Sure! Let's go through the problem step by step to determine the specific heat capacity of copper.
Given data:
- Mass of the copper rod, [tex]\( m = 200.0 \text{ g} \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 20.0^{\circ} \text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = 40.0^{\circ} \text{C} \)[/tex]
- Heat added, [tex]\( q = 1540 \text{ J} \)[/tex]
The formula to calculate the heat added is:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
Here, [tex]\( \Delta T \)[/tex] is the change in temperature, and [tex]\( C_p \)[/tex] is the specific heat capacity.
First, let's determine the temperature change [tex]\( \Delta T \)[/tex]:
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \][/tex]
[tex]\[ \Delta T = 40.0^{\circ} \text{C} - 20.0^{\circ} \text{C} \][/tex]
[tex]\[ \Delta T = 20.0^{\circ} \text{C} \][/tex]
Now, we can rearrange the heat equation to solve for the specific heat capacity [tex]\( C_p \)[/tex]:
[tex]\[ C_p = \frac{q}{m \cdot \Delta T} \][/tex]
Substituting the given values:
[tex]\[ C_p = \frac{1540 \text{ J}}{200.0 \text{ g} \cdot 20.0^{\circ} \text{C}} \][/tex]
Calculating the specific heat capacity:
[tex]\[ C_p = \frac{1540}{200.0 \cdot 20.0} \][/tex]
[tex]\[ C_p = \frac{1540}{4000.0} \][/tex]
[tex]\[ C_p = 0.385 \text{ J} / (\text{g} \cdot {}^{\circ} \text{C}) \][/tex]
Therefore, the specific heat capacity of copper is [tex]\( 0.385 \text{ J} / (\text{g} \cdot {}^{\circ} \text{C}) \)[/tex].
So, the correct option from the provided choices is:
[tex]\[ 0.385 \text{ J} / (\text{g}, {}^{\circ} \text{C}) \][/tex]
