To determine the amount of heat energy required to heat a cube of iron, we follow these steps:
1. Identify the given values:
- Specific heat capacity ([tex]\(C_p\)[/tex]): [tex]\(0.450 \, \text{J/g}^\circ\text{C}\)[/tex]
- Mass ([tex]\(m\)[/tex]): [tex]\(55.8 \, \text{g}\)[/tex]
- Initial temperature ([tex]\(T_{\text{initial}}\)[/tex]): [tex]\(25.0^\circ\text{C}\)[/tex]
- Final temperature ([tex]\(T_{\text{final}}\)[/tex]): [tex]\(175.0^\circ\text{C}\)[/tex]
2. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[
\Delta T = T_{\text{final}} - T_{\text{initial}}
\][/tex]
Substituting the given values:
[tex]\[
\Delta T = 175.0^\circ\text{C} - 25.0^\circ\text{C} = 150.0^\circ\text{C}
\][/tex]
3. Use the formula for heat energy ([tex]\(q\)[/tex]):
[tex]\[
q = m \cdot C_p \cdot \Delta T
\][/tex]
Where:
- [tex]\(q\)[/tex] is the heat energy
- [tex]\(m\)[/tex] is the mass
- [tex]\(C_p\)[/tex] is the specific heat capacity
- [tex]\(\Delta T\)[/tex] is the temperature change
4. Substitute the known values into the formula:
[tex]\[
q = 55.8 \, \text{g} \cdot 0.450 \, \text{J/g}^\circ\text{C} \cdot 150.0^\circ\text{C}
\][/tex]
5. Perform the calculation:
[tex]\[
q = 55.8 \times 0.450 \times 150.0
\][/tex]
[tex]\[
q = 3766.5 \, \text{J}
\][/tex]
Therefore, the heat energy required to heat the cube of iron is [tex]\(3766.5 \, \text{J}\)[/tex]. Rounded to three significant figures, the answer is:
[tex]\[
\boxed{3766.5 \, \text{J}}
\][/tex]
