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Sagot :
To determine the change in energy ([tex]\(\Delta E_{\text{rxn}}\)[/tex]) for the combustion of naphthalene in a bomb calorimeter, we need to follow a step-by-step process involving the given data. Let's go through the calculations:
### Step 1: Calculate the Change in Temperature ([tex]\(\Delta T\)[/tex])
We are given the initial temperature ([tex]\(T_{\text{initial}}\)[/tex]) and the final temperature ([tex]\(T_{\text{final}}\)[/tex]) of the bomb calorimeter:
- Initial temperature, [tex]\(T_{\text{initial}} = 25.10^\circ C\)[/tex]
- Final temperature, [tex]\(T_{\text{final}} = 31.56^\circ C\)[/tex]
The change in temperature ([tex]\(\Delta T\)[/tex]) is calculated as:
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 31.56^\circ C - 25.10^\circ C = 6.46^\circ C \][/tex]
### Step 2: Calculate the Heat Absorbed by the Calorimeter ([tex]\(q_{\text{calorimeter}}\)[/tex])
The heat capacity of the bomb calorimeter ([tex]\(C_{\text{calorimeter}}\)[/tex]) is provided:
[tex]\[ C_{\text{calorimeter}} = 5.0 \, \text{kJ/}^\circ\text{C} \][/tex]
The heat absorbed by the calorimeter ([tex]\(q_{\text{calorimeter}}\)[/tex]) is given by:
[tex]\[ q_{\text{calorimeter}} = C_{\text{calorimeter}} \times \Delta T \][/tex]
Substituting the values:
[tex]\[ q_{\text{calorimeter}} = 5.0 \, \text{kJ/}^\circ\text{C} \times 6.46^\circ C = 32.30 \, \text{kJ} \][/tex]
### Step 3: Calculate the Number of Moles of Naphthalene
Given the mass of naphthalene ([tex]\(m_{\text{naphthalene}}\)[/tex]) is:
[tex]\[ m_{\text{naphthalene}} = 0.820 \, \text{g} \][/tex]
The molar mass of naphthalene ([tex]\(C_{10}H_8\)[/tex]) can be calculated as:
[tex]\[ \text{Molar mass of } C_{10}H_8 = (10 \times 12.01 \, \text{g/mol}) + (8 \times 1.008 \, \text{g/mol}) = 128.18 \, \text{g/mol} \][/tex]
The number of moles of naphthalene ([tex]\(n_{\text{naphthalene}}\)[/tex]) is:
[tex]\[ n_{\text{naphthalene}} = \frac{m_{\text{naphthalene}}}{\text{Molar mass of } C_{10}H_8} = \frac{0.820 \, \text{g}}{128.18 \, \text{g/mol}} = 0.00640 \, \text{mol} \][/tex]
### Step 4: Calculate [tex]\(\Delta E_{\text{rxn}}\)[/tex] per Mole
Finally, we determine the change in energy per mole of naphthalene burned:
[tex]\[ \Delta E_{\text{rxn}} = \frac{q_{\text{calorimeter}}}{n_{\text{naphthalene}}} \][/tex]
Substituting the values:
[tex]\[ \Delta E_{\text{rxn}} = \frac{32.30 \, \text{kJ}}{0.00640 \, \text{mol}} = 5048 \, \text{kJ/mol} \][/tex]
Thus, the change in energy ([tex]\(\Delta E_{\text{rxn}}\)[/tex]) for the combustion of naphthalene is:
[tex]\[ \Delta E_{\text{rxn}} = 5048 \, \text{kJ/mol} \][/tex]
### Summary
1. Change in Temperature, [tex]\(\Delta T\)[/tex]: [tex]\(6.46^\circ C\)[/tex]
2. Heat absorbed by the Calorimeter, [tex]\(q_{\text{calorimeter}}\)[/tex]: [tex]\(32.30 \, \text{kJ}\)[/tex]
3. Number of Moles of Naphthalene: [tex]\(0.00640 \, \text{mol}\)[/tex]
4. [tex]\(\Delta E_{\text{rxn}}\)[/tex]: [tex]\(5048 \, \text{kJ/mol}\)[/tex]
These calculations help us understand the thermal energy released in the combustion reaction of naphthalene.
