To find the specific heat capacity ([tex]\(C_p\)[/tex]) of copper using the given data, we can use the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- [tex]\(q\)[/tex] is the heat added,
- [tex]\(m\)[/tex] is the mass of the substance,
- [tex]\(C_p\)[/tex] is the specific heat capacity,
- [tex]\(\Delta T\)[/tex] is the change in temperature.
We are given:
- Heat added ([tex]\(q\)[/tex]) = 4200.0 Joules,
- The mass of the copper rod ([tex]\(m\)[/tex]) = 200.0 grams,
- Initial temperature = [tex]\(20.0^{\circ} C\)[/tex],
- Final temperature = [tex]\(75.0^{\circ} C\)[/tex].
First, we need to calculate the change in temperature ([tex]\(\Delta T\)[/tex]):
[tex]\[ \Delta T = \text{Final temperature} - \text{Initial temperature} \][/tex]
[tex]\[ \Delta T = 75.0^{\circ} C - 20.0^{\circ} C \][/tex]
[tex]\[ \Delta T = 55.0^{\circ} C \][/tex]
Next, we rearrange the formula to solve for the specific heat capacity ([tex]\(C_p\)[/tex]):
[tex]\[ C_p = \frac{q}{m \cdot \Delta T} \][/tex]
Substitute the given values into the equation:
[tex]\[ C_p = \frac{4200.0 \text{ Joules}}{200.0 \text{ grams} \times 55.0^{\circ} C} \][/tex]
[tex]\[ C_p = \frac{4200.0}{11000.0} \text{ J/(g}\cdot^\circ\text{C)} \][/tex]
[tex]\[ C_p \approx 0.38181818181818183 \text{ J/(g}\cdot^\circ\text{C)} \][/tex]
Thus, the specific heat capacity ([tex]\(C_p\)[/tex]) of copper is approximately [tex]\(0.385 \text{ J/(g}\cdot^\circ\text{C)}\)[/tex].
From the given multiple-choice options, the closest match to our calculated value is:
[tex]\[ \boxed{0.385 \text{ J/(g}\cdot^\circ\text{C)} } \][/tex]
