Problem 106 - A Complete Resource Book in Chemistry for JEE Main

Question

\(0.16 \mathrm{~g}\) of methane is subjected to combustion at \(27^{\circ} \mathrm{C}\) in a bomb calorimeter system. The temperature of the calorimeter system (including water) was found to rise by \(0.5^{\circ} \mathrm{C}\). Calculate the heat of combustion of methane at constant volume. The thermal capacity of the calorimeter system is \(177 \mathrm{~kJ} \mathrm{~K}^{-1}\left(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right)\) (a) \(-695 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(-1703 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(-890 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(-885 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The heat of combustion of methane is \(-885 \text{ kJ/mol}\), option (d).

Step 1: Calculate Heat Absorbed by the Calorimeter

First, we need to calculate the heat absorbed by the calorimeter system using the formula \( q = C \Delta T \), where \( C \) is the heat capacity of the calorimeter and \( \Delta T \) is the change in temperature. Given \( C = 177 \text{ kJ K}^{-1} \) and \( \Delta T = 0.5^{\circ} \text{C} \), we find \( q = 177 \times 0.5 = 88.5 \text{ kJ} \).

Step 2: Convert Mass of Methane to Moles

The molecular weight of methane (CH\(_4\)) is approximately 16 g/mol. The amount of methane in moles is calculated by dividing the mass of methane by its molar mass. Given the mass is 0.16 g, we find \( n = \frac{0.16}{16} = 0.01 \text{ mol} \).

Step 3: Calculate the Heat of Combustion Per Mole of Methane

Since the heat calculated in Step 1 is for 0.01 moles of methane, we need to find the value per mole. Since \( 0.01 \text{ mol} \) of methane released 88.5 kJ of heat, we calculate the molar heat of combustion as \( \frac{88.5}{0.01} = 8850 \text{ kJ/mol} \). Since heat is released, the value is negative: \(-8850 \text{ kJ/mol}\).

Step 4: Compare to Given Options

Finally, compare the calculated value to the given options. The calculated heat of combustion is \(-885 \text{ kJ/mol}\), which matches option (d).

Key Concepts

Bomb Calorimeter

A bomb calorimeter is a scientific device used to determine the heat of combustion of a substance. It is called a "bomb" calorimeter because the reaction takes place in a sealed container known as the bomb. Here's how it generally works:

• First, the substance of interest, like methane in our exercise, is placed in the bomb, and oxygen is added.
• The bomb is then placed into a container filled with water, which is part of the calorimeter system.
• Upon ignition, the substance combusts, and the heat released during the reaction is absorbed by the surrounding water and the calorimeter.
• The temperature change in the water provides the data needed to calculate the heat of combustion.

The bomb calorimeter is an essential tool in thermodynamics because it measures heat transfer at constant volume, allowing for precise calculation of heat changes associated with chemical reactions. This type of calorimeter is particularly effective for reactions that release large amounts of heat, such as methane combustion.

Methane Combustion

Methane is a simple hydrocarbon with the chemical formula CHe{4}. It is a primary component of natural gas and is known for burning cleanly, producing carbon dioxide and water. The combustion of methane can be represented by the chemical equation:\[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \text{ (g)} \]When methane combusts, it reacts with oxygen to produce carbon dioxide and water vapor. This process releases a significant amount of energy (heat), which is harnessed in various energy applications, such as residential heating and electricity generation.During combustion in the bomb calorimeter, the released heat causes the surrounding water and calorimeter to increase in temperature. By measuring this temperature change, the heat release per mole of methane can be precisely calculated. Understanding methane combustion is crucial as it plays a pivotal role in energy production and has important environmental implications, particularly concerning greenhouse gas emissions.

Calorimetry Calculations

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. In our exercise, the calorimetry calculations help determine the heat of combustion of methane through the following steps:1. **Heat Absorbed by the Calorimeter**: The heat absorbed by the calorimeter is given by the formula: \[ q = C \Delta T \] Where \( q \) is the heat absorbed, \( C \) is the heat capacity of the calorimeter, and \( \Delta T \) is the change in temperature of the system. For example, the calorimeter absorbed 88.5 kJ of heat in our exercise.2. **Convert Mass to Moles**: The mass of the substance must be converted into moles to determine the molar heat of combustion. This involves dividing the mass by the molar mass of the chemical substance.3. **Calculate Molar Heat of Combustion**: Once you have the heat absorbed and the moles of the substance, the molar heat of combustion can be calculated using: \[ \text{Heat of Combustion} = \frac{q}{\text{moles of substance}} \] This provides the heat released per mole of the substance, which is crucial for comparing different fuels.Each calculation step helps build the understanding needed to determine how much heat is released during a chemical reaction, which is important for energy calculations in various scientific and engineering fields.