Problem 164 - Master Resource Book in JEE Main Physics

Question

\(1 \mathrm{~kg}\) of diatomic gas is at a pressure of \(8 \times 10^{4} \mathrm{Nm}^{-2}\). The density of the gas is \(4 \mathrm{~kg} \mathrm{~m}^{-3}\). What is the energy of the gas due to its thermal motion? [AIEEE 2009] (a) \(5 \times 10^{4} \mathrm{~J}\) (b) \(6 \times 10^{4} \mathrm{~J}\) \((\mathrm{c}) 7 \times 10^{4} \mathrm{~J}\) (d) \(3 \times 10^{4} \mathrm{~J}\)

Step-by-Step Solution

Answer

The energy of the gas is \(5 \times 10^4 \mathrm{~J}\) (option a).

Step 1: Understand the Problem

We are given a diatomic gas with a density of \(4 \, \text{kg/m}^3\) and a pressure of \(8 \times 10^4 \, \text{Nm}^{-2}\). We need to calculate the energy due to thermal motion of \(1 \, \text{kg}\) of this gas.

Step 2: Calculate Volume from Density

Using the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), we find the volume \( V \) by rearranging: \( V = \frac{\text{Mass}}{\text{Density}} = \frac{1 \, \text{kg}}{4 \, \text{kg/m}^3} = 0.25 \, \text{m}^3 \).

Step 3: Apply Ideal Gas Law

For an ideal gas, \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature. We need to find \( n \) given \( P \) and \( V\).

Step 4: Determine Moles of Gas

The number of moles \( n \) is given by \( n = \frac{\text{Mass}}{\text{Molar Mass}} \). For diatomic molecules like oxygen or nitrogen, the molar mass is about \(28 \text{ g/mol}\) or \(0.028 \, \text{kg/mol}\). Thus, \( n = \frac{1}{0.028} = 35.71 \text{ moles}\).

Step 5: Calculate Energy Using Formula

For diatomic gases, energy due to thermal motion can be calculated by \( E = \frac{5}{2} nRT \). Simplifying using ideal gas law, \( E = \frac{5}{2} PV \). Substituting the known values, \( E = \frac{5}{2} \times 8 \times 10^4 \, \text{Nm}^{-2} \times 0.25 \, \text{m}^3 = 5 \times 10^4 \, \text{J}\).

Step 6: Choose the Correct Option

Comparing our result with the given options, the closest value is \(5 \times 10^4 \, \text{J}\). Thus, the correct answer is option (a).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in thermodynamics that relates pressure, volume, and temperature for a mole of gas using the formula \( PV = nRT \). This formula provides a simple way to understand how gases behave under different conditions. Here,

• \( P \) represents pressure in Pascals (Nm^-2).
• \( V \) is the volume in cubic meters (m³).
• \( n \) is the number of moles of gas.
• \( R \) is the universal gas constant, approximately 8.314 J/(mol K).
• \( T \) stands for temperature in Kelvin.

Using this law, you can understand the interplay of these variables. Specifically, it allows you to calculate any missing variable if the others are provided. It is essential in solving problems where you need to ascertain the properties of working gases.

Diatomic Gas

A diatomic gas consists of molecules composed of two atoms. Common examples include oxygen (O₂) and nitrogen (N₂). These gases have unique properties due to their molecular structure, which allows them rotational and vibrational motion in addition to translation. This complexity affects how they store energy. For instance,

• Diatomic gases have higher heat capacities.
• They experience energy distribution in five degrees of freedom at room temperature.

The energy in a diatomic gas is largely dependent on these various motions, making them interesting subjects of study in thermodynamics.

Density and Volume Relationship

The density and volume relationship is a simple yet powerful concept. It helps bridge the gap between mass and volume with the equation \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). From this, you can rearrange to solve for volume: \[ V = \frac{\text{Mass}}{\text{Density}} \].
This relationship is crucial, especially when dealing with gases in confined spaces, as it allows you to calculate the space a gas will occupy given a specific mass. For example, in our exercise, knowing the mass and density allows us to find the volume, a key step towards using the Ideal Gas Law to find pressure or temperature.

Energy Calculation

In thermodynamics, calculating energy is fundamental, and for diatomic gases, the formula \( E = \frac{5}{2} nRT \) is used. This represents the internal energy due to thermal motion. Here,

• \( n \) signifies the number of moles.
• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.

However, by using the Ideal Gas Law, you can reformulate it to \( E = \frac{5}{2} PV \). This relation is particularly useful when you know pressure and volume but not the temperature directly. It succinctly captures how much energy is stored thermally in a gas under given conditions.

Thermodynamics Concepts

Thermodynamics is the study of heat, work, and energy conversion. It covers various principles, including the kinetic theory of gases and the laws of thermodynamics. Key concepts include:

• Thermal energy and how it relates to temperature and internal energy.
• The significance of understanding molecular motion in gases—translational, rotational, and vibrational.
• Application of laws like the First Law of Thermodynamics, which is essentially the principle of energy conservation.

These principles help explain how energy is transferred and transformed in physical systems, providing the basis for calculating and predicting energy changes in gases and other materials.