Question

Isooctane, \(\mathrm{C}_{8} \mathrm{H}_{18}\), is a component of gasoline. When \(0.500 \mathrm{~g}\) of isooctane is burned, \(24.06 \mathrm{~kJ}\) of heat is given off. If \(10.00 \mathrm{mg}\) of isooctane is burned in a bomb calorimeter (heat capacity \(=5175 \mathrm{~J} /{ }^{\circ} \mathrm{C}\) ) initially at \(23.6^{\circ} \mathrm{C}\), what is the temperature of the calorimeter when reaction is complete?

Short answer

Answer: The final temperature of the calorimeter when the reaction is complete is 23.693°C.

Step-by-step solution

Calculate the heat released by the reaction

To find the heat released, we can use the information given: Heat released by burning \(0.500 \mathrm{g}\) of isooctane = \(24.06 \mathrm{k J}\) Amount of isooctane used in the reaction: \(10.00 \mathrm{~mg}\) First, we need to convert the mass of isooctane burned in the bomb calorimeter to grams: \(10.00 \mathrm{~mg}\) = \(0.0100 \mathrm{~g}\) Next, calculate the heat released when burning \(0.0100 \mathrm{~g}\) of isooctane by using the given proportion: \(\frac{24.06 \mathrm{~kJ}}{0.500 \mathrm{~g}} = \frac{Q}{0.0100 \mathrm{~g}}\) Solve for \(Q\): \(Q = \frac{24.06 \mathrm{~kJ}}{0.500 \mathrm{~g}} \times 0.0100 \mathrm{~g} = 0.4812 \mathrm{~kJ}\) Now, convert the heat released to Joules: Heat released, \(Q = 0.4812 \mathrm{~kJ} \times \frac{1000 \mathrm{~J}}{1 \mathrm{kJ}} = 481.2 \mathrm{~J}\)

Calculate the temperature increase in the calorimeter

Using the heat released (481.2 J), we can calculate the temperature increase in the calorimeter: \(\Delta T = \frac{Q}{C}\) where \(C\) is the heat capacity of the calorimeter (\(5175 \mathrm{~J}/^{\circ} \mathrm{C}\)), and \(\Delta T\) is the temperature increase. \(\Delta T = \frac{481.2 \mathrm{~J}}{5175 \mathrm{~J}/^{\circ} \mathrm{C}} = 0.0930 \, ^{\circ} \mathrm{C}\)

Find the final temperature of the calorimeter

Finally, we can find the final temperature of the calorimeter by adding the initial temperature and the temperature increase: Final temperature = Initial temperature + \(\Delta T\) = \(23.6 ^{\circ} \mathrm{C} + 0.0930 \, ^{\circ} \mathrm{C} = 23.693 \, ^{\circ} \mathrm{C}\) Therefore, the temperature of the calorimeter when the reaction is complete is \(23.693 ^{\circ} \mathrm{C}\).

Key concepts

Energy Transfer Calculations

Understanding the basics of energy transfer is essential for comprehending many processes in chemistry, one of which is calorimetry, a technique used to measure the heat of chemical reactions. Heat, a form of energy, always flows from a hotter object to a cooler one, and the ability to calculate this transfer is fundamental in thermochemistry.

In the exercise provided, we calculate the quantity of heat (Q) released by the combustion of a specific amount of isooctane, a component of gasoline. To do this accurately, one must utilize the energy released per gram of isooctane and scale it to the smaller amount used in this experiment. Through proportional calculations, we determine the energy exchanged during the reaction in Joules or kilojoules.

The mathematical relationship here is straightforward: if burning x grams of a substance releases y Joules, then burning 1 gram would release y/x Joules, and we can scale this to any mass of the substance. By using this type of energy transfer calculation, students can predict the outcome of various reactions, such as the heat produced or absorbed during a reaction, which is absolutely crucial in fields like energy production and management.

Heat Capacity

Heat capacity is a concept that often puzzles students, but it is simply a material's ability to absorb heat energy. The higher the heat capacity, the more energy it can absorb without significantly increasing in temperature. It's measured in joules per degree Celsius (J/°C) in the SI system, indicating the amount of energy needed to raise the temperature of the substance by one degree Celsius.

Understanding heat capacity is crucial when working with calorimetry experiments. It allows scientists and students alike to predict how much the temperature of a substance will increase when a known amount of energy is added. In our exercise, the bomb calorimeter has a heat capacity of 5175 J/°C, which means for every 5175 joules of heat energy introduced to the system, the temperature will rise by 1°C. This property serves as a vital factor in calculating the final temperature of the calorimeter after the energy from the burning isooctane gets absorbed.

By dividing the heat released by the heat capacity (Q/C), we find the temperature change. This outcome is an excellent application of the concept, showing how heat capacity plays a part in understanding and controlling reactions in real-world scenarios.

Thermochemistry

Thermochemistry is a branch of chemistry concerned with the study of the energy changes that occur during chemical reactions and physical transformations. It is based on the first law of thermodynamics, which asserts that energy cannot be created or destroyed, only transformed.

In the context of calorimetry, thermochemistry focuses on the heat absorbed or released by a system. This heat is often observed as a temperature change in a substance. The science of thermochemistry explains why certain reactions are exothermic (releasing heat) or endothermic (absorbing heat).

The exercise demonstrates a classic thermochemical problem: calculating the temperature change resulting from the exothermic reaction of burning isooctane. We have a known initial temperature, the amount of heat released, and the calorimeter's ability to absorb heat (its heat capacity). With these parameters, we can apply thermochemical principles to determine the new equilibrium temperature after the reaction has ceased, which is key to understanding how much energy is produced during the combustion of gasoline, an essential part of energy systems and engines.

In summary, the integration of energy transfer calculations, heat capacity, and thermochemical concepts allows students to paint a full picture of the heat dynamics in chemistry, preparing them for more complex applications in the field.