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) \(\overline{\mathrm{K} . \mathrm{E}}_{\mathrm{CO}}<\overline{\mathrm{K} . \mathrm{E}}_{\mathrm{N}_{2}}\) (b) \(\overline{\mathrm{K} . \mathrm{E} \cdot \mathrm{co}}>\overline{\mathrm{K}} \cdot \mathrm{E}_{\mathrm{N}_{2}}\) (c) \(\mathrm{K} \cdot \mathrm{E}_{. \mathrm{CO}}=\mathrm{K} \cdot \mathrm{E}_{\mathrm{N}_{2}}\) (d) cannot be predicted unless volumes of the gases are given.

Step-by-Step Solution

Answer

The correct expression is (c): \( \mathrm{K} \. \mathrm{E}_{\mathrm{CO}}=\mathrm{K} \. \mathrm{E}_{\mathrm{N}_{2}} \).

Step 1: Understand average molar kinetic energy expression

The average molar kinetic energy of a gas is given by the expression \( \overline{K.E} = \frac{3}{2}RT \), where \( R \) is the universal gas constant and \( T \) is the temperature. It shows that the average molar kinetic energy of gaseous molecules depends only on temperature and is the same for different gases at the same temperature.

Step 2: Determine the implication for different gases

Since the average kinetic energy expression depends only on temperature and not on the type of gas, it implies it is the same for all gases at a given temperature.

Step 3: Analyze the possible answers

We need to compare the kinetic energies of \( \text{CO} \) and \( \text{N}_2 \) at the same temperature using the expression \( \overline{K.E} = \frac{3}{2}RT \). Since neither the gas type nor their mass appears in the expression, \( \overline{K.E}_{CO} = \overline{K.E}_{N_2} \).

Step 4: Choose the correct option

Given the reasoning in the previous steps, the correct representation of the relationship between the average molar kinetic energy of \( \text{CO} \) and \( \text{N}_2 \) at the same temperature is option (c): \( \mathrm{K} \. \mathrm{E}_{\mathrm{CO}}=\mathrm{K} \. \mathrm{E}_{\mathrm{N}_{2}} \).

Key Concepts

Universal Gas Constant

The universal gas constant, denoted by \( R \), is a critical component in the field of chemistry, especially when dealing with gases. This constant represents a universal factor that applies to all gases when they are in ideal conditions. The value of \( R \) is approximately 8.314 J/(mol·K), and it acts as a bridge connecting various properties of gases, including temperature, pressure, and volume.

In the context of kinetic energy, the universal gas constant is part of the formula for calculating the average molar kinetic energy of gases. This formula is \( \overline{K.E} = \frac{3}{2}RT \), where \( T \) is the temperature. This highlights how \( R \) helps us understand the physical behavior of gases as it aids in quantifying how energy is distributed among the molecules.

Overall, \( R \) is indispensable for calculating and understanding properties related to gases, making it a cornerstone concept in chemistry and physics whenever we're discussing gaseous states or processes.

Temperature Dependence

Temperature plays a key role in determining the average kinetic energy of gases. In the formula \( \overline{K.E} = \frac{3}{2}RT \), temperature \( T \) is directly proportional to the average molar kinetic energy of gaseous molecules.

This direct proportionality means that as temperature increases, the kinetic energy of the molecules also increases, and vice versa. This is because temperature is essentially a measure of the average speed at which molecules are moving. When molecules gain energy from heat, they move faster, thereby increasing the kinetic energy.

Thus, temperature dependence is a fundamental concept when analyzing the behavior of gases. It explains how and why the energy within a system of gas molecules changes with varying temperature, regardless of the type of gas present.

Gaseous Molecules

Gaseous molecules are in a constant state of motion, and knowing how they behave under various conditions helps us understand many phenomena. One fundamental aspect is the kinetic theory of gases, which explains how gas molecules move and interact.

The average molar kinetic energy of gaseous molecules can be calculated using \( \overline{K.E} = \frac{3}{2}RT \), illustrating that it's only dependent on the temperature and the universal gas constant. This independence from the type or mass of gas means that, at the same temperature, \( \text{CO} \) and \( \text{N}_2 \) molecules will have identical average molar kinetic energies.

This principle underscores a critical understanding – the nature of the gas (e.g., whether it's oxygen or nitrogen) does not affect the kinetic energy at a given temperature within ideal conditions. This simplifies many calculations and predictions involving gas behaviors in chemical reactions and physical processes by focusing on temperature as a primary variable.