Sure, let's solve this step by step.
Given:
- The enthalpy of fusion ([tex]\(\Delta H_{\text{fus}}\)[/tex]) for [tex]\(\text{NaCl}\)[/tex] is 30.2 kJ/mol.
- The amount of heat ([tex]\( q \)[/tex]) required to melt the sample is 732.6 kJ.
We need to find the mass of the sample.
First, we use the formula:
[tex]\[ q = n \Delta H_{\text{fus}} \][/tex]
where [tex]\( n \)[/tex] is the number of moles.
Rearranging the formula to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{q}{\Delta H_{\text{fus}}} \][/tex]
Substituting the given values:
[tex]\[ n = \frac{732.6 \, \text{kJ}}{30.2 \, \text{kJ/mol}} \][/tex]
[tex]\[ n = 24.2583 \, \text{mol} \][/tex]
Next, we need to find the mass of this amount of [tex]\(\text{NaCl}\)[/tex]. We know the molar mass of [tex]\(\text{NaCl}\)[/tex] is 58.44 g/mol.
The mass can be calculated using:
[tex]\[ \text{mass} = n \times \text{molar mass} \][/tex]
Substituting the values:
[tex]\[ \text{mass} = 24.2583 \, \text{mol} \times 58.44 \, \text{g/mol} \][/tex]
[tex]\[ \text{mass} = 1417.65 \, \text{g} \][/tex]
Therefore, the mass of the [tex]\(\text{NaCl}\)[/tex] sample that needs 732.6 kJ of heat to melt completely is approximately 1417.7 g.
The correct answer from the choices provided is:
[tex]\[ \boxed{1,417.7 \, \text{g}} \][/tex]
