To find the entropy change of the universe when iron melts, we need to analyze the entropy changes of both the system (iron) and the surroundings. Here's the step-by-step solution:
1. Given Values:
- Enthalpy of fusion [tex]\( \Delta H_{fusion} = 12,500 \frac{J}{mol} \)[/tex]
- Melting temperature [tex]\( T_{m} = 1,452 K \)[/tex]
- Entropy of iron [tex]\( \Delta S_{iron} = 162.8 \frac{J}{mol \cdot K} \)[/tex]
2. Calculate the Entropy Change of the Surroundings:
The entropy change of the surroundings ([tex]\(\Delta S_{surroundings}\)[/tex]) can be found using the formula:
[tex]\[
\Delta S_{surroundings} = -\frac{\Delta H_{fusion}}{T_m}
\][/tex]
Substituting the given values:
[tex]\[
\Delta S_{surroundings} = -\frac{12,500 \frac{J}{mol}}{1,452 K} \approx -8.609 \frac{J}{mol \cdot K}
\][/tex]
3. Calculate the Entropy Change of the Universe:
The total entropy change of the universe ([tex]\(\Delta S_{universe}\)[/tex]) is the sum of the entropy change of the system (iron) and the entropy change of the surroundings:
[tex]\[
\Delta S_{universe} = \Delta S_{iron} + \Delta S_{surroundings}
\][/tex]
Substituting the calculated entropy change of the surroundings and given entropy of iron:
[tex]\[
\Delta S_{universe} = 162.8 \frac{J}{mol \cdot K} + (-8.609 \frac{J}{mol \cdot K})
\][/tex]
Simplifying the expression:
[tex]\[
\Delta S_{universe} \approx 162.8 \frac{J}{mol \cdot K} - 8.609 \frac{J}{mol \cdot K} = 154.191 \frac{J}{mol \cdot K}
\][/tex]
Therefore, the entropy change of the universe when iron melts is approximately [tex]\( 154.191 \frac{J}{mol \cdot K} \)[/tex].
