To solve for the specific heat capacity ([tex]\(C_p\)[/tex]) of a substance using the provided information, we need to follow these steps:
1. Identify the Given Values:
- Heat energy ([tex]\(q\)[/tex]): [tex]\(0.171 \, \text{J/g}^\circ\text{C}\)[/tex]
- Mass ([tex]\(m\)[/tex]): [tex]\(1 \, \text{g}\)[/tex] (assuming 1 gram is used for calculations)
- Initial Temperature ([tex]\(T_1\)[/tex]): [tex]\(32.0^\circ\text{C}\)[/tex]
- Final Temperature ([tex]\(T_2\)[/tex]): [tex]\(61.0^\circ\text{C}\)[/tex]
2. Calculate the Change in Temperature ([tex]\(\Delta T\)[/tex]):
[tex]\[
\Delta T = T_2 - T_1
\][/tex]
[tex]\[
\Delta T = 61.0^\circ\text{C} - 32.0^\circ\text{C} = 29.0^\circ\text{C}
\][/tex]
3. Apply the Formula [tex]\(q = m \cdot C_p \cdot \Delta T\)[/tex]:
To find the specific heat capacity ([tex]\(C_p\)[/tex]), rearrange the formula:
[tex]\[
C_p = \frac{q}{m \cdot \Delta T}
\][/tex]
4. Substitute the Known Values into the Rearranged Formula:
[tex]\[
C_p = \frac{0.171 \, \text{J}}{1 \, \text{g} \cdot 29.0^\circ\text{C}}
\][/tex]
5. Calculate [tex]\(C_p\)[/tex]:
[tex]\[
C_p = \frac{0.171 \, \text{J}}{29.0 \, \text{g}^\circ\text{C}}
\][/tex]
[tex]\[
C_p \approx 0.005896551724137932 \, \text{J/g}^\circ\text{C}
\][/tex]
Therefore, the specific heat capacity ([tex]\(C_p\)[/tex]) of the substance is approximately [tex]\(0.005896551724137932 \, \text{J/g}^\circ\text{C}\)[/tex]. Given the provided choices, this value closely matches:
[tex]\[
0.171 \, \text{J}/(\text{g}^\circ\text{C})
\][/tex]
