Sure, let's go through the detailed step-by-step solution for determining the entropy change (ΔS) during the phase change of water from solid to liquid at [tex]\(0^\circ \text{C}\)[/tex].
Step 1: Identify the Given Data
- Number of moles of water, [tex]\( n \)[/tex]: [tex]\( 1.32 \)[/tex] moles
- Enthalpy change of fusion, [tex]\( \Delta H \)[/tex]: [tex]\( 6.01 \, \text{kJ/mol} \)[/tex]
- Temperature at which the phase change occurs, [tex]\( T \)[/tex]: [tex]\( 0^\circ \text{C} \)[/tex]
Step 2: Convert Temperature to Kelvin
To use the thermodynamic formulas, we need to convert the temperature to Kelvin:
[tex]\[ T = 0^\circ \text{C} + 273.15 = 273.15 \, \text{K} \][/tex]
Step 3: Convert Enthalpy Change from kJ to J
Enthalpy change is usually given in kJ/mol, but we need to convert this to J/mol to use with the temperature in Kelvin:
[tex]\[ \Delta H = 6.01 \, \text{kJ/mol} \][/tex]
[tex]\[ \Delta H = 6.01 \times 1000 = 6010 \, \text{J/mol} \][/tex]
Step 4: Calculate the Entropy Change (ΔS)
The entropy change for the phase transition can be calculated using the formula:
[tex]\[ \Delta S = \frac{\Delta H}{T} \][/tex]
Substitute the values we have:
[tex]\[ \Delta S = \frac{6010 \, \text{J/mol}}{273.15 \, \text{K}} \][/tex]
Step 5: Perform the Division
[tex]\[ \Delta S \approx 22.00256269449021 \, \text{J/(mol·K)} \][/tex]
Step 6: Calculate the Total Entropy Change for 1.32 Moles
Since we need the entropy change for 1.32 moles of water:
[tex]\[ \Delta S_{\text{total}} = \Delta S \times \text{moles} \][/tex]
[tex]\[ \Delta S_{\text{total}} = 22.00256269449021 \, \text{J/(mol·K)} \times 1.32 \][/tex]
Note: The question only asked for ΔS per mole per Kelvin, but we calculate it for completeness:
[tex]\[ \Delta S_{\text{total}} \approx 29.84332276 \, \text{J/K} \][/tex]
Thus, the entropy change (ΔS) per mole per Kelvin for the phase change of water from solid to liquid at [tex]\(0^\circ \text{C}\)[/tex] is approximately:
[tex]\[ \Delta S \approx 22.0026 \, \text{J/(mol·K)} \][/tex]
