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Sagot :
To compute the specific heat values of each metal, we'll use the given data and the provided formula for specific heat capacity:
[tex]\[ c_{\text{metal}} = \frac{-c_{\text{water}} \cdot m_{\text{water}} \cdot \Delta T_{\text{water}}}{m_{\text{metal}} \cdot \Delta T_{\text{metal}}} \][/tex]
where:
- [tex]\( c_{\text{water}} = 4.184 \, \text{J/g}^\circ \text{C} \)[/tex]
- [tex]\( m_{\text{water}} \)[/tex] is the mass of water in grams
- [tex]\( m_{\text{metal}} \)[/tex] is the mass of the metal in grams
- [tex]\( \Delta T_{\text{water}} \)[/tex] is the change in temperature of the water in degrees Celsius
- [tex]\( \Delta T_{\text{metal}} \)[/tex] is the change in temperature of the metal in degrees Celsius
Let's calculate the specific heat for each metal and round to the nearest hundredth place.
### Aluminum (Al)
Given:
- [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]
[tex]\[ c_{\text{Al}} = \frac{-4.184 \cdot 39.85 \cdot 4.7}{11.98 \cdot (-72.9)} \][/tex]
[tex]\[ c_{\text{Al}} \approx 0.9 \, \text{J/g}^\circ \text{C} \][/tex]
### Copper (Cu)
Given:
- [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]
[tex]\[ c_{\text{Cu}} = \frac{-4.184 \cdot 40.13 \cdot 1.9}{12.14 \cdot (-75.4)} \][/tex]
[tex]\[ c_{\text{Cu}} \approx 0.35 \, \text{J/g}^\circ \text{C} \][/tex]
### Iron (Fe)
Given:
- [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]
[tex]\[ c_{\text{Fe}} = \frac{-4.184 \cdot 40.24 \cdot 2.4}{12.31 \cdot (-75.1)} \][/tex]
[tex]\[ c_{\text{Fe}} \approx 0.44 \, \text{J/g}^\circ \text{C} \][/tex]
### Lead (Pb)
Given:
- [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]
[tex]\[ c_{\text{Pb}} = \frac{-4.184 \cdot 39.65 \cdot 0.7}{12.46 \cdot (-76.7)} \][/tex]
[tex]\[ c_{\text{Pb}} \approx 0.12 \, \text{J/g}^\circ \text{C} \][/tex]
### Summary
- Aluminum (Al): [tex]\( 0.9 \, \text{J/g}^\circ \text{C} \)[/tex]
- Copper (Cu): [tex]\( 0.35 \, \text{J/g}^\circ \text{C} \)[/tex]
- Iron (Fe): [tex]\( 0.44 \, \text{J/g}^\circ \text{C} \)[/tex]
- Lead (Pb): [tex]\( 0.12 \, \text{J/g}^\circ \text{C} \)[/tex]
[tex]\[ c_{\text{metal}} = \frac{-c_{\text{water}} \cdot m_{\text{water}} \cdot \Delta T_{\text{water}}}{m_{\text{metal}} \cdot \Delta T_{\text{metal}}} \][/tex]
where:
- [tex]\( c_{\text{water}} = 4.184 \, \text{J/g}^\circ \text{C} \)[/tex]
- [tex]\( m_{\text{water}} \)[/tex] is the mass of water in grams
- [tex]\( m_{\text{metal}} \)[/tex] is the mass of the metal in grams
- [tex]\( \Delta T_{\text{water}} \)[/tex] is the change in temperature of the water in degrees Celsius
- [tex]\( \Delta T_{\text{metal}} \)[/tex] is the change in temperature of the metal in degrees Celsius
Let's calculate the specific heat for each metal and round to the nearest hundredth place.
### Aluminum (Al)
Given:
- [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]
[tex]\[ c_{\text{Al}} = \frac{-4.184 \cdot 39.85 \cdot 4.7}{11.98 \cdot (-72.9)} \][/tex]
[tex]\[ c_{\text{Al}} \approx 0.9 \, \text{J/g}^\circ \text{C} \][/tex]
### Copper (Cu)
Given:
- [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]
[tex]\[ c_{\text{Cu}} = \frac{-4.184 \cdot 40.13 \cdot 1.9}{12.14 \cdot (-75.4)} \][/tex]
[tex]\[ c_{\text{Cu}} \approx 0.35 \, \text{J/g}^\circ \text{C} \][/tex]
### Iron (Fe)
Given:
- [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]
[tex]\[ c_{\text{Fe}} = \frac{-4.184 \cdot 40.24 \cdot 2.4}{12.31 \cdot (-75.1)} \][/tex]
[tex]\[ c_{\text{Fe}} \approx 0.44 \, \text{J/g}^\circ \text{C} \][/tex]
### Lead (Pb)
Given:
- [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]
[tex]\[ c_{\text{Pb}} = \frac{-4.184 \cdot 39.65 \cdot 0.7}{12.46 \cdot (-76.7)} \][/tex]
[tex]\[ c_{\text{Pb}} \approx 0.12 \, \text{J/g}^\circ \text{C} \][/tex]
### Summary
- Aluminum (Al): [tex]\( 0.9 \, \text{J/g}^\circ \text{C} \)[/tex]
- Copper (Cu): [tex]\( 0.35 \, \text{J/g}^\circ \text{C} \)[/tex]
- Iron (Fe): [tex]\( 0.44 \, \text{J/g}^\circ \text{C} \)[/tex]
- Lead (Pb): [tex]\( 0.12 \, \text{J/g}^\circ \text{C} \)[/tex]
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