Certainly! Let's break down the steps needed to find [tex]\(\Delta E_{\text{molar}}\)[/tex] for the combustion of hexane in the bomb calorimeter.
### Step 1: Calculate the change in temperature
We start by determining the change in temperature ([tex]\(\Delta T\)[/tex]) resulting from the combustion:
[tex]\[
\Delta T = T_{\text{final}} - T_{\text{initial}}
\][/tex]
[tex]\[
\Delta T = 38.84^{\circ} C - 25.84^{\circ} C = 13.00^{\circ} C
\][/tex]
### Step 2: Calculate the heat absorbed by the calorimeter
Next, use the heat capacity of the calorimeter to find the heat absorbed ([tex]\(q_{\text{calorimeter}}\)[/tex]):
[tex]\[
q_{\text{calorimeter}} = \text{heat capacity of calorimeter} \times \Delta T
\][/tex]
[tex]\[
q_{\text{calorimeter}} = 5.78 \, \text{kJ}/^{\circ}\text{C} \times 13.00^{\circ} \text{C} = 75.14 \, \text{kJ}
\][/tex]
### Step 3: Convert the mass of hexane to moles
We need to convert the given mass of hexane to moles using the molar mass of hexane ([tex]\(C_6H_{14}\)[/tex]). The molar mass of hexane is approximately 86.18 g/mol:
[tex]\[
\text{moles of hexane} = \frac{\text{mass of hexane}}{\text{molar mass of hexane}}
\][/tex]
[tex]\[
\text{moles of hexane} = \frac{1.548 \, \text{g}}{86.18 \, \text{g/mol}} = 0.01796 \, \text{mol}
\][/tex]
### Step 4: Calculate [tex]\(\Delta E_{\text{molar}}\)[/tex]
Finally, calculate [tex]\(\Delta E_{\text{molar}}\)[/tex], the energy change per mole of hexane:
[tex]\[
\Delta E_{\text{molar}} = \frac{q_{\text{calorimeter}}}{\text{moles of hexane}}
\][/tex]
[tex]\[
\Delta E_{\text{molar}} = \frac{75.14 \, \text{kJ}}{0.01796 \, \text{mol}} = 4183 \, \text{kJ/mol}
\][/tex]
Rounding off to three significant figures:
[tex]\[
\Delta E_{\text{molar}} = 4180 \, \text{kJ/mol}
\][/tex]
Therefore, [tex]\(\Delta E_{\text{molar}}\)[/tex] for the combustion of hexane in this example is [tex]\(4180 \, \text{kJ/mol}\)[/tex].
