Problem 27 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

At constant volume, \(4 \mathrm{~mol}\) of an ideal gas when heated from \(300 \mathrm{~K}\) to \(500 \mathrm{~K}\) changes its internal energy by \(5000 \mathrm{~J}\). The molar heat capacity at constant volume is [Main Jan. 08, 2020 (II)]

Step-by-Step Solution

Answer

The molar heat capacity \( C_v \) is 6.25 J/mol.K.

Step 1: Understand the Problem

We are given the change in internal energy of an ideal gas and need to find its molar heat capacity at constant volume. The formula relating change in internal energy, temperature, and molar heat capacity at constant volume is \( \Delta U = n C_v \Delta T \).

Step 2: Insert Known Values into Formula

Given values include \( \Delta U = 5000 \text{ J} \), \( n = 4 \text{ mol} \), \( \Delta T = 500 \text{ K} - 300 \text{ K} = 200 \text{ K} \). Substitute these into the formula: \[ 5000 = 4 \times C_v \times 200 \].

Step 3: Solve for Molar Heat Capacity \(C_v\)

Rearrange the formula to solve for \( C_v \): \[ C_v = \frac{5000}{4 \times 200} \].

Step 4: Compute the Numerical Value

Calculate \( C_v \) using the rearranged formula: \[ C_v = \frac{5000}{800} = 6.25 \text{ J/mol.K} \].

Key Concepts

Ideal Gas

An ideal gas is a theoretical model that simplifies the behavior of gases under various conditions. While no real gas behaves exactly like an ideal gas, this model helps us understand how gases react to changes in temperature, volume, and pressure. Ideal gases are described by the ideal gas law, which combines the laws of Boyle, Charles, and Avogadro. This law states:

• All gas particles are in constant motion.
• There are no attractive or repulsive forces between particles.
• The volume of the gas particles is negligible compared to the volume of the container.

These assumptions allow scientists to predict how gases will behave when the gas's pressure, temperature, or volume is changed.

Internal Energy

Internal energy is the total energy that is stored within a system. For ideal gases, this energy is primarily due to the kinetic energy of the gas molecules. The internal energy depends on:

• The number of gas particles, or moles, in the system.
• The temperature of the gas, which affects the speed and kinetic energy of the particles.

In the context of the exercise, the internal energy change was calculated to be 5000 J when the gas was heated. This energy change reflects how the temperature increase provided more kinetic energy to the gas molecules, increasing the system's internal energy.

Constant Volume

When considering a gas system, analyzing it at constant volume means that the volume does not change during the process. This is important because it simplifies the calculations of changes in energy. If the volume is constant, there is no work done by the gas (as work is calculated by pressure-volume changes), which simplifies the measurement of energy changes.
In practice, a constant volume process means that any heat added to the gas directly increases its internal energy and temperature, as there is no energy due to work being exchanged.

Temperature Change

Temperature change is a crucial factor in predicting how gases behave. In thermodynamics, a change in temperature will typically lead to a change in the kinetic energy of the molecules, and thus a change in the internal energy of the gas.

• For ideal gases, temperature is a direct measure of the average kinetic energy of the gas's particles.
• Temperature change can be calculated easily using the difference between final and initial temperatures.

In the scenario you studied, the temperature rose from 300 K to 500 K, resulting in an increase in internal energy. This temperature change was instrumental in deriving the molar heat capacity at constant volume.