Let's find [tex]\(\Delta E_{\text{comb}}\)[/tex] for the combustion of naphthalene step-by-step. Here's the detailed solution:
1. Identify Given Variables:
- Mass of naphthalene, [tex]\( \text{mass}_{\text{naphthalene}} = 0.820 \, \text{grams} \)[/tex]
- Initial temperature, [tex]\( T_{\text{initial}} = 25.10^\circ \text{C} \)[/tex]
- Final temperature, [tex]\( T_{\text{final}} = 31.56^\circ \text{C} \)[/tex]
- Heat capacity of the calorimeter, [tex]\( C_{\text{calorimeter}} = 5.11 \, \text{kJ/}^\circ \text{C} \)[/tex]
- Molar mass of naphthalene, [tex]\( \text{MM}_{\text{naphthalene}} = 128.17 \, \text{g/mol} \)[/tex]
2. Calculate the Temperature Change:
[tex]\[
\Delta T = T_{\text{final}} - T_{\text{initial}}
\][/tex]
Substituting the given temperatures:
[tex]\[
\Delta T = 31.56^\circ \text{C} - 25.10^\circ \text{C} = 6.46^\circ \text{C}
\][/tex]
3. Calculate the Energy Change in kJ (ΔE):
[tex]\[
\Delta E = C_{\text{calorimeter}} \times \Delta T
\][/tex]
Substituting the given values:
[tex]\[
\Delta E = 5.11 \, \text{kJ/}^\circ \text{C} \times 6.46^\circ \text{C} = 33.01 \, \text{kJ}
\][/tex]
4. Convert the Mass of Naphthalene to Moles:
[tex]\[
\text{Moles}_{\text{naphthalene}} = \frac{\text{mass}_{\text{naphthalene}}}{\text{MM}_{\text{naphthalene}}}
\][/tex]
Substituting the given values:
[tex]\[
\text{Moles}_{\text{naphthalene}} = \frac{0.820 \, \text{grams}}{128.17 \, \text{g/mol}} = 0.00640 \, \text{moles}
\][/tex]
5. Calculate the Change in Energy per Mole:
[tex]\[
\Delta E_{\text{comb}} = \frac{\Delta E}{\text{Moles}_{\text{naphthalene}}}
\][/tex]
Substituting the calculated values:
[tex]\[
\Delta E_{\text{comb}} = \frac{33.01 \, \text{kJ}}{0.00640 \, \text{moles}} = 5159.72 \, \text{kJ/mol}
\][/tex]
6. Result:
Rounding to three significant figures:
[tex]\[
\Delta E_{\text{comb}} = 5159.718 \, \text{kJ/mol}
\][/tex]
Therefore, the change in energy for the combustion of naphthalene is [tex]\( \Delta E_{\text{comb}} = 5159.718 \, \text{kJ/mol} \)[/tex].
