Problem 17 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Helium atom is two times heavier than a hydrogen molecule. At 298 \(\mathrm{K}\), the average kinetic energy of a helium atom is (a) two times that of a hydrogen molecule. (b) same as that of a hydrogen molecule. (c) four times that of a hydrogen molecule. (d) half that of a hydrogen molecule.

Step-by-Step Solution

Answer

(b) Same as that of a hydrogen molecule.

Step 1: Understand the Relationship of Kinetic Energy

The average kinetic energy of a gas particle depends only on the temperature and is given by the formula \( \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin. This implies that the average kinetic energy of a gas particle is the same for all gases at the same temperature.

Step 2: Apply the Kinetic Energy Formula to Both Gases

Since the average kinetic energy formula \( \frac{3}{2}kT \) holds, we can say that at 298 K, both helium and hydrogen will have the same kinetic energy because their energies depend only on the temperature, not mass or any other factor.

Step 3: Compare the Kinetic Energies

Since the temperature is the same for both helium atoms and hydrogen molecules, the average kinetic energy of helium atoms is the same as that of hydrogen molecules, regardless of the fact that helium atoms are heavier.

Step 4: Select the Correct Option

Based on our analysis, the correct answer is option (b), which states that the average kinetic energy of a helium atom is the same as that of a hydrogen molecule at 298 K.

Key Concepts

Boltzmann Constant

The Boltzmann constant, denoted as \( k \), is a fundamental physical constant that plays a crucial role in the kinetic theory of gases. It relates the average kinetic energy of particles in a gas to the temperature of the gas. Mathematically, it can be expressed as:\[ k = 1.38 \times 10^{-23} \text{ J/K} \]This tiny value signifies that at the level of individual atoms and molecules, the energy is exceedingly small.

• It is the bridge between the macroscopic and microscopic worlds, allowing us to connect the temperature (a macroscopic property) with the kinetic energy of particles (a microscopic property).
• In calculations involving gases, the Boltzmann constant ensures that the units are consistent, particularly when dealing with temperature in Kelvin.

Overall, the Boltzmann constant is a fundamental component that aids in understanding how heat and energy manifest at microscopic levels.

Temperature Dependence of Kinetic Energy

The temperature of a gas is directly related to the average kinetic energy of its particles. This relationship is vital for understanding how gases behave under different thermal conditions. The formula given by:\[ KE = \frac{3}{2}kT \]where \( KE \) is the average kinetic energy per particle, \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin, explains this dependency.

• As the temperature increases, the average kinetic energy of the gas particles also increases. This means they move faster and collide more vigorously.
• Conversely, a decrease in temperature leads to a decrease in kinetic energy, causing the particles to move more slowly.

Temperature affects all gases in the same way, regardless of their individual masses, highlighting the universal nature of this kinetic relationship.

Gas Laws at Constant Temperature

Gas laws, such as Boyle's Law and Charles's Law, describe how gases behave under various conditions. When the temperature is held constant, some interesting behaviors arise:

• Boyle's Law: This principle states that for a given mass of gas at a constant temperature, the volume of the gas is inversely proportional to the pressure. This means if the pressure increases, the volume decreases, and vice versa.
• Charles's Law: Although this law is more typically associated with variations in temperature, at constant pressure, it shows how the volume of a gas is directly proportional to its temperature. With a fixed temperature, gases maintain a steady volume when pressure is applied.

Keeping the temperature constant implies that the kinetic energy remains unchanged, allowing for a stable analysis of pressure and volume changes in gases. This understanding is crucial when examining how real-life applications, like gas storage tanks and balloons, respond under constant temperature scenarios.