Answered

Step 8: Compute the Specific Heat of Each Metal

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

Aluminum: [tex]$c=$[/tex] [tex]$\square \, \text{J/g} \, ^{\circ} \text{C}$[/tex]

Copper: [tex]$c=$[/tex] [tex]$\square \, \text{J/g} \, ^{\circ} \text{C}$[/tex]

Iron: [tex]$c=$[/tex] [tex]$\square \, \text{J/g} \, ^{\circ} \text{C}$[/tex]

Lead: [tex]$c=$[/tex] [tex]$\square \, \text{J/g} \, ^{\circ} \text{C}$[/tex]

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

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 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]

Other Questions