Question

Ethyl alcohol has \(\Delta H_{\text {fusion }}=5.02 \mathrm{~kJ} / \mathrm{mol}\) and melts at \(-114.1^{\circ} \mathrm{C}\). What is the value of \(\Delta S_{\text {fusion }}\) for ethyl alcohol?

Short answer

\(\Delta S_{\text{fusion}} \approx 31.57 \, \text{J/mol K}\)

Step-by-step solution

Understanding the Relationship

We are given the enthalpy of fusion, \(\Delta H_{\text{fusion}} = 5.02 \, \text{kJ/mol}\), and we need to find the entropy of fusion, \(\Delta S_{\text{fusion}}\). The change in entropy during fusion can be found using the formula: \(\Delta S = \frac{\Delta H}{T}\), where \(T\) is the temperature in Kelvin.

Converting Temperature to Kelvin

Convert the melting point from Celsius to Kelvin: \(-114.1^{\circ}\text{C} = -114.1 + 273.15 = 159.05 \, \text{K}\).

Calculating Entropy of Fusion

Use the formula \(\Delta S = \frac{\Delta H}{T}\): \[\Delta S_{\text{fusion}} = \frac{5.02 \, \text{kJ/mol}}{159.05 \, \text{K}}\]Convert 5.02 kJ to J: \(5.02 \, \text{kJ} = 5020 \, \text{J}\).\[\Delta S_{\text{fusion}} = \frac{5020 \, \text{J/mol}}{159.05 \, \text{K}} \approx 31.57 \, \text{J/mol K}\]

Key concepts

Entropy of Fusion

The entropy of fusion (\( \Delta S_{\text{fusion}} \)) refers to the change in entropy when a substance transitions from a solid to a liquid at its melting point. Entropy, a measure of disorder or randomness, increases during this phase change due to the increased molecular motion and distribution of energy. In simpler terms, as a solid melts into a liquid, its particles move more freely, leading to greater disorder. To calculate \( \Delta S_{\text{fusion}} \), we use the relationship between enthalpy and entropy given by the formula:

• \( \Delta S = \frac{\Delta H}{T} \)

Where:

• \( \Delta H \) is the enthalpy of fusion, representing the heat absorbed during melting.
• \( T \) is the temperature in Kelvin at which the fusion occurs.

This formula helps us understand how much entropy changes when we know the heat absorbed and the melting temperature.

Temperature Conversion

Temperature conversion is crucial when using formulas involving thermodynamic calculations. For entropy and enthalpy problems, the temperature should always be in Kelvin. This is because Kelvin (\(^\circ \text{K} \)) is the SI unit for temperature, ensuring accurate calculations in thermodynamic equations. To convert a temperature from Celsius (\(^\circ \text{C} \)) to Kelvin:

• Add 273.15 to the Celsius temperature.

For example, if the melting point of ethyl alcohol is \(-114.1^\circ \text{C}\), you would convert it to Kelvin by:

• \(-114.1 + 273.15 = 159.05 \, \text{K}\)

Ensuring this conversion is done correctly is pivotal for subsequent calculations, as using Celsius in such calculations would yield incorrect results due to the scale difference.

Enthalpy-Entropy Relationship

The enthalpy-entropy relationship is a fundamental concept in thermodynamics, illustrating how energy and disorder relate during phase changes. During fusion, the enthalpy change (\( \Delta H_{\text{fusion}} \)), represents the energy needed to convert a solid into a liquid at its melting point, without any temperature change. Entropy change (\( \Delta S_{\text{fusion}} \)) rises as the molecular disorder increases. The mathematical connection between them is:

• \( \Delta S = \frac{\Delta H}{T} \)

This equation highlights how temperature (\( T \)) affects entropy change for a given enthalpy change. For instance, when calculating the entropy of fusion for ethyl alcohol, where \( \Delta H_{\text{fusion}} = 5.02 \, \text{kJ/mol} \) and the temperature is \( 159.05 \, \text{K} \), we first convert everything to consistent units, which results in:

• \( \Delta S_{\text{fusion}} = \frac{5020 \text{ J/mol}}{159.05 \text{ K}} \approx 31.57 \text{ J/mol K} \)

Understanding this relationship not only helps solve problems like the one provided but also deepens our understanding of phase transitions and energy dynamics.