Problem 40 - Chemistry and Chemical Reactivity

Question

The enthalpy of vaporization of liquid diethyl ether, \(\left(\mathrm{C}_{2} \mathrm{H}_{5}\right)_{2} \mathrm{O},\) is \(26.0 \mathrm{kJ} / \mathrm{mol}\) at the boiling point of \(35.0^{\circ} \mathrm{C} .\) Calculate \(\Delta S^{\circ}\) for a vapor-to-liquid transformation at \(35.0^{\circ} \mathrm{C}.\)

Step-by-Step Solution

Answer

The \( \Delta S^\circ \) for vapor-to-liquid transformation is \(-84.38 \text{ J/mol K}\).

Step 1: Understand the Question

We are asked to find the change in entropy \( \Delta S^\circ \) for a phase transition from vapor to liquid for diethyl ether at its boiling point, given the enthalpy of vaporization \( \Delta H_{\text{vap}} \).

Step 2: Recall the Clausius-Clapeyron Relation

At the boiling point, the vaporization occurs at equilibrium, and we use the formula for entropy change \( \Delta S^\circ = \frac{\Delta H_{\text{vap}}}{T} \).

Step 3: Convert Temperature to Kelvin

The temperature given is in Celsius. Convert it to Kelvin by adding 273.15. Thus, \( T = 35.0 + 273.15 = 308.15 \text{ K} \).

Step 4: Insert Values into the Formula

Insert the known values into the formula: \( \Delta S^\circ = \frac{26.0 \text{ kJ/mol}}{308.15 \text{ K}} \).

Step 5: Perform the Calculation

Calculate \( \Delta S^\circ = \frac{26.0 \times 1000 \text{ J/mol}}{308.15 \text{ K}} = 84.38 \text{ J/mol K} \).

Step 6: Interpret the Result

The calculation applies to a vapor-to-liquid process, hence \( \Delta S^\circ = -84.38 \text{ J/mol K} \) since the entropy decreases.

Key Concepts

Enthalpy of Vaporization

The enthalpy of vaporization is a key concept when studying phase transitions, particularly the transition from liquid to gas. It is denoted as \( \Delta H_{\text{vap}} \) and represents the amount of energy needed to convert one mole of a liquid into a gas at constant pressure. This energy is necessary to overcome intermolecular forces that hold liquid molecules together.
The stronger these forces, the higher the enthalpy of vaporization. For example, water has a comparatively high enthalpy of vaporization at standard conditions because its molecules are strongly attracted to each other through hydrogen bonding.
Enthalpy of vaporization is usually expressed in kilojoules per mole \( \text{kJ/mol} \). In the exercise, diethyl ether has an enthalpy of vaporization of 26.0 \( \text{kJ/mol} \), which indicates the energy required to vaporize it at its boiling point. Understanding this concept is crucial as it allows us to calculate how much energy is involved in phase transitions.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is an essential formula in thermodynamics, particularly in explaining how vapor pressure changes with temperature. It relates the enthalpy of vaporization to the entropy change during a phase transition at equilibrium. The equation is utilized for finding the change in entropy \( \Delta S \) during vaporization or condensation at the boiling point.
When a liquid boils, its vapor pressure equals the external pressure. At this point, the Clausius-Clapeyron equation simplifies to \( \Delta S^\circ = \frac{\Delta H_{\text{vap}}}{T} \), where \( T \) is the temperature in Kelvin. This equation tells us that the change in entropy depends on both the enthalpy of vaporization and the temperature.
In the problem, using the given enthalpy of vaporization and the temperature converted to Kelvin, one can determine the change in entropy for the phase transition, which gives insight into the disorder or randomness introduced when diethyl ether vaporizes.

Temperature Conversion to Kelvin

Temperature conversion to Kelvin is a crucial step when dealing with thermodynamic equations, as they require absolute temperatures. Kelvin is the SI unit for temperature and the calculations necessitate using this scale because it begins from absolute zero, where theoretically, all molecular movement ceases.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For instance, in the exercise, the boiling point of diethyl ether is 35.0°C. By converting this to Kelvin:

• 35.0 + 273.15 = 308.15 K

This conversion allows us to use the temperature in the Clausius-Clapeyron equation to find changes in entropy during the phase transition.
Ensuring that temperature is in Kelvin is essential for accurate calculations in thermodynamics since many thermodynamic relationships depend on this absolute scale.