Answered

Use your data, the equation to the right, and the specific heat of water ([tex]4.184 \, J/g \, ^{\circ}C[/tex]) to compute the specific heat values of each metal. Use a calculator and round to the nearest hundredth place.

Aluminum: [tex]c = \square \, J/g \, ^{\circ}C[/tex]
Copper: [tex]c = \square \, J/g \, ^{\circ}C[/tex]
Iron: [tex]c = \square \, J/g \, ^{\circ}C[/tex]
Lead: [tex]c = \square \, J/g \, ^{\circ}C[/tex]

\begin{tabular}{|c|c|c|c|c|}
\cline{2-5}
\multicolumn{1}{c|}{} & Al & Cu & Fe & Pb \\
\hline
[tex]$m_{\text{water}}(g)$[/tex] & 39.85 & 40.13 & 40.24 & 39.65 \\
\hline
[tex]$m_{\text{metal}}(g)$[/tex] & 11.98 & 12.14 & 12.31 & 12.46 \\
\hline
[tex]$\Delta T_{\text{water}}(^{\circ}C)$[/tex] & 4.7 & 1.9 & 2.4 & 0.7 \\
\hline
[tex]$\Delta T_{\text{metal}}(^{\circ}C)$[/tex] & -72.9 & -75.4 & -75.1 & -76.7 \\
\hline
\end{tabular}

Be sure to record all of these values in the data table.

[tex] c_{\text{metal}} = \frac{-c_{\text{water}} \, m_{\text{water}} \, \Delta T_{\text{water}}}{m_{\text{metal}} \, \Delta T_{\text{metal}}}[/tex]



Answer :

GinnyAnswer
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]