Problem 44 - 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) a liquid to vapor usansformation and (b) a vapor to liquid transformation at \(35.0^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

1. Liquid to vapor: 84.37 J/mol⋅K. 2. Vapor to liquid: -84.37 J/mol⋅K.

Step 1: Formula for Entropy Change

The change in entropy, \[ \Delta S^{\circ} \]for a phase transition is given by the formula:\[ \Delta S^{\circ} = \frac{\Delta H^{\circ}}{T} \]where \( \Delta H^{\circ} \) is the enthalpy of vaporization and \( T \) is the temperature in Kelvin.

Step 2: Converting Temperature

First, we need to convert the temperature from Celsius to Kelvin. The formula to convert Celsius to Kelvin is:\[ T(K) = T(^{\circ}C) + 273.15 \]For this problem:\[ T = 35.0 + 273.15 = 308.15 \text{ K} \]

Step 3: Calculation of Entropy Change for Liquid to Vapor

For part (a), we need \( \Delta S^{\circ} \) when the liquid transforms into vapor. Using our formula and given that \( \Delta H^{\circ} = 26.0 \text{kJ/mol} \) (or \(26000 \text{J/mol}\), since 1 kJ = 1000 J):\[ \Delta S^{\circ}_{\text{liquid to vapor}} = \frac{26000 \text{ J/mol}}{308.15 \text{ K}} \]Calculating the above:\[ \Delta S^{\circ}_{\text{liquid to vapor}} \approx 84.37 \text{ J/mol⋅K} \]

Step 4: Calculation of Entropy Change for Vapor to Liquid

For part (b), the vapor to liquid transformation, the change in entropy will be the negative of the liquid to vapor transformation because it is the reverse process:\[ \Delta S^{\circ}_{\text{vapor to liquid}} = -\Delta S^{\circ}_{\text{liquid to vapor}} \]Thus:\[ \Delta S^{\circ}_{\text{vapor to liquid}} = -84.37 \text{ J/mol⋅K} \]

Step 5: Conclusion of the Calculation

We calculated \( \Delta S^{\circ}\) for both transformations: 1. From liquid to vapor: \( \Delta S^{\circ} = 84.37 \text{ J/mol⋅K} \)2. From vapor to liquid: \( \Delta S^{\circ} = -84.37 \text{ J/mol⋅K} \)

Key Concepts

Entropy Change

Entropy change is a crucial concept when studying thermodynamics and phase transitions. It represents the degree of disorder or randomness in a system.
During a phase transition, such as from liquid to vapor, entropy typically increases as the molecules move from a more ordered liquid state to a more disordered gaseous state.
Entropy change (\( \Delta S^{\circ} \)) for a phase transition can be calculated using the formula:

• \( \Delta S^{\circ} = \frac{\Delta H^{\circ}}{T} \)

Where \( \Delta H^{\circ} \) is the enthalpy of vaporization and \( T \) is temperature in Kelvin.
This formula helps determine how much the entropy changes as a function of temperature and the energy needed for the transformation.

Phase Transition

Phase transitions are changes between different states of matter—solid, liquid, and gas. In our exercise, we focus on the transition from liquid to gas (vaporization) and gas to liquid (condensation).
During vaporization, molecules break free from the intermolecular forces that hold them in the liquid state, moving to a more disordered gaseous state. This results in an increase in entropy.

• Vaporization: Liquid \( \rightarrow \) Gas
• Condensation: Gas \( \rightarrow \) Liquid

The reverse process, condensation, has the opposite effect on entropy.
Instead of increasing, entropy decreases because molecules are going from a random gaseous state to a structured liquid state. These transitions require specific energy changes—enthalpy changes—which can be used to calculate the corresponding changes in entropy.

Temperature Conversion

Converting temperature values is a common requirement in solving thermodynamic problems.
For the entropy change formula, temperature must be in Kelvin to maintain consistency in units.
The relationship between degrees Celsius and Kelvin is straightforward. This conversion is essential to perform accurate calculations in thermodynamics-related problems.
Using the formula:

• \( T(K) = T(^{\circ}C) + 273.15 \)

This formula allows us to convert a temperature of \( 35.0^{\circ}C \) to Kelvin, resulting in \( 308.15 \text{ K} \).

Celsius to Kelvin

Converting Celsius to Kelvin is a fundamental skill in physics and chemistry.
This adjustment aligns temperature scales so that they correspond with other thermodynamic equations, like the one calculating entropy change.
Kelvin is the standard unit for temperature in calculations because absolute temperatures are needed to ensure consistency, preventing negative values in thermodynamic equations.

• Add 273.15 to the Celsius temperature.
• Ensures calculations are consistent with absolute scale usage.

By converting \( 35.0^{\circ}C \) to Kelvin, we understand that temperature impacts how molecules behave and the energy transformations during processes like vaporization.