Question

At the transition temperature of \(95.4^{\circ} \mathrm{C}\), the enthalpy of transition from rhombic to monoclinic sulfur is \(0.38 \mathrm{kJ} \mathrm{mol}^{-1}\) a. Calculate the entropy of transition under these conditions. b. At its melting point, \(119^{\circ} \mathrm{C}\), the enthalpy of fusion of monoclinic sulfur is \(1.23 \mathrm{kJ} \mathrm{mol}^{-1}\). Calculate the entropy of fusion. c. The values given in parts (a) and (b) are for 1 mol of sulfur; however, in crystalline and liquid sulfur, the molecule is present as \(S_{8}\). Convert the values of the enthalpy and entropy of fusion in parts (a) and (b) to those appropriate for \(S_{8}\)

Short answer

The entropy of transition is approximately 1.031 J/mol·K, and the entropy of fusion is approximately 3.137 J/mol·K. For S8, the enthalpy and entropy values are as follows: - Part (a): \(\Delta H_\text{S8} (a) = 3.04\,\text{kJ/mol}\) and \(\Delta S_\text{S8} (a) \approx 8.248\,\text{J/mol·K}\) - Part (b): \(\Delta H_\text{S8} (b) = 9.84\,\text{kJ/mol}\) and \(\Delta S_\text{S8} (b) \approx 25.096\,\text{J/mol·K}\)

Step-by-step solution

Calculate the entropy of transition

To calculate the entropy of transition, we can use the following formula: \[ \Delta S = \frac{\Delta H}{T} \] Where: \(\Delta S\) = Entropy of transition (J/mol·K) \(\Delta H\) = Enthalpy of transition (J/mol) \(T\) = Transition temperature (K) First, let's convert the given temperature from Celsius to Kelvin: \[ T = 95.4 + 273.15 = 368.55\,\text{K} \] Now we can calculate the entropy of transition: \[ \Delta S (a) = \frac{0.38\,\text{kJ/mol}}{368.55\,\text{K}} = \frac{380\,\text{J/mol}}{368.55\,\text{K}} \approx 1.031\,\text{J/mol·K} \]

Calculate the entropy of fusion

Using the same formula as in Step 1, we can calculate the entropy of fusion. First, convert the melting point from Celsius to Kelvin: \[ T_\text{melt} = 119 + 273.15 = 392.15\,\text{K} \] Now, calculate the entropy of fusion: \[ \Delta S (b) = \frac{1.23\,\text{kJ/mol}}{392.15\,\text{K}} = \frac{1230\,\text{J/mol}}{392.15\,\text{K}} \approx 3.137\,\text{J/mol·K} \]

Convert the enthalpy and entropy of fusion for S8

Since there are 8 sulfur atoms in a molecule of S8, we will multiply the enthalpy and entropy of fusion values by 8: For Part (a): \[ \Delta H_\text{S8} (a) = 8 \times 0.38\,\text{kJ/mol} = 3.04\,\text{kJ/mol} \] \[ \Delta S_\text{S8} (a) = 8 \times 1.031\,\text{J/mol·K} \approx 8.248\,\text{J/mol·K} \] For Part (b): \[ \Delta H_\text{S8} (b) = 8 \times 1.23\,\text{kJ/mol} = 9.84\,\text{kJ/mol} \] \[ \Delta S_\text{S8} (b) = 8 \times 3.137\,\text{J/mol·K} \approx 25.096\,\text{J/mol·K} \] So, the enthalpy and entropy values for the transition and fusion of S8 are as follows: Part (a): \(\Delta H_\text{S8} (a) = 3.04\,\text{kJ/mol}\) and \(\Delta S_\text{S8} (a) \approx 8.248\,\text{J/mol·K}\) Part (b): \(\Delta H_\text{S8} (b) = 9.84\,\text{kJ/mol}\) and \(\Delta S_\text{S8} (b) \approx 25.096\,\text{J/mol·K}\)

Key concepts

Transition Temperature

The transition temperature refers to the specific temperature at which a substance shifts from one phase to another. For sulfur, this happens when it changes from one crystalline form to another, specifically from rhombic to monoclinic sulfur. To work with these temperatures in calculations, they must be in the Kelvin scale.

• Converting from Celsius to Kelvin involves adding 273.15 to the Celsius temperature.
• This conversion allows us to use these temperatures in equations such as those for calculating entropy or enthalpy changes.

In our example, the transition temperature for sulfur from rhombic to monoclinic form is 95.4°C, which is equivalent to 368.55 K. Understanding this concept is fundamental as it impacts how we calculate the thermodynamic properties that describe the phase transitions. Each phase change requires or releases a specific amount of energy, reflected in enthalpy and entropy changes under the given temperature conditions.

Enthalpy of Fusion

Enthalpy of fusion is the amount of energy required to change a substance from solid to liquid at its melting point, without changing its temperature. It is expressed in kJ/mol. The process of fusion absorbs energy to overcome the forces holding the solid together.

• This energy is necessary for the transition from a structured, orderly solid to a more disordered liquid state.
• The unit kJ/mol provides a measure for the energy change per mole of substance.

In the context of sulfur, the enthalpy of fusion for monoclinic sulfur at its melting point of 119°C (or 392.15 K) is 1.23 kJ/mol. This value is critical to understanding the energy dynamics involved in melting sulfur, giving insights into how different phases absorb and utilize energy during transformations.

Entropy of Fusion

Entropy of fusion is a measure of the change in disorder or randomness as a substance transforms from a solid to a liquid. Like enthalpy, entropy is also tied to temperature, and for phase transitions, is computed as the ratio of enthalpy of fusion to the temperature in Kelvin. It is measured in J/mol·K.

• Entropy increases during fusion because a liquid has a larger disorder than a solid.
• The formula used is: \[ \Delta S = \frac{\Delta H}{T} \]
• This relationship allows us to quantify the increase in entropy during melting.

For sulfur melting at 392.15 K, the calculated entropy of fusion is approximately 3.137 J/mol·K. This value helps chemists understand the degree of disorder introduced during the melting process, reflecting a fundamental aspect of chemical thermodynamics.

Molecular Sulfur (S8)

Sulfur commonly exists as a molecule comprised of eight atoms, known as S8. These S8 rings are the structural unit in both solid and liquid states. When making calculations of enthalpy and entropy for transitions in sulfur, it is essential to account for the number of atoms.

• To adjust for the molecular nature of sulfur, multiply the calculated values per mole of sulfur atoms by 8.
• This adjustment yields data that accurately reflects the behavior of molecular sulfur in different states.

In our exercise, converting the enthalpy and entropy for transitions per mole of S8 gives: - For the enthalpy of transition: 3.04 kJ/mol - For entropy: approximately 8.248 J/mol·K - For enthalpy of fusion: 9.84 kJ/mol - For entropy: approximately 25.096 J/mol·K These conversions are critical for correctly interpreting sulfur's thermodynamic properties in real-world applications, ensuring that calculations accommodate the molecule's true structure.