Problem 7 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following expressions correctly represents the relationship between the average molar kinetic energy, K.E. of \(\mathrm{CO}\) and \(\mathrm{N}_{2}\) molecules at the same temperature? (a) \(\overline{\mathrm{K} . \mathrm{E}}_{\cdot \mathrm{CO}}<\overline{\mathrm{K} \cdot \mathrm{E}}_{\mathrm{N}_{2}}\) (b) \(\overline{\mathrm{K} . \mathrm{E}_{\mathrm{CO}}}>\overline{\mathrm{K}} \cdot \mathrm{E}_{\mathrm{N}_{2}}\) (c) \(\mathrm{K} . \mathrm{E}_{\mathrm{CO}}=\mathrm{K} \cdot \mathrm{E}_{\mathrm{N}_{2}}\) (d) cannot be predicted unless volumes of the gases are given.

Step-by-Step Solution

Answer

The correct expression is (c): \( \mathrm{K.E}_{\mathrm{CO}} = \mathrm{K.E}_{\mathrm{N}_2} \).

Step 1: Understanding Kinetic Energy in Gases

The average molar kinetic energy of gases is related to the temperature, not directly to the kind of gas. This principle is explained by the kinetic molecular theory, where the formula for average kinetic energy per molecule in a gas is given by \(\frac{3}{2} k T\), where \(k\) is the Boltzmann constant and \(T\) is the temperature in Kelvin. Kinetic energy is independent of the type of gas.

Step 2: Analyzing the Options

Since kinetic energy is dependent on temperature and not on the type of gas, for gases at the same temperature, their average kinetic energies are equal. This implies that the expression for kinetic energy should reflect equality.

Step 3: Choosing the Correct Expression

Given the options, only option (c) states that the kinetic energy of \(\mathrm{CO}\) molecules is equal to that of \(\mathrm{N}_2\) molecules, which matches the known principle that gases at the same temperature have equal average kinetic energies regardless of the type of gas.

Key Concepts

Average Molar Kinetic Energy

In the study of gases, understanding the concept of average molar kinetic energy is crucial. This refers to the average energy of motion found in one mole of gas particles, influenced by the kinetic molecular theory. It's important to note that the average molar kinetic energy is temperature dependent and not influenced by the type of gas. This energy can be represented by a formula that involves the Boltzmann constant. The formula for the average kinetic energy per mole of gas at a given temperature is derived from:

• Kinetic energy per molecule: \( \frac{3}{2} k T \).
• Kinetic energy per mole: multiplying by Avogadro's number \( N_A \), resulting in \( \frac{3}{2} R T \).
• Here, \( R \) is the universal gas constant.

This means that regardless of whether you are dealing with oxygen, nitrogen, or any other gas, the average molar kinetic energy will be the same if they are at the same temperature.

Kinetic Energy Formula

The kinetic energy for an individual gas molecule is derived from basic principles in physics. The central formula that governs this concept is:\[K.E. = \frac{1}{2} m v^2\]Where:

• \( m \) is the mass of the gas molecule.
• \( v \) is the velocity of the molecule.

This formula calculates the kinetic energy based on mass and velocity, both crucial factors in physics. However, in gases, when considering averages across many molecules, temperature becomes the pivotal factor. For gases, this typical formula focuses on temperature rather than the specific mass and velocity of individual molecules, as they collectively follow the laws of thermodynamics.
While the basic kinetic energy formula involves physical parameters, in gases the macroscopic property of temperature simplifies our calculations, giving us the average kinetic energy through:\[K.E. = \frac{3}{2} k T\]

Temperature Dependence of Kinetic Energy

The relationship between temperature and kinetic energy is a foundational concept in thermodynamics and the kinetic molecular theory. Temperature directly affects the average kinetic energy of a gas, defining their speed and movement intensities. Here are the key points:

• As temperature increases, so does the average kinetic energy of gas molecules.
• The formula \( \frac{3}{2} k T \) reveals how intimately connected temperature and kinetic energy are, with \( T \) in Kelvin ensuring proportionality.
• This makes it independent of the type of gases under consideration. Only the temperature informs the kinetic energy under equal conditions.

This understanding guides predictions about gas behavior and allows us to compare different gases' energy directly by observing their temperature. Thus, learning this aspect aids in grasping more advanced concepts in chemistry and physics, underscoring the fundamental concept that at the same temperature, all gases share the same average kinetic energy.