We will use the principle of conservation of energy to solve this calorimetry problem. The heat released by the burning oil is absorbed by the water and the equivalent metal parts of the calorimeter.

Given data:

• Volume of oil: $0.50$ mL
• Caloric value of oil: $-8.1$ kcal/mL (negative sign indicates heat released)
• Mass of water: $2.15$ kg
• Mass of equivalent metal parts: $0.50$ kg
• Initial temperature: $25^\circ C$
• Specific heat capacity of water: $1$ kcal/kg·°C
• Total mass of water and equivalent metal parts:
  $m_{\text{total}} = 2.15 + 0.50 = 2.65 \text{ kg}$

Step 1: Calculate the total heat released by burning oil

$Q_{\text{oil}} = (\text{caloric value}) \times (\text{volume of oil})$ $Q_{\text{oil}} = (-8.1 \, \text{kcal/mL}) \times (0.50 \, \text{mL})$ $Q_{\text{oil}} = -4.05 \, \text{kcal}$

Step 2: Use heat transfer equation

Since the heat is fully absorbed by the water and calorimeter, we apply:

$Q = m c \Delta T$

where
$Q$ is heat absorbed,
$m$ is the total mass,
$c$ is the specific heat of water,
$\Delta T$ is the change in temperature.

Rearranging for $\Delta T$:

$\Delta T = \frac{Q}{m c}$

Substituting values:

$\Delta T = \frac{4.05}{2.65 \times 1}$

$\Delta T = \frac{4.05}{2.65} \approx 1.53^\circ C$

Step 3: Final Temperature

$T_{\text{final}} = T_{\text{initial}} + \Delta T$

$T_{\text{final}} = 25 + 1.53$

$T_{\text{final}} \approx 26.53^\circ C$

Final Answer:

The temperature after equilibrium is attained is approximately
$\mathbf{26.53^\circ C}$

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Would you like me to break anything down further?

Here are five related questions you might find interesting:

1. How would the final temperature change if the calorimeter had a different heat capacity?
2. What happens if some heat is lost to the surroundings?
3. How would the result change if we burned more oil?
4. Can we use this method to determine the energy content of different fuels?
5. How does a bomb calorimeter improve the accuracy of such experiments?

Tip: Always ensure the calorimeter is well-insulated to minimize heat loss and improve accuracy! 🔥