Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the value of \(\Delta \mathrm{E}\), when \(64 \mathrm{~g}\) oxygen is heated from \(0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) at constant volume? \(\left(\mathrm{C}_{\mathrm{v}}\right.\) on an average is \(5 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) ) (a) \(1500 \mathrm{~J}\) (b) \(1800 \mathrm{~J}\) (c) \(2000 \mathrm{~J}\) (d) \(2200 \mathrm{~J}\)

Step-by-Step Solution

Answer

The value of \( \Delta E \) is 2000 J, so the correct answer is (c) 2000 J.

Step 1: Determine Number of Moles

To find the number of moles of oxygen, use the formula \ \[ n = \frac{\text{mass}}{\text{molar mass}} \]. \In this case, the molar mass of oxygen (O\(_2\)) is 32 g/mol. Therefore, the number of moles is: \ \[ n = \frac{64 \, \text{g}}{32 \, \text{g/mol}} = 2 \, \text{moles} \].

Step 2: Calculate Change in Energy

The change in energy at constant volume is calculated using the formula \ \[ \Delta E = n \cdot C_v \cdot \Delta T \], where \ \( C_v \) is the heat capacity at constant volume, given as 5 J K\(^{-1}\) mol\(^{-1}\), and \ \( \Delta T \) is the change in temperature. \( \Delta T = 100^{\circ} \text{C} - 0^{\circ} \text{C} = 100 \, \text{K} \). \Plug the values into the formula: \ \[ \Delta E = 2 \, \text{mol} \cdot 5 \, \text{J K}^{-1} \text{mol}^{-1} \cdot 100 \, \text{K} = 1000 \, \text{J} \per \text{mol} \times 2 \] \( \Delta E = 1000 \, \text{J} \cdot 2 = 2000 \, \text{J} \).

Step 3: Identify the Correct Answer

From the calculations, the change in energy \( \Delta E \) is 2000 J. Therefore, the correct answer is option (c) 2000 J.

Key Concepts

Heat Capacity

Heat capacity is an essential concept in thermodynamics. It refers to the amount of heat energy required to change the temperature of a substance by one degree. There are two key types of heat capacity: specific heat capacity and molar heat capacity.
Specific heat capacity is concerned with the amount of heat needed for a unit mass of a substance. This means how much heat you need to change the temperature of 1 gram or 1 kilogram of a substance by 1 Kelvin or 1 Celsius respectively.

Molar heat capacity, on the other hand, is related to the amount of heat needed for one mole of a substance. In our example, the heat capacity considered is for a constant volume process, denoted as \( C_v \). This represents the heat capacity of one mole of a substance when the volume remains unchanged during heating.

• Constant Volume: The system's volume does not change as it is heated.
• Heat Capacity Symbol: \( C_v \) is used for the heat capacity at constant volume.

The heat capacity at constant volume is particularly crucial when analyzing reactions in closed environments, like gas-filled rigid containers.
The units for molar heat capacity at constant volume \( C_v \) are typically in Joules per Kelvin per mole \( \text{J K}^{-1} \text{mol}^{-1} \). It shows how much energy is needed per mole to raise the temperature by one degree in a fixed volume scenario.

Change in Energy

The change in energy of a system, often referred to as \( \Delta E \), is a measure of how much energy the system has gained or lost. When a substance is heated, the change in energy is calculated by the amount of heat absorbed at a constant volume.

For gases, this is particularly relevant when considering reactions in thermally insulated systems where no expansion work is done. The mathematical formula used for calculating the change in energy at constant volume is:

\[ \Delta E = n \cdot C_v \cdot \Delta T \]
This formula involves:

• \( n \): The number of moles of the substance.
• \( C_v \): The heat capacity at constant volume, provided as \( 5 \, \text{J K}^{-1} \text{mol}^{-1} \) in this scenario.
• \( \Delta T \): The change in temperature, which is \( 100^{\circ} \text{C} \) in this exercise.

By substituting the values into the formula, we calculate how much energy the system absorbs or releases when heated.

It's important to remember that this formula assumes no heat is lost to the surroundings, making it an idealized calculation often used in theoretical contexts.

Moles Calculation

Calculating the number of moles in a given problem is vital for determining changes in chemical and physical systems. Moles provide a way to relate quantities of substances to the microscopic particles that constitute them, like atoms and molecules.

In our exercise example, we calculated the number of moles using the formula:
\[ n = \frac{\text{mass}}{\text{molar mass}} \]
This is a simple and straightforward method to find out how many moles of a substance you have based on its total mass and its molar mass.

• **Mass of Substance:** 64 grams of oxygen in this case.
• **Molar Mass:** The molar mass of oxygen as \( 32 \, \text{g/mol} \) because oxygen is \( O_2 \), with the atomic mass of each oxygen being \(16 \text{ g/mol}\).

Using the calculation, we find that \( \frac{64 \, \text{g}}{32 \, \text{g/mol}} = 2 \text{ moles} \).

This calculation provides the basis for further analysis, such as using the determined number of moles to compute changes in energy during heating or cooling processes. Understanding how to compute moles helps you link the macroscopic quantities, like grams, to the microscopic amounts, like atoms or molecules, which are fundamental in chemical reactions.