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Sagot :
To calculate the specific heat ([tex]\(c\)[/tex]) of each metal, we will use the following equation:
[tex]\[ c_{\text{metal}} = \frac{-c_{\text{water}} m_{\text{water}} \Delta T_{\text{water}}}{m_{\text{metal}} \Delta T_{\text{metal}}} \][/tex]
Let's plug in the given values step-by-step:
1. Aluminum (Al):
[tex]\[ m_{\text{water}} = 39.85 \text{ g}, \quad m_{\text{metal}} = 11.98 \text{ g}, \quad \Delta T_{\text{water}} = 4.7 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -72.9 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (39.85 \, \text{g}) \times (4.7 \, ^\circ \text{C})}{(11.98 \, \text{g}) \times (-72.9 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.9 \, \text{J/g}^\circ \text{C} \][/tex]
2. Copper (Cu):
[tex]\[ m_{\text{water}} = 40.13 \text{ g}, \quad m_{\text{metal}} = 12.14 \text{ g}, \quad \Delta T_{\text{water}} = 1.9 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -75.4 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (40.13 \, \text{g}) \times (1.9 \, ^\circ \text{C})}{(12.14 \, \text{g}) \times (-75.4 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.35 \, \text{J/g}^\circ \text{C} \][/tex]
3. Iron (Fe):
[tex]\[ m_{\text{water}} = 40.24 \text{ g}, \quad m_{\text{metal}} = 12.31 \text{ g}, \quad \Delta T_{\text{water}} = 2.4 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -75.1 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (40.24 \, \text{g}) \times (2.4 \, ^\circ \text{C})}{(12.31 \, \text{g}) \times (-75.1 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.44 \, \text{J/g}^\circ \text{C} \][/tex]
4. Lead (Pb):
[tex]\[ m_{\text{water}} = 39.65 \text{ g}, \quad m_{\text{metal}} = 12.46 \text{ g}, \quad \Delta T_{\text{water}} = 0.7 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -76.7 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (39.65 \, \text{g}) \times (0.7 \, ^\circ \text{C})}{(12.46 \, \text{g}) \times (-76.7 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.12 \, \text{J/g}^\circ \text{C} \][/tex]
So, summarizing the specific heat values:
- Aluminum (Al): [tex]\(c \approx 0.9 \, \text{J/g}^\circ \text{C}\)[/tex]
- Copper (Cu): [tex]\(c \approx 0.35 \, \text{J/g}^\circ \text{C}\)[/tex]
- Iron (Fe): [tex]\(c \approx 0.44 \, \text{J/g}^\circ \text{C}\)[/tex]
- Lead (Pb): [tex]\(c \approx 0.12 \, \text{J/g}^\circ \text{C}\)[/tex]
[tex]\[ c_{\text{metal}} = \frac{-c_{\text{water}} m_{\text{water}} \Delta T_{\text{water}}}{m_{\text{metal}} \Delta T_{\text{metal}}} \][/tex]
Let's plug in the given values step-by-step:
1. Aluminum (Al):
[tex]\[ m_{\text{water}} = 39.85 \text{ g}, \quad m_{\text{metal}} = 11.98 \text{ g}, \quad \Delta T_{\text{water}} = 4.7 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -72.9 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (39.85 \, \text{g}) \times (4.7 \, ^\circ \text{C})}{(11.98 \, \text{g}) \times (-72.9 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.9 \, \text{J/g}^\circ \text{C} \][/tex]
2. Copper (Cu):
[tex]\[ m_{\text{water}} = 40.13 \text{ g}, \quad m_{\text{metal}} = 12.14 \text{ g}, \quad \Delta T_{\text{water}} = 1.9 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -75.4 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (40.13 \, \text{g}) \times (1.9 \, ^\circ \text{C})}{(12.14 \, \text{g}) \times (-75.4 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.35 \, \text{J/g}^\circ \text{C} \][/tex]
3. Iron (Fe):
[tex]\[ m_{\text{water}} = 40.24 \text{ g}, \quad m_{\text{metal}} = 12.31 \text{ g}, \quad \Delta T_{\text{water}} = 2.4 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -75.1 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (40.24 \, \text{g}) \times (2.4 \, ^\circ \text{C})}{(12.31 \, \text{g}) \times (-75.1 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.44 \, \text{J/g}^\circ \text{C} \][/tex]
4. Lead (Pb):
[tex]\[ m_{\text{water}} = 39.65 \text{ g}, \quad m_{\text{metal}} = 12.46 \text{ g}, \quad \Delta T_{\text{water}} = 0.7 ^\circ \text{C}, \quad \Delta T_{\text{metal}} = -76.7 ^\circ \text{C} \][/tex]
[tex]\[ c_{\text{metal}} = \frac{- (4.184 \, \text{J/g}^\circ \text{C}) \times (39.65 \, \text{g}) \times (0.7 \, ^\circ \text{C})}{(12.46 \, \text{g}) \times (-76.7 \, ^\circ \text{C})} \][/tex]
[tex]\[ c_{\text{metal}} \approx 0.12 \, \text{J/g}^\circ \text{C} \][/tex]
So, summarizing the specific heat values:
- Aluminum (Al): [tex]\(c \approx 0.9 \, \text{J/g}^\circ \text{C}\)[/tex]
- Copper (Cu): [tex]\(c \approx 0.35 \, \text{J/g}^\circ \text{C}\)[/tex]
- Iron (Fe): [tex]\(c \approx 0.44 \, \text{J/g}^\circ \text{C}\)[/tex]
- Lead (Pb): [tex]\(c \approx 0.12 \, \text{J/g}^\circ \text{C}\)[/tex]
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