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

Question

Read the following statement and explanation and answer as per the options given below : Assertion : The pressure of a fixed amount of an ideal gas is proportional to its temperature Reason : Frequency of collisions and their impact both increase in proportion to the square root of temperature. (a) If both assertion and reason are CORRECT, and reason is the CORRECT explanation of the assertion. (b) If both assertion and reason are CORRECT, but reason is NOT the CORRECT explanation of the assertion. (c) If assertion is CORRECT, but reason is INCORRECT. (d) If assertion is INCORRECT, but reason is CORRECT.

Step-by-Step Solution

Answer

(c) If assertion is CORRECT, but reason is INCORRECT.

Step 1: Analyze the Assertion

The assertion states that the pressure of a fixed amount of an ideal gas is proportional to its temperature. According to the ideal gas law, \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of gas, \( R \) is the ideal gas constant, and \( T \) is temperature. For a fixed amount of gas at constant volume, \( P \) is directly proportional to \( T \). Therefore, the assertion is correct.

Step 2: Analyze the Reason

The reason suggests that the frequency of collisions and their impact increase in proportion to the square root of the temperature. This refers to the kinetic theory of gases, which states that the average kinetic energy of gas molecules is proportional to the absolute temperature and depends on the square root of temperature for velocity calculations. Thus, while the frequency and impact increase with temperature, it is not due to the square root relation directly. The reason is therefore incorrect for explaining the pressure-temperature relationship.

Step 3: Compare Assertion and Reason

Since the assertion is correct based on the ideal gas law, but the reason provided does not correctly explain the assertion as it incorrectly describes the relationship as being based on the square root of temperature (which applies to molecular velocity, not pressure directly), the correct choice is that the reason is incorrect.

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases provides a framework that explains the behavior of gases by considering their molecular composition. It is based on several assumptions, which help us understand how gases move and how their properties emerge. One fundamental idea is that gas molecules are in constant random motion. This motion is related to the temperature of the gas.
At a molecular level, the temperature of a gas correlates with its particles' kinetic energy. In simple terms, higher temperatures mean that particles move faster due to increased kinetic energy.

• Gas particles do not attract or repel each other significantly; they collide elastically, meaning no energy is lost during collisions.
• The volume occupied by the gas molecules themselves is negligible compared to the space between them; this is because gases can expand to fill any container.

The kinetic theory also provides insight into how gases exert pressure. When gas molecules collide with the walls of a container, they exert force, resulting in pressure. This pressure is related to the kinetic energy of the molecules that increases with higher temperatures.

Pressure-Temperature Relationship

When we examine the pressure-temperature relationship in gases, the ideal gas law can be a helpful tool. It is summarized by the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) for the ideal gas constant, and \( T \) for temperature in Kelvin.
For a fixed amount of gas at constant volume, the equation simplifies to show that pressure \( P \) is directly proportional to temperature \( T \). This means that if the volume doesn't change, increasing the temperature of the gas will also increase its pressure because the molecules move faster and hit the walls more forcefully and frequently.

• Keep in mind that the relationship applies to ideal gases, which assumes perfect gas behavior without intermolecular forces or particle volume.
• This direct relation helps explain how systems like pressure cookers function, where higher temperatures increase pressure for faster cooking.

The assertion in the exercise reflects this relationship correctly, aligning with the ideal gas law's principles that higher temperatures result in higher gas pressures.

Average Kinetic Energy

The concept of average kinetic energy is crucial for understanding gas behavior in relation to temperature. In terms of gases, kinetic energy is the energy due to the motion of the gas particles. Each gas particle moves randomly, and their speeds can vary, but the average kinetic energy is a useful measure across the whole gas.
According to the kinetic molecular theory, the average kinetic energy of gas molecules is directly proportional to the absolute temperature (measured in Kelvin). Mathematically, this can be expressed as \( KE_{avg} = \frac{3}{2}kT \) where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature.

• This equation indicates that as temperature increases, the average kinetic energy of the molecules also increases. In practical terms, hotter gases have particles that move more swiftly.
• The misconception that the reason in the exercise described, connecting pressure with the square root of temperature, actually pertains to the speed of molecular motion, not kinetic energy or pressure directly.

The reason provided in the exercise confuses these concepts, as it inaccurately tries to link them, highlighting the importance of understanding each separately.