To find the molar heat of fusion for iron, we need to follow a systematic approach. Let's solve the problem step by step using the given data and the principles of thermodynamics.
Given Data:
1. Mass of iron ([tex]\( Fe \)[/tex]) [tex]\( = 200.0 \)[/tex] grams
2. Energy released when iron freezes [tex]\( = 9,840 \)[/tex] calories
Additional Information:
- Molar mass of iron ([tex]\( Fe \)[/tex]) [tex]\( = 55.845 \)[/tex] grams per mole
Here's the step-by-step solution:
### Step 1: Calculate the number of moles of iron
To determine the number of moles ([tex]\( n \)[/tex]) of iron, we can use the formula:
[tex]\[ n = \frac{\text{mass}}{\text{molar mass}} \][/tex]
Substituting the given values:
[tex]\[ n = \frac{200.0 \text{ g}}{55.845 \text{ g/mol}} \][/tex]
### Step 2: Solve for [tex]\( n \)[/tex]
Perform the division:
[tex]\[ n \approx 3.581341212284 \text{ mol} \][/tex]
### Step 3: Use the formula [tex]\( q = n \Delta H \)[/tex]
We are given the energy released ([tex]\( q \)[/tex]) and the number of moles ([tex]\( n \)[/tex]). We need to solve for the molar heat of fusion ([tex]\( \Delta H \)[/tex]).
Rearranging the formula to solve for [tex]\( \Delta H \)[/tex]:
[tex]\[ \Delta H = \frac{q}{n} \][/tex]
Substitute the known values:
[tex]\[ \Delta H = \frac{9,840 \text{ cal}}{3.581341212284 \text{ mol}} \][/tex]
### Step 4: Solve for [tex]\( \Delta H \)[/tex]
Perform the division:
[tex]\[ \Delta H \approx 2747.574 \text{ cal/mol} \][/tex]
### Step 5: Compare with the given choices
The calculated value of [tex]\( \Delta H \)[/tex] is approximately [tex]\( 2747.574 \text{ cal/mol} \)[/tex]. Comparing this result with the given options:
- [tex]\( 2747.7 \text{ cal/mol} \)[/tex]
- [tex]\( 2811.7 \text{ cal/mol} \)[/tex]
- [tex]\( 3280.0 \text{ cal/mol} \)[/tex]
The value [tex]\( 2747.574 \text{ cal/mol} \)[/tex] is closest to [tex]\( 2747.7 \text{ cal/mol} \)[/tex].
### Conclusion
The molar heat of fusion for iron is [tex]\( 2747.7 \text{ cal/mol} \)[/tex]. This matches the given first choice perfectly.
