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Sagot :
To calculate the specific heat values of each metal using the provided data and the equation [tex]\( c_{\text{metal}} = \frac{-c_{\text{water}} m_{\text{water}} \Delta T_{\text{water}}}{m_{\text{metal}} \Delta T_{\text{metal}}} \)[/tex], we'll follow these steps for each metal:
Given:
- [tex]\( c_{\text{water}} = 4.184 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
### Aluminum (Al)
1. Collect data:
- [tex]\( m_{\text{water}} = 39.85 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 11.98 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 4.7 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -72.9 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 39.85 \times 4.7}{11.98 \times -72.9} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.90 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Copper (Cu)
1. Collect data:
- [tex]\( m_{\text{water}} = 40.13 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.14 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 1.9 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.4 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 40.13 \times 1.9}{12.14 \times -75.4} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.35 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Iron (Fe)
1. Collect data:
- [tex]\( m_{\text{water}} = 40.24 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.31 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 2.4 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.1 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 40.24 \times 2.4}{12.31 \times -75.1} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.44 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Lead (Pb)
1. Collect data:
- [tex]\( m_{\text{water}} = 39.65 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.46 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 0.7 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -76.7 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 39.65 \times 0.7}{12.46 \times -76.7} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.12 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Final Answers:
- Aluminum (Al): [tex]\( c = 0.90 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
- Copper (Cu): [tex]\( c = 0.35 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
- Iron (Fe): [tex]\( c = 0.44 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
- Lead (Pb): [tex]\( c = 0.12 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
Make sure to record these computed values in the provided data table.
Given:
- [tex]\( c_{\text{water}} = 4.184 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
### Aluminum (Al)
1. Collect data:
- [tex]\( m_{\text{water}} = 39.85 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 11.98 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 4.7 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -72.9 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 39.85 \times 4.7}{11.98 \times -72.9} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.90 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Copper (Cu)
1. Collect data:
- [tex]\( m_{\text{water}} = 40.13 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.14 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 1.9 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.4 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 40.13 \times 1.9}{12.14 \times -75.4} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.35 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Iron (Fe)
1. Collect data:
- [tex]\( m_{\text{water}} = 40.24 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.31 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 2.4 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.1 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 40.24 \times 2.4}{12.31 \times -75.1} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.44 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Lead (Pb)
1. Collect data:
- [tex]\( m_{\text{water}} = 39.65 \, \text{g} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.46 \, \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 0.7 \, ^\circ \text{C} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -76.7 \, ^\circ \text{C} \)[/tex]
2. Use the formula:
[tex]\[ c_{\text{metal}} = \frac{- 4.184 \times 39.65 \times 0.7}{12.46 \times -76.7} \][/tex]
3. Calculate:
[tex]\[ c_{\text{metal}} \approx 0.12 \, \text{J}/\text{g}^{\circ} \text{C} \][/tex]
### Final Answers:
- Aluminum (Al): [tex]\( c = 0.90 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
- Copper (Cu): [tex]\( c = 0.35 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
- Iron (Fe): [tex]\( c = 0.44 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
- Lead (Pb): [tex]\( c = 0.12 \, \text{J}/\text{g}^{\circ} \text{C} \)[/tex]
Make sure to record these computed values in the provided data table.
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