### Step 1: Calculate the Change in Temperature ([tex]\(\Delta T\)[/tex])
We are given the initial temperature ([tex]\(T_{\text{initial}}\)[/tex]) and the final temperature ([tex]\(T_{\text{final}}\)[/tex]) of the bomb calorimeter:
- Initial temperature, [tex]\(T_{\text{initial}} = 25.10^\circ C\)[/tex]
- Final temperature, [tex]\(T_{\text{final}} = 31.56^\circ C\)[/tex]
The change in temperature ([tex]\(\Delta T\)[/tex]) is calculated as:
[tex]\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 31.56^\circ C - 25.10^\circ C = 6.46^\circ C \][/tex]
### Step 2: Calculate the Heat Absorbed by the Calorimeter ([tex]\(q_{\text{calorimeter}}\)[/tex])
The heat capacity of the bomb calorimeter ([tex]\(C_{\text{calorimeter}}\)[/tex]) is provided:
[tex]\[ C_{\text{calorimeter}} = 5.0 \, \text{kJ/}^\circ\text{C} \][/tex]
The heat absorbed by the calorimeter ([tex]\(q_{\text{calorimeter}}\)[/tex]) is given by:
[tex]\[ q_{\text{calorimeter}} = C_{\text{calorimeter}} \times \Delta T \][/tex]
Substituting the values:
[tex]\[ q_{\text{calorimeter}} = 5.0 \, \text{kJ/}^\circ\text{C} \times 6.46^\circ C = 32.30 \, \text{kJ} \][/tex]
### Step 3: Calculate the Number of Moles of Naphthalene
Given the mass of naphthalene ([tex]\(m_{\text{naphthalene}}\)[/tex]) is:
[tex]\[ m_{\text{naphthalene}} = 0.820 \, \text{g} \][/tex]
The molar mass of naphthalene ([tex]\(C_{10}H_8\)[/tex]) can be calculated as:
[tex]\[ \text{Molar mass of } C_{10}H_8 = (10 \times 12.01 \, \text{g/mol}) + (8 \times 1.008 \, \text{g/mol}) = 128.18 \, \text{g/mol} \][/tex]
The number of moles of naphthalene ([tex]\(n_{\text{naphthalene}}\)[/tex]) is:
[tex]\[ n_{\text{naphthalene}} = \frac{m_{\text{naphthalene}}}{\text{Molar mass of } C_{10}H_8} = \frac{0.820 \, \text{g}}{128.18 \, \text{g/mol}} = 0.00640 \, \text{mol} \][/tex]
### Step 4: Calculate [tex]\(\Delta E_{\text{rxn}}\)[/tex] per Mole
Finally, we determine the change in energy per mole of naphthalene burned:
[tex]\[ \Delta E_{\text{rxn}} = \frac{q_{\text{calorimeter}}}{n_{\text{naphthalene}}} \][/tex]
Substituting the values:
[tex]\[ \Delta E_{\text{rxn}} = \frac{32.30 \, \text{kJ}}{0.00640 \, \text{mol}} = 5048 \, \text{kJ/mol} \][/tex]
Thus, the change in energy ([tex]\(\Delta E_{\text{rxn}}\)[/tex]) for the combustion of naphthalene is:
[tex]\[ \Delta E_{\text{rxn}} = 5048 \, \text{kJ/mol} \][/tex]
### Summary
1. Change in Temperature, [tex]\(\Delta T\)[/tex]: [tex]\(6.46^\circ C\)[/tex]
2. Heat absorbed by the Calorimeter, [tex]\(q_{\text{calorimeter}}\)[/tex]: [tex]\(32.30 \, \text{kJ}\)[/tex]
3. Number of Moles of Naphthalene: [tex]\(0.00640 \, \text{mol}\)[/tex]
4. [tex]\(\Delta E_{\text{rxn}}\)[/tex]: [tex]\(5048 \, \text{kJ/mol}\)[/tex]
These calculations help us understand the thermal energy released in the combustion reaction of naphthalene.
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