Sure! Let's break down the solution step-by-step to find the specific heat ([tex]\(C_p\)[/tex]) of the substance.
### Given Data:
1. Mass ([tex]\(m\)[/tex]) of the substance: [tex]\(0.465 \, \text{kg}\)[/tex]
2. Heat energy ([tex]\(q\)[/tex]) added: [tex]\(3,000.0 \, \text{J}\)[/tex]
3. Initial temperature ([tex]\(T_i\)[/tex]): [tex]\(0.0^\circ \text{C}\)[/tex]
4. Final temperature ([tex]\(T_f\)[/tex]): [tex]\(100.0^\circ \text{C}\)[/tex]
### Step-by-Step Solution:
1. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[
\Delta T = T_f - T_i = 100.0^\circ \text{C} - 0.0^\circ \text{C} = 100.0^\circ \text{C}
\][/tex]
2. Use the formula [tex]\(q = m C_p \Delta T\)[/tex] to solve for [tex]\(C_p\)[/tex]:
[tex]\[
q = m C_p \Delta T
\][/tex]
Rearranging to solve for [tex]\(C_p\)[/tex]:
[tex]\[
C_p = \frac{q}{m \Delta T}
\][/tex]
Plugging in the given values:
[tex]\[
C_p = \frac{3,000.0 \, \text{J}}{0.465 \, \text{kg} \times 100.0^\circ \text{C}}
\][/tex]
[tex]\[
C_p = \frac{3,000.0 \, \text{J}}{46.5 \, \text{kg}^\circ \text{C}}
\][/tex]
[tex]\[
C_p = 64.51612903225806 \, \text{J/(kg}^\circ \text{C)}
\][/tex]
3. Convert the specific heat from [tex]\(\text{J/(kg}^\circ \text{C)}\)[/tex] to [tex]\(\text{J/(g}^\circ \text{C)}\)[/tex]:
Since there are [tex]\(1,000 \, \text{g}\)[/tex] in [tex]\(1 \, \text{kg}\)[/tex], divide the specific heat by [tex]\(1,000\)[/tex]:
[tex]\[
C_p = \frac{64.51612903225806 \, \text{J/(kg}^\circ \text{C)}}{1,000}
\][/tex]
[tex]\[
C_p = 0.06451612903225806 \, \text{J/(g}^\circ \text{C)}
\][/tex]
### Conclusion:
None of the provided choices correctly match the specific heat calculated, which is approximately [tex]\(0.0645 \, \text{J/(g}^\circ \text{C)}\)[/tex]. Thus, the specific heat of the substance is [tex]\(0.0645 \, \text{J/(g}^\circ \text{C)}\)[/tex].
