A copper rod with a mass of [tex]$200.0 \, \text{g}$[/tex] has an initial temperature of [tex]$20.0^{\circ} \text{C}$[/tex] and is heated to [tex]$40.0^{\circ} \text{C}$[/tex]. If [tex]$1,540 \, \text{J}$[/tex] of heat are needed to heat the rod, what is the specific heat of copper?

Use the formula: [tex] q = m C_p \Delta T [/tex].

A. [tex] 0.0130 \, \text{J} / (\text{g} \cdot {}^{\circ} \text{C}) [/tex]
B. [tex] 0.0649 \, \text{J} / (\text{g} \cdot {}^{\circ} \text{C}) [/tex]
C. [tex] 0.193 \, \text{J} / (\text{g} \cdot {}^{\circ} \text{C}) [/tex]
D. [tex] 0.385 \, \text{J} / (\text{g} \cdot {}^{\circ} \text{C}) [/tex]



Answer :

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]