To solve the problem of calculating the heat of combustion of benzoic acid at constant volume and constant pressure, we will follow these steps:
### Given Data:
- Mass of benzoic acid (m) = 0.5 g
- Temperature rise (ΔT) = 0.55 °C
- Thermal capacity of the calorimeter (C) = 23.85 kJ
- Molar mass of benzoic acid (C₇H₆O₂) = 122 g/mol
### Step 1: Calculate the number of moles of benzoic acid
To find the number of moles (n) of benzoic acid:
\[
n = \frac{m}{\text{Molar mass}} = \frac{0.5 \, \text{g}}{122 \, \text{g/mol}} = 0.0041 \, \text{mol}
\]
### Step 2: Calculate the heat absorbed by the calorimeter (q) at constant volume
The heat absorbed by the calorimeter can be calculated using the formula:
\[
q = C \times \Delta T
\]
Substituting the values:
\[
q = 23.85 \, \text{kJ/°C} \times 0.55 \, °C = 13.1425 \, \text{kJ}
\]
### Step 3: Calculate the heat of combustion at constant volume (ΔU)
Since we are calculating the heat of combustion at constant volume, we have:
\[
\Delta U = \frac{q}{n}
\]
Substituting the values:
\[
\Delta U = \frac{13.1425 \, \text{kJ}}{0.0041 \, \text{mol}} = 3202.56 \, \text{kJ/mol}
\]
### Step 4: Calculate the change in gaseous moles (Δn)
For the combustion of benzoic acid, the balanced reaction is:
\[
C_7H_6O_2 + 7O_2 \rightarrow 7CO_2 + 3H_2O
\]
From the reaction, we see:
- Reactants: 1 (C₇H₆O₂) + 7 (O₂) = 8 moles
- Products: 7 (CO₂) + 3 (H₂O) = 10 moles
Thus, the change in gaseous moles (Δn) is:
\[
\Delta n = \text{moles of products} - \text{moles of reactants} = 10 - 8 = 2
\]
### Step 5: Calculate the heat of combustion at constant pressure (ΔH)
Using the relation:
\[
\Delta H = \Delta U + \Delta n \cdot R \cdot T
\]
Where:
- R = 8.314 J/(mol·K) = 0.008314 kJ/(mol·K)
- T = 15 °C = 288 K
Substituting the values:
\[
\Delta H = 3202.56 \, \text{kJ/mol} + (2 \cdot 0.008314 \, \text{kJ/(mol·K)} \cdot 288 \, \text{K})
\]
Calculating the second term:
\[
\Delta n \cdot R \cdot T = 2 \cdot 0.008314 \cdot 288 = 4.788 \, \text{kJ}
\]
Now substituting back:
\[
\Delta H = 3202.56 \, \text{kJ/mol} + 4.788 \, \text{kJ} = 3207.348 \, \text{kJ/mol}
\]
### Final Answers:
- Heat of combustion at constant volume (ΔU) = -3202.56 kJ/mol
- Heat of combustion at constant pressure (ΔH) = -3207.348 kJ/mol
