Answer :
To compute the specific heat values for Copper (Cu), Iron (Fe), and Lead (Pb), we need to use the following given equation for each metal:
[tex]\[ c_{\text{metal}} = \frac{-c_{\text{water}} m_{\text{water}} \Delta T_{\text{water}}}{m_{\text{metal}} \Delta T_{\text{metal}}} \][/tex]
Given:
- The specific heat of water, [tex]\( c_{\text{water}} = 4.184 \text{ J/g} ^{\circ} \text{C} \)[/tex]
Let's calculate the specific heat values for each metal step-by-step.
### Copper (Cu):
1. Mass of water ([tex]\(m_{\text{water}}\)[/tex]): 40.13 g
2. Mass of metal ([tex]\(m_{\text{metal}}\)[/tex]): 12.14 g
3. Change in temperature of water ([tex]\(\Delta T_{\text{water}}\)[/tex]): 1.9 [tex]\(^{\circ}\)[/tex]C
4. Change in temperature of metal ([tex]\(\Delta T_{\text{metal}}\)[/tex]): -75.4 [tex]\(^{\circ}\)[/tex]C
Substitute these values into the equation:
[tex]\[ c_{\text{Cu}} = \frac{-4.184 \times 40.13 \times 1.9}{12.14 \times -75.4} \][/tex]
#### Simplification (performed by calculator):
[tex]\[ c_{\text{Cu}} \approx 0.35 \text{ J/g} ^{\circ} \text{C} \][/tex]
### Iron (Fe):
1. Mass of water ([tex]\(m_{\text{water}}\)[/tex]): 40.24 g
2. Mass of metal ([tex]\(m_{\text{metal}}\)[/tex]): 12.31 g
3. Change in temperature of water ([tex]\(\Delta T_{\text{water}}\)[/tex]): 2.4 [tex]\(^{\circ}\)[/tex]C
4. Change in temperature of metal ([tex]\(\Delta T_{\text{metal}}\)[/tex]): -75.1 [tex]\(^{\circ}\)[/tex]C
Substitute these values into the equation:
[tex]\[ c_{\text{Fe}} = \frac{-4.184 \times 40.24 \times 2.4}{12.31 \times -75.1} \][/tex]
#### Simplification (performed by calculator):
[tex]\[ c_{\text{Fe}} \approx 0.44 \text{ J/g} ^{\circ} \text{C} \][/tex]
### Lead (Pb):
1. Mass of water ([tex]\(m_{\text{water}}\)[/tex]): 39.65 g
2. Mass of metal ([tex]\(m_{\text{metal}}\)[/tex]): 12.46 g
3. Change in temperature of water ([tex]\(\Delta T_{\text{water}}\)[/tex]): 0.7 [tex]\(^{\circ}\)[/tex]C
4. Change in temperature of metal ([tex]\(\Delta T_{\text{metal}}\)[/tex]): -76.7 [tex]\(^{\circ}\)[/tex]C
Substitute these values into the equation:
[tex]\[ c_{\text{Pb}} = \frac{-4.184 \times 39.65 \times 0.7}{12.46 \times -76.7} \][/tex]
#### Simplification (performed by calculator):
[tex]\[ c_{\text{Pb}} \approx 0.12 \text{ J/g} ^{\circ} \text{C} \][/tex]
Therefore, the specific heat values are:
- Copper ([tex]\(Cu\)[/tex]): [tex]\( 0.35 \text{ J/g} ^{\circ} \text{C} \)[/tex]
- Iron ([tex]\(Fe\)[/tex]): [tex]\( 0.44 \text{ J/g} ^{\circ} \text{C} \)[/tex]
- Lead ([tex]\(Pb\)[/tex]): [tex]\( 0.12 \text{ J/g} ^{\circ} \text{C} \)[/tex]
[tex]\[ c_{\text{metal}} = \frac{-c_{\text{water}} m_{\text{water}} \Delta T_{\text{water}}}{m_{\text{metal}} \Delta T_{\text{metal}}} \][/tex]
Given:
- The specific heat of water, [tex]\( c_{\text{water}} = 4.184 \text{ J/g} ^{\circ} \text{C} \)[/tex]
Let's calculate the specific heat values for each metal step-by-step.
### Copper (Cu):
1. Mass of water ([tex]\(m_{\text{water}}\)[/tex]): 40.13 g
2. Mass of metal ([tex]\(m_{\text{metal}}\)[/tex]): 12.14 g
3. Change in temperature of water ([tex]\(\Delta T_{\text{water}}\)[/tex]): 1.9 [tex]\(^{\circ}\)[/tex]C
4. Change in temperature of metal ([tex]\(\Delta T_{\text{metal}}\)[/tex]): -75.4 [tex]\(^{\circ}\)[/tex]C
Substitute these values into the equation:
[tex]\[ c_{\text{Cu}} = \frac{-4.184 \times 40.13 \times 1.9}{12.14 \times -75.4} \][/tex]
#### Simplification (performed by calculator):
[tex]\[ c_{\text{Cu}} \approx 0.35 \text{ J/g} ^{\circ} \text{C} \][/tex]
### Iron (Fe):
1. Mass of water ([tex]\(m_{\text{water}}\)[/tex]): 40.24 g
2. Mass of metal ([tex]\(m_{\text{metal}}\)[/tex]): 12.31 g
3. Change in temperature of water ([tex]\(\Delta T_{\text{water}}\)[/tex]): 2.4 [tex]\(^{\circ}\)[/tex]C
4. Change in temperature of metal ([tex]\(\Delta T_{\text{metal}}\)[/tex]): -75.1 [tex]\(^{\circ}\)[/tex]C
Substitute these values into the equation:
[tex]\[ c_{\text{Fe}} = \frac{-4.184 \times 40.24 \times 2.4}{12.31 \times -75.1} \][/tex]
#### Simplification (performed by calculator):
[tex]\[ c_{\text{Fe}} \approx 0.44 \text{ J/g} ^{\circ} \text{C} \][/tex]
### Lead (Pb):
1. Mass of water ([tex]\(m_{\text{water}}\)[/tex]): 39.65 g
2. Mass of metal ([tex]\(m_{\text{metal}}\)[/tex]): 12.46 g
3. Change in temperature of water ([tex]\(\Delta T_{\text{water}}\)[/tex]): 0.7 [tex]\(^{\circ}\)[/tex]C
4. Change in temperature of metal ([tex]\(\Delta T_{\text{metal}}\)[/tex]): -76.7 [tex]\(^{\circ}\)[/tex]C
Substitute these values into the equation:
[tex]\[ c_{\text{Pb}} = \frac{-4.184 \times 39.65 \times 0.7}{12.46 \times -76.7} \][/tex]
#### Simplification (performed by calculator):
[tex]\[ c_{\text{Pb}} \approx 0.12 \text{ J/g} ^{\circ} \text{C} \][/tex]
Therefore, the specific heat values are:
- Copper ([tex]\(Cu\)[/tex]): [tex]\( 0.35 \text{ J/g} ^{\circ} \text{C} \)[/tex]
- Iron ([tex]\(Fe\)[/tex]): [tex]\( 0.44 \text{ J/g} ^{\circ} \text{C} \)[/tex]
- Lead ([tex]\(Pb\)[/tex]): [tex]\( 0.12 \text{ J/g} ^{\circ} \text{C} \)[/tex]
