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Consider the reaction:

[tex]\[ C_{12}H_{22}O_{11}(s) + 12 O_2(g) \rightarrow 12 CO_2(g) + 11 H_2O(l) \][/tex]

In which 10.0 g of sucrose, [tex]\[ C_{12}H_{22}O_{11} \][/tex], was burned in a bomb calorimeter with a heat capacity of [tex]\[ 7.50 \text{ kJ/} ^{\circ} \text{C} \][/tex]. The temperature increase inside the calorimeter was found to be [tex]\[ 22.0^{\circ} \text{C} \][/tex].

Calculate the change in internal energy, [tex]\[ \Delta E \][/tex], for this reaction per mole of sucrose.

Express the change in internal energy in kilojoules per mole to three significant figures.

Sagot :

GinnyAnswer
Given the problem, we need to calculate the change in internal energy, [tex]\(\Delta E\)[/tex], for the combustion reaction of sucrose in a bomb calorimeter. We are provided with the following data:
- Mass of sucrose: 10.0 grams
- Heat capacity of the calorimeter: 7.50 kJ/°C
- Temperature increase: 22.0 °C
- Molar mass of sucrose ([tex]\(\text{C}_{12}\text{H}_{22}\text{O}_{11}\)[/tex]): 342.30 g/mol

We proceed in a step-by-step manner as follows:

1. Calculate the heat absorbed by the calorimeter:

Given the heat capacity of the calorimeter ([tex]\(C\)[/tex]) and the temperature increase ([tex]\(\Delta T\)[/tex]), we use the formula:
[tex]\[ q = C \times \Delta T \][/tex]

Substituting the given values:
[tex]\[ q = 7.50 \, \text{kJ/°C} \times 22.0 \, \text{°C} = 165.0 \, \text{kJ} \][/tex]

Hence, the heat absorbed by the calorimeter is 165.0 kJ.

2. Calculate the number of moles of sucrose burned:

To find the number of moles, we use the molar mass of sucrose and the given mass:
[tex]\[ \text{moles of sucrose} = \frac{\text{mass of sucrose}}{\text{molar mass of sucrose}} \][/tex]

Substituting the values:
[tex]\[ \text{moles of sucrose} = \frac{10.0 \, \text{g}}{342.30 \, \text{g/mol}} \approx 0.0292 \, \text{mol} \][/tex]

So, 10.0 grams of sucrose corresponds to approximately 0.0292 moles.

3. Calculate the change in internal energy per mole of sucrose ([tex]\(\Delta E\)[/tex]):

The heat absorbed by the calorimeter represents the total heat released by the combustion of the given amount of sucrose. To find [tex]\(\Delta E\)[/tex] per mole, we divide the total heat absorbed by the number of moles of sucrose:
[tex]\[ \Delta E \, (\text{per mole}) = \frac{q}{\text{moles of sucrose}} \][/tex]

Substituting in the values:
[tex]\[ \Delta E \, (\text{per mole}) = \frac{165.0 \, \text{kJ}}{0.029214139643587496 \, \text{mol}} \approx 5647.95 \, \text{kJ/mol} \][/tex]

Thus, the change in internal energy [tex]\(\Delta E\)[/tex] for the reaction per mole of sucrose is [tex]\(5647.95\)[/tex] kJ/mol.

Expressing this to three significant figures, we get:
[tex]\[ \Delta E \approx 5650 \, \text{kJ/mol} \][/tex]

Therefore, the change in internal energy for this reaction per mole of sucrose is approximately [tex]\(5650 \, \text{kJ/mol}\)[/tex].
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