-Clapeyron Equation

The Clausius-Clapeyron Equation is a fundamental principle that connects the change in vapor pressure of a substance with temperature to its enthalpy of vaporization. This equation is especially useful for predicting the boiling points of substances under different conditions. It takes the form \( \ln \frac{P_2}{P_1} = \frac{\Delta H_v}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) \). Here, \( P_1 \) and \( P_2 \) are the vapor pressures at temperatures \( T_1 \) and \( T_2 \) respectively.
\( \Delta H_v \) denotes the enthalpy of vaporization, and \( R \) is the ideal gas constant, typically valued at 8.314 J/mol·K.
In the context of the exercise, using this equation allows determination of the boiling point when the enthalpy of vaporization and the entropy change are known.
By understanding how pressure changes with temperature, one can predict how a substance behaves when it transitions from a liquid to a gas, an essential aspect for many scientific and engineering processes.

Enthalpy of Vaporization

Enthalpy of vaporization, symbolized as \( \Delta H_v \), is the amount of energy required to convert a unit of liquid into vapor without changing its temperature. This energy quantifies the strength of intermolecular forces in a liquid as the phase changes from liquid to gas.
For bromine in its gaseous form, the given value of the enthalpy of formation is 30.91 kJ/mol. This value indicates how much energy bromine needs to overcome the attractive forces holding its molecules in a liquid state.
Knowing \( \Delta H_v \) allows us to calculate the boiling point by integrating it with entropy values via the Clausius-Clapeyron relation.
The section of the exercise demonstrates calculating \( \Delta H_v \), converting it into Joules for consistency with other values, and using it to compute the transition from liquid to gas at the boiling point.

Entropy Change

Entropy change, denoted \( \Delta S \), measures the disorder or random distribution of energy between molecules during a phase change.
For bromine, the entropy in the liquid state is 152.2 J/mol·K and in the gaseous state is 245.4 J/mol·K. When bromine boils, the entropy changes from the liquid entropy to the gaseous entropy. The change is simply \( \Delta S = S_g - S_l = 245.4 - 152.2 = 93.2 \) J/mol·K.
This change in entropy relates directly to the boiling point through the equation \( \Delta S = \frac{\Delta H_v}{T_b} \), where \( T_b \) is the boiling temperature.
By understanding how to calculate \( \Delta S \), one learns how energy disperses in a system, providing insights into the stability of phases and conditions under which transitions occur.

Bromine

Bromine (chemical symbol Br), a halogen element, commonly exists as a reddish-brown liquid at room temperature.
It easily vaporizes, providing an opportunity to apply thermodynamic equations such as the Clausius-Clapeyron equation to predict its boiling point.
In this problem, bromine's physical properties, specifically its enthalpy of vaporization and entropy values, enable estimation of its normal boiling point at 331.7 K.
This exercise illustrates using detailed thermodynamic data to find practical answers about real substances.
Aside from theoretical calculations, understanding bromine's behavior is valuable in chemistry and industries where bromine is used, such as the manufacture of flame retardants and certain types of medicines.