Question

The heat capacity of a bomb calorimeter was determined by burning \(6.79 \mathrm{~g}\) methane (energy of combustion \(=-802 \mathrm{~kJ} /\) \(\mathrm{mol} \mathrm{CH}_{4}\) ) in the bomb. The temperature changed by \(10.8^{\circ} \mathrm{C} .\) a. What is the heat capacity of the bomb? b. A \(12.6-\mathrm{g}\) sample of acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}\), produced a temperature increase of \(16.9^{\circ} \mathrm{C}\) in the same calorimeter. What is the energy of combustion of acetylene (in \(\mathrm{kJ} / \mathrm{mol}\) )?

Short answer

The heat capacity of the bomb calorimeter is \(C_{bomb} = \frac{-802\text{ kJ/mol} \times \frac{6.79\text{ g}}{16.05 \text{ g/mol}}}{10.8^{\circ}\text{C}}\). The energy of combustion of acetylene is \(\frac{C_{bomb} \times 16.9 ^{\circ}\text{C}}{\frac{12.6\text{ g}}{26.04 \text{ g/mol}}}\).

Step-by-step solution

Calculate heat released by methane

We begin by finding the number of moles of methane (CH4) and then calculating the heat released by it. The molar mass of CH4 is 12.01 + 4(1.01) = 16.05 g/mol. Moles of methane = \(\frac{6.79\text{ g}}{16.05 \text{ g/mol}}\)

Calculate the heat generated from the burning methane

Using the energy of combustion given, we can calculate the total heat \(q_{CH4}\) generated by burning methane: \(q_{CH4} = -802\text{ kJ/mol} \times \frac{6.79\text{ g}}{16.05 \text{ g/mol}}\)

Calculate the heat capacity of the bomb

Knowing the heat generated and the temperature change, we can now calculate the heat capacity of the bomb calorimeter, C_bomb. The formula for heat capacity is: C_bomb = \(\frac{q_{CH4}}{\Delta T}\)

Calculate moles of acetylene

Now we can move on to part b of the problem. To find the energy of combustion for acetylene (C2H2), we first need to calculate the number of moles. The molar mass of C2H2 is 2(12.01) + 2(1.01) = 26.04 g/mol. Moles of acetylene = \(\frac{12.6\text{ g}}{26.04 \text{ g/mol}}\)

Calculate the heat generated by burning acetylene

Using the heat capacity of the bomb and the change in temperature for acetylene, we can calculate the total heat generated by burning acetylene, \(q_{C2H2}\), as: \(q_{C2H2} = C_bomb \times 16.9 ^{\circ}\text{C}\)

Calculate the energy of combustion of acetylene

Lastly, we can calculate the energy of combustion of acetylene by dividing \(q_{C2H2}\) by the moles of acetylene: \( \text{Energy of combustion} = \frac{q_{C2H2}} {\text{Moles of acetylene}}\)

Key concepts

Heat Capacity

Heat capacity is a fundamental property that measures how much heat energy is required to change the temperature of an object. In the context of a bomb calorimeter, the heat capacity helps us understand how well it can absorb the energy released during a reaction, like combustion.
In simple terms, heat capacity ( C ) is the amount of heat needed to raise the temperature of a system by one degree Celsius.
For the bomb calorimeter, it is calculated using the formula:

• C = \( \frac{q}{\Delta T} \)

where \( q \) is the heat exchanged and \( \Delta T \) is the change in temperature.
Knowing the heat capacity of a calorimeter allows scientists to determine the energy changes in reactions accurately, as any increase in temperature can be translated back to energy (in Joules or kJ ).
This aids in understanding how energetic particular reactions can be and allows us to compare different reactions effectively.

Energy of Combustion

The energy of combustion refers to the total energy released when a substance undergoes complete combustion with oxygen.
It is an indicator of how much heat a substance can produce when burned.
In the bomb calorimeter setup, we often look at this property to understand the efficiency of any given fuel.

• When methane burns, for example, the energy released is -802 kJ/mol . This negative sign indicates an exothermic reaction, meaning heat is given off.
• By measuring the temperature change in the calorimeter, one can calculate how much energy is released using the combustion energy per mole of the substance.

Such measurements are crucial for evaluating the potential of different fuels to produce energy, helping in sectors like energy generation and chemical synthesis.

Methane Combustion

Methane combustion is a process in which methane ( CH_4 ) is combined with oxygen ( O_2 ) to produce carbon dioxide ( CO_2 ), water ( H_2O), and energy.
This common biochemical reaction is represented as:

• \( CH_4 + 2 O_2 \rightarrow CO_2 + 2 H_2O + \ ext{energy} \)

One of the primary reasons methane is used as a fuel is due to its high energy of combustion, making it efficient and practical.
The reaction releases large amounts of energy as heat, which can be harnessed for heating or powering engines.
This factor makes it favorable for domestic and industrial applications.
It's important to note that methane's combustion is clean, producing less CO_2 compared to other hydrocarbons, thereby contributing less to global warming.

Acetylene Combustion

Acetylene ( C_2H_2) combustion is another exothermic reaction where acetylene reacts with oxygen to form carbon dioxide and water:

• \( 2 C_2H_2 + 5 O_2 \rightarrow 4 CO_2 + 2 H_2O + \ ext{energy} \)

Acetylene is known for its incredibly high flame temperature, which is why it is commonly used in welding and cutting applications.
To accurately determine its energy of combustion, a bomb calorimeter is used to capture the heat produced during the reaction.
By monitoring how this reaction alters the calorimeter's temperature, one can calculate both the heat emitted and the energy per mole of acetylene.
This knowledge is vital for industries that need precise control over heat production, ensuring efficiency and safety in operations.