Question

Isooctane is a primary component of gasoline and gives gasoline its octane rating. Burning \(1.00 \mathrm{~mL}\) of isooctane \((d=0.688 \mathrm{~g} / \mathrm{mL})\) releases \(33.0 \mathrm{~kJ}\) of heat. When \(10.00 \mathrm{~mL}\) of is ooctane is burned in a bomb calorime- ter, the temperature in the bomb rises from \(23.2^{\circ} \mathrm{C}\) to \(66.5^{\circ} \mathrm{C}\). What is the heat capacity of the bomb calorimeter?

Short answer

Answer: The heat capacity of the bomb calorimeter is 7.62 kJ/°C.

Step-by-step solution

Calculate the heat released when burning 10.00 mL of isooctane

We are given that burning 1.00 mL of isooctane releases 33.0 kJ of heat. Therefore, we can find the heat released when burning 10.00 mL of isooctane by multiplying the heat released for 1.00 mL by 10.00 mL: \(q = 33.0 \mathrm{~kJ/mL} \times 10.00 \mathrm{~mL} = 330.0 \mathrm{~kJ}\)

Calculate the change in temperature

The initial temperature of the bomb calorimeter is \(23.2^{\circ} \mathrm{C}\), and the final temperature is \(66.5^{\circ} \mathrm{C}\). To find the change in temperature, we can subtract the initial temperature from the final temperature: \(\Delta T = T_f - T_i = 66.5^{\circ} \mathrm{C} - 23.2^{\circ} \mathrm{C} = 43.3^{\circ} \mathrm{C}\)

Calculate the heat capacity of the bomb calorimeter

Now that we have the heat released (\(q = 330.0 \mathrm{~kJ}\)) and the change in temperature (\(\Delta T = 43.3^{\circ} \mathrm{C}\)), we can use the formula \(q = C \cdot \Delta T\) to find the heat capacity (C) of the bomb calorimeter: \(C = \frac{q}{\Delta T} = \frac{330.0 \mathrm{~kJ}}{43.3^{\circ} \mathrm{C}} = 7.62 \frac{\mathrm{kJ}}{^\circ \mathrm{C}}\) The heat capacity of the bomb calorimeter is 7.62 kJ/\(^{\circ} \mathrm{C}\).

Key concepts

Heat Capacity

Heat capacity is a fundamental concept in calorimetry that measures the amount of heat required to change a substance's temperature by a certain amount. It is an essential factor when dealing with thermal energy transfer and can be found in the formula:
\[ q = C \cdot \Delta T \]where:

• \( q \) is the heat absorbed or released
• \( C \) is the heat capacity
• \( \Delta T \) is the change in temperature

Heat capacity depends on the material and the amount of substance present. In calorimetry, knowing the heat capacity of a device, like a bomb calorimeter, helps in determining the energy changes during chemical reactions.
In our exercise, the heat capacity of a bomb calorimeter is calculated to understand energy changes when isooctane is burned. By determining how much thermal energy causes the recorded temperature changes, we gain insight into the process's efficiency.

Bomb Calorimeter

A bomb calorimeter is a device used in calorimetry to measure the heat of combustion of a substance. It's a crucial tool in chemistry and physics for studying energy transformations. Here's how it typically works:

• The substance, in this case, isooctane, is placed in a sealed container called the "bomb."
• The bomb is filled with oxygen to support combustion.
• The bomb is submerged in a water-filled container, and the initial temperature is recorded.
• After ignition, as the substance burns, the heat is transferred to the surrounding water.
• The temperature rise in the water is measured to calculate the total energy released.

The bomb calorimeter can offer precise measurements because the entire system is insulated, minimizing heat loss to the environment. This accuracy is essential for determining precise enthalpy changes in chemical reactions.

Isooctane Combustion

Isooctane is a significant compound in the gasoline industry, contributing to its octane rating, which indicates the fuel's ability to resist knocking during combustion. When isooctane combusts, it reacts with oxygen to produce carbon dioxide, water, and heat.
In our study, burning 1.00 mL of isooctane releases 33.0 kJ of heat. This controlled combustion is performed in a bomb calorimeter, which can safely manage the high energy output. The amount of heat given off during the burning process is calculated to determine the substance's energy content.
This study's focus on the energy yielded by isooctane combustion helps in understanding its efficiency as a fuel. The data can also be used to calculate energy requirements and emissions for various applications.

Temperature Change

Temperature change is a direct indication of energy transfer in calorimetry. It's determined by measuring the initial and final temperatures using thermometers. The temperature change, represented by \( \Delta T \), is a key variable in calculating heat capacity.
In the given exercise, the initial temperature in the bomb calorimeter was \(23.2^{\circ} \mathrm{C}\) and rose to \(66.5^{\circ} \mathrm{C}\). These values are subtracted to find the actual change in temperature:
\[ \Delta T = 66.5^{\circ} \mathrm{C} - 23.2^{\circ} \mathrm{C} = 43.3^{\circ} \mathrm{C} \]This recorded temperature rise is crucial as it helps determine the amount of heat absorbed by the calorimeter, indicated by the equation \( q = C \cdot \Delta T \), which further assists in calculating the device's heat capacity.