Answer :
To compute the specific heat values of each metal, we will use the formula provided:
[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] and [tex]\( \Delta T_{\text{water}} \)[/tex] are the mass and temperature change of water, respectively
- [tex]\( m_{\text{metal}} \)[/tex] and [tex]\( \Delta T_{\text{metal}} \)[/tex] are the mass and temperature change of the metal, respectively
Now let's go through the computations for each metal step-by-step.
### For Aluminum (Al)
Given:
- [tex]\( m_{\text{water}} = 39.85 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 4.7 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 11.98 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -72.9 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Al}} = \frac{-4.184 \cdot 39.85 \cdot 4.7}{11.98 \cdot (-72.9)} \][/tex]
The calculated specific heat capacity for Aluminum is:
[tex]\[ c_{\text{Al}} = 0.90 \ \text{J/g}^\circ\text{C} \][/tex]
### For Copper (Cu)
Given:
- [tex]\( m_{\text{water}} = 40.13 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 1.9 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.14 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.4 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Cu}} = \frac{-4.184 \cdot 40.13 \cdot 1.9}{12.14 \cdot (-75.4)} \][/tex]
The calculated specific heat capacity for Copper is:
[tex]\[ c_{\text{Cu}} = 0.35 \ \text{J/g}^\circ\text{C} \][/tex]
### For Iron (Fe)
Given:
- [tex]\( m_{\text{water}} = 40.24 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 2.4 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.31 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.1 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Fe}} = \frac{-4.184 \cdot 40.24 \cdot 2.4}{12.31 \cdot (-75.1)} \][/tex]
The calculated specific heat capacity for Iron is:
[tex]\[ c_{\text{Fe}} = 0.44 \ \text{J/g}^\circ\text{C} \][/tex]
### For Lead (Pb)
Given:
- [tex]\( m_{\text{water}} = 39.65 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 0.7 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.46 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -76.7 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Pb}} = \frac{-4.184 \cdot 39.65 \cdot 0.7}{12.46 \cdot (-76.7)} \][/tex]
The calculated specific heat capacity for Lead is:
[tex]\[ c_{\text{Pb}} = 0.12 \ \text{J/g}^\circ\text{C} \][/tex]
So, the specific heat capacities of the metals are:
- Aluminum (Al): [tex]\(0.90 \ \text{J/g}^\circ\text{C}\)[/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] and [tex]\( \Delta T_{\text{water}} \)[/tex] are the mass and temperature change of water, respectively
- [tex]\( m_{\text{metal}} \)[/tex] and [tex]\( \Delta T_{\text{metal}} \)[/tex] are the mass and temperature change of the metal, respectively
Now let's go through the computations for each metal step-by-step.
### For Aluminum (Al)
Given:
- [tex]\( m_{\text{water}} = 39.85 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 4.7 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 11.98 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -72.9 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Al}} = \frac{-4.184 \cdot 39.85 \cdot 4.7}{11.98 \cdot (-72.9)} \][/tex]
The calculated specific heat capacity for Aluminum is:
[tex]\[ c_{\text{Al}} = 0.90 \ \text{J/g}^\circ\text{C} \][/tex]
### For Copper (Cu)
Given:
- [tex]\( m_{\text{water}} = 40.13 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 1.9 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.14 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.4 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Cu}} = \frac{-4.184 \cdot 40.13 \cdot 1.9}{12.14 \cdot (-75.4)} \][/tex]
The calculated specific heat capacity for Copper is:
[tex]\[ c_{\text{Cu}} = 0.35 \ \text{J/g}^\circ\text{C} \][/tex]
### For Iron (Fe)
Given:
- [tex]\( m_{\text{water}} = 40.24 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 2.4 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.31 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -75.1 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Fe}} = \frac{-4.184 \cdot 40.24 \cdot 2.4}{12.31 \cdot (-75.1)} \][/tex]
The calculated specific heat capacity for Iron is:
[tex]\[ c_{\text{Fe}} = 0.44 \ \text{J/g}^\circ\text{C} \][/tex]
### For Lead (Pb)
Given:
- [tex]\( m_{\text{water}} = 39.65 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{water}} = 0.7 \ ^\circ\text{C} \)[/tex]
- [tex]\( m_{\text{metal}} = 12.46 \ \text{g} \)[/tex]
- [tex]\( \Delta T_{\text{metal}} = -76.7 \ ^\circ\text{C} \)[/tex]
Using the formula:
[tex]\[ c_{\text{Pb}} = \frac{-4.184 \cdot 39.65 \cdot 0.7}{12.46 \cdot (-76.7)} \][/tex]
The calculated specific heat capacity for Lead is:
[tex]\[ c_{\text{Pb}} = 0.12 \ \text{J/g}^\circ\text{C} \][/tex]
So, the specific heat capacities of the metals are:
- Aluminum (Al): [tex]\(0.90 \ \text{J/g}^\circ\text{C}\)[/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]
