To determine the specific heat ([tex]\(C_p\)[/tex]) of the substance, we can use the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- [tex]\( q \)[/tex] is the heat added (in Joules),
- [tex]\( m \)[/tex] is the mass of the substance (in kg or g),
- [tex]\( C_p \)[/tex] is the specific heat capacity (in [tex]\( \text{J/g}^\circ\text{C} \)[/tex]),
- [tex]\(\Delta T \)[/tex] is the change in temperature (in [tex]\(^\circ\text{C}\)[/tex]).
Let’s break this down step-by-step:
1. Identify given values:
- [tex]\( q = 3000.0 \text{ J} \)[/tex]
- [tex]\( m = 0.465 \text{ kg} \)[/tex]
- Initial temperature = [tex]\( 50.0^\circ \text{C} \)[/tex]
- Final temperature = [tex]\( 100.0^\circ \text{C} \)[/tex]
2. Calculate the temperature change ([tex]\(\Delta T\)[/tex]):
[tex]\[
\Delta T = 100.0^\circ\text{C} - 50.0^\circ\text{C} = 50.0^\circ\text{C}
\][/tex]
3. Convert the mass [tex]\(m\)[/tex] from kilograms to grams:
[tex]\[
m = 0.465 \text{ kg} \times 1000 \frac{\text{g}}{\text{kg}} = 465 \text{ g}
\][/tex]
4. Rearrange the specific heat formula to solve for [tex]\( C_p \)[/tex]:
[tex]\[
C_p = \frac{q}{m \cdot \Delta T}
\][/tex]
5. Substitute the known values into the equation:
[tex]\[
C_p = \frac{3000.0 \text{ J}}{465 \text{ g} \times 50.0^\circ\text{C}}
\][/tex]
6. Calculate the specific heat [tex]\( C_p \)[/tex]:
[tex]\[
C_p = \frac{3000.0}{465 \times 50.0}
\][/tex]
[tex]\[
C_p = \frac{3000.0}{23250}
\][/tex]
[tex]\[
C_p \approx 0.129 \text{ J/g}^\circ\text{C}
\][/tex]
Therefore, the specific heat of the substance is approximately [tex]\(0.129 \text{ J/g}^\circ\text{C}\)[/tex].
Among the given options:
- [tex]\(0.00775 \text{ J/g}^\circ\text{C}\)[/tex]
- [tex]\(0.0600 \text{ J/g}^\circ\text{C}\)[/tex]
- [tex]\(0.129 \text{ J/g}^\circ\text{C}\)[/tex]
- [tex]\(0.155 \text{ J/g}^\circ\text{C}\)[/tex]
The correct answer is [tex]\( 0.129 \text{ J/g}^\circ\text{C} \)[/tex].
