Certainly! Let's solve this step-by-step.
### Problem Statement:
A copper rod with a mass of \(200.0 \, g\) is heated from an initial temperature of \(20.0^{\circ}C\) to a final temperature of \(40.0^{\circ}C\). It absorbs \(1,540 \, J\) of heat during this process. We need to determine the specific heat capacity (\(C_p\)) of the copper rod using the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
#### Given:
1. Mass (\(m\)): \(200.0 \, g\)
2. Initial temperature (\(T_i\)): \(20.0^{\circ}C\)
3. Final temperature (\(T_f\)): \(40.0^{\circ}C\)
4. Heat added (\(q\)): \(1,540 \, J\)
### Step-by-Step Solution:
1. Calculate the change in temperature (\(\Delta T\)):
[tex]\[ \Delta T = T_f - T_i \][/tex]
[tex]\[ \Delta T = 40.0^{\circ}C - 20.0^{\circ}C \][/tex]
[tex]\[ \Delta T = 20.0^{\circ}C \][/tex]
2. Rearrange the formula to solve for the specific heat capacity \((C_p)\):
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
[tex]\[ C_p = \frac{q}{m \cdot \Delta T} \][/tex]
3. Substitute the values into the formula:
[tex]\[ q = 1,540 \, J \][/tex]
[tex]\[ m = 200.0 \, g \][/tex]
[tex]\[ \Delta T = 20.0^{\circ}C \][/tex]
[tex]\[ C_p = \frac{1,540 \, J}{200.0 \, g \cdot 20.0^{\circ}C} \][/tex]
4. Calculate the specific heat capacity \(C_p\):
[tex]\[ C_p = \frac{1,540}{200.0 \cdot 20.0} \][/tex]
[tex]\[ C_p = \frac{1,540}{4,000} \][/tex]
[tex]\[ C_p = 0.385 \, \text{J/(g·°C)} \][/tex]
### Conclusion:
The specific heat capacity of copper is \(0.385 \, \text{J/(g·°C)}\).
From the provided options:
- \(0.0130 \, \text{J/(g·°C)}\)
- \(0.0649 \, \text{J/(g·°C)}\)
- \(0.193 \, \text{J/(g·°C)}\)
- \(0.385 \, \text{J/(g·°C)}\)
The correct answer is:
[tex]\[ \boxed{0.385 \, \text{J/(g·°C)} } \][/tex]
