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

Step-by-Step Solution

Answer

At the same temperature, \( \overline{\text{K.E.}}_{CO} = \overline{\text{K.E.}}_{N_2} \); therefore, option (c) is correct.

Step 1: Understanding the Question

We need to determine which statement correctly reflects the relationship between the average molar kinetic energy of CO and \( \text{N}_2 \) at the same temperature.

Step 2: Recall the Kinetic Energy Formula

The formula for the average molar kinetic energy \( (\text{K.E.}) \) of gas molecules is \( \overline{\text{K.E.}} = \frac{3}{2}RT \), where \( R \) is the gas constant and \( T \) is the absolute temperature.

Step 3: Analyzing the Expression

According to the formula, the average kinetic energy depends only on the temperature \( T \), not on the type of gas. As the temperature is constant for both \( \text{CO} \) and \( \text{N}_2 \), their average kinetic energies must be equal.

Step 4: Choosing the Correct Option

Since \( \overline{\text{K.E.}}_{CO} = \overline{\text{K.E.}}_{N_2} \), the correct option reflecting this equality is (c) \( \mathrm{K} \cdot \mathrm{E}_{\mathrm{C} O}=\mathrm{K} \cdot \mathrm{E}_{\mathrm{N}_{2}} \).

Key Concepts

Average Molar Kinetic Energy

The concept of average molar kinetic energy is a fundamental aspect of the Kinetic Molecular Theory. It's essential to understand that kinetic energy is the energy a gas particle possesses due to its motion. But what about the average molar kinetic energy at the molecular level? This refers to the average energy connected with a mole of gas particles. The average molar kinetic energy of gas molecules is given by the formula \( \overline{\text{K.E.}} = \frac{3}{2}RT \), where:

• \( R \) is the universal gas constant.
• \( T \) is the temperature measured in Kelvin.

This relationship shows us that the average kinetic energy is solely dependent on the temperature, and not on the type of gas. That means, regardless of whether it’s carbon monoxide (CO) or nitrogen (\( \text{N}_2 \)), if they are at the same temperature, their average molar kinetic energies are equal. This concept is crucial in understanding how gases behave and interact under various conditions.

Gas Law

The Gas Law plays an essential role in helping us understand the behavior of gases. One important aspect of this is how gases share properties at the same temperature. According to the ideal gas law \( PV = nRT \), gases behave ideally under a set of conditions and their properties are dependent on different variables such as pressure (\( P \)), volume (\( V \)), and the number of moles (\( n \)).

In ideal conditions, one crucial takeaway is that gases at the same temperature will have the same average kinetic energy, irrespective of their molecular weights or compositions. This means that lighter and heavier gas molecules can have the same kinetic energy if they are at the same temperature. Therefore, when comparing gases like CO and \( \text{N}_2 \), their kinetic energies are equal at equal temperatures, reinforcing concepts like the kinetic molecular theory.

Temperature Dependence

Temperature is a key player when it comes to the kinetic energy of gas molecules. As temperature increases, so does the kinetic energy of the molecules. This is because temperature is a measure of the average kinetic energy of the molecules within a substance.

• Higher temperatures mean higher molecular speeds.
• The relationship is linear, meaning if you double the temperature, you double the kinetic energy.

For gases like CO and \( \text{N}_2 \), at the same temperature, their molecules move at similar average speeds, contributing to equal kinetic energies. This principle helps clarify why temperature affects the kinetic behavior of molecules. By understanding temperature dependence, one can predict and manipulate conditions to achieve desired chemical or physical transformations. It demonstrates how heat energy contributes to molecular motion, a fundamental principle in both chemistry and physics.