Problem 37 - A Complete Resource Book in Chemistry for JEE Main

Question

In a constant volume calorimeter \(3.5 \mathrm{~g}\) of a gas (mol.wt.28) was burnt in excess \(\mathrm{O}_{2}\) at \(298 \mathrm{~K}\). The increase in temperature is \(0.45 \mathrm{~K}\) due to combustion. If Heat capacity of calorimeter is \(2.5 \mathrm{~K} 5 / \mathrm{K}\) the value of \(\Delta E\) is (a) \(90 \mathrm{~kJ} / \mathrm{m}\) (b) \(9 \mathrm{~kJ} / \mathrm{m}\) (c) \(45 \mathrm{~kJ} / \mathrm{m}\) (d) \(18 \mathrm{~kJ} / \mathrm{m}\)

Step-by-Step Solution

Answer

The value of \( \Delta E \) is 9 kJ/mol, corresponding to option (b).

Step 1: Calculate the number of moles of gas

To find the number of moles of the gas, we use the formula \( n = \frac{m}{M} \), where \( m \) is the mass and \( M \) is the molar mass. Here, \( m = 3.5 \; \mathrm{g} \) and \( M = 28 \; \mathrm{g\/mol} \). Hence, \( n = \frac{3.5}{28} = 0.125 \; \mathrm{mol} \).

Step 2: Calculate the heat absorbed by the calorimeter

The heat absorbed is given by \( q = C \times \Delta T \), where \( C \) is the heat capacity of the calorimeter, and \( \Delta T \) is the change in temperature. Here, \( C = 2.5 \; \mathrm{kJ/K} \) and \( \Delta T = 0.45 \; \mathrm{K} \). Therefore, \( q = 2.5 \times 0.45 = 1.125 \; \mathrm{kJ} \).

Step 3: Calculate the change in internal energy per mole

Since the reaction occurs in a constant volume calorimeter, the heat absorbed equals the change in internal energy \( \Delta E \) of the system. The total \( \Delta E \) for 0.125 mol is \( 1.125 \; \mathrm{kJ} \). Therefore, for one mole, \( \Delta E = \frac{1.125}{0.125} = 9 \; \mathrm{kJ/mol} \).

Step 4: Choose the correct answer

Compare the calculated \( \Delta E \) value with the provided options: (a) \( 90 \; \mathrm{kJ/mol} \), (b) \( 9 \; \mathrm{kJ/mol} \), (c) \( 45 \; \mathrm{kJ/mol} \), (d) \( 18 \; \mathrm{kJ/mol} \). The correct answer is (b) \( 9 \; \mathrm{kJ/mol} \).

Key Concepts

Calorimetry

Calorimetry is the science of measuring heat changes associated with chemical reactions, physical changes, or heat capacity. It helps us understand how energy is transferred during these processes. A calorimeter is an instrument used in this measurement. It isolates the reaction from the surroundings to ensure accurate readings.

There are different types of calorimeters, such as bomb calorimeters, which maintain constant volume, and coffee cup calorimeters, which are at constant pressure. In our exercise, a constant volume calorimeter is used, which means that any heat absorbed or released is directly related to the change in internal energy of the system.

• This type allows for precise measurement because volume remains unchanged, allowing only heat transfer to be measured.
• The process typically involves a known weight of reactant and the temperature change resulting from the reaction.

This absorption or release of heat leads us to measure the change in internal energy for a reaction at constant volume.

Heat Capacity

Heat capacity is a fundamental concept in calorimetry, which pertains to the amount of heat energy required to change the temperature of a substance by a specific amount. It is an intrinsic property of bodies and substances, denoting how much they "resist" temperature change.

There are two types of heat capacities: specific heat capacity (if considering per gram) and molar heat capacity (if considering per mole). In our problem, we are dealing with the heat capacity of the calorimeter, which is given as 2.5 kJ/K. This indicates how much heat the entire calorimeter system can absorb for every Kelvin of temperature rise.

• Heat capacity helps in calculating the energy transferred during a reaction. Knowing it allows for the determination of the energy change in terms of the temperature change observed.
• It's particularly useful when instruments like calorimeters are used, as it helps us understand and quantify the thermal energy involved in the reaction process.

By applying this principle, we determined the amount of heat (q) transferred in our given exercise.

Internal Energy Change

Internal energy change, symbolized as \( \Delta E \), represents the total change in energy within a system after a process, such as a chemical reaction. At constant volume, as in the bomb calorimeter, the internal energy change is observed as the heat exchanged with the surroundings.

In our specific exercise, the gas was combusted in a constant volume in the presence of excess oxygen. Because the volume is constant, any thermal change measured is equated directly to \( \Delta E \) since no work is done by volume changes.

• This means the calorimeter's temperature rise directly tells us the internal energy change for the reaction.
• Understanding \( \Delta E \) helps predict the energy exchange patterns and reactant behavior under said conditions.

The calculated \( \Delta E \) ties into determining how energetically efficient or demanding certain reactions are, which is particularly useful in fields that deal with energy resource management and chemical manufacturing.