Problem 85 - A Complete Resource Book in Chemistry for JEE Main

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) same as that of a hydrogen molecule (b) half that of a hydrogen molecule (c) two times that of a hydrogen molecule (d) four times that of a hydrogen molecule

Step-by-Step Solution

Answer

(a) The average kinetic energy of a helium atom is the same as that of a hydrogen molecule at the same temperature.

Step 1: Understand the Relationship

The average kinetic energy of a gas particle depends only on the temperature if the gas behaves ideally, according to the equation \( K.E. = \frac{3}{2} k T \), where \( k \) is the Boltzmann constant and \( T \) is the temperature. Since both helium and hydrogen are at the same temperature (298 K), this simplifies our comparison.

Step 2: Compare the Kinetic Energies

Since the average kinetic energy for both a helium atom and a hydrogen molecule depends only on temperature and not mass, the average kinetic energy per particle is the same at the same temperature according to the equation from Step 1.

Step 3: Choose the Correct Answer

Based on Step 2, the average kinetic energy of a helium atom is the same as that of a hydrogen molecule because kinetic energy for ideal gases at the same temperature is independent of mass.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and the number of moles of an ideal gas. The formula \[ PV = nRT \] helps us understand how gases respond to changes in various conditions. Here,

• \(P\) stands for pressure,
• \(V\) is the volume,
• \(n\) represents the number of moles,
• \(R\) is the universal gas constant,
• \(T\) is the temperature in Kelvin.

For gases behaving ideally, this equation is crucial because it enables us to predict how one property of a gas will change when another is altered. However, it's important to note that the Ideal Gas Law doesn't tell us much about individual particles' kinetic energy. Instead, for kinetic energy discussions, the temperature variable, \(T\), becomes the primary focus.

Temperature Dependence

Temperature plays a pivotal role in determining the kinetic energy of gas molecules. For an ideal gas, the average kinetic energy \[ K.E. = \frac{3}{2} k T \] is directly proportional to the temperature \(T\) measured in Kelvin. Here, \(k\) is the Boltzmann constant. This formula signifies that as the temperature of a gas increases, so does the average kinetic energy of its particles. Conversely, a decrease in temperature lowers the kinetic energy. One crucial insight from this formula is that the kinetic energy does not depend on the mass of the gas molecules. Two different gases at the same temperature have the same average kinetic energy per particle, which helps us understand phenomena like the average kinetic energy being the same for both helium and hydrogen molecules when at 298 K.

Boltzmann Constant

The Boltzmann constant, denoted by \(k\), is a key physical constant that arises in many areas of physics, especially in the study of statistical mechanics and thermodynamics. It bridges microscopic and macroscopic physics

• The value of the Boltzmann constant is approximately \(1.38 \times 10^{-23} \text{J/K}\).
• It allows us to relate the average kinetic energy of individual gas particles to the temperature of the gas.

In the formula for kinetic energy \( K.E. = \frac{3}{2} k T \), the Boltzmann constant ensures that temperature, initially a macroscopic concept, can be used to describe the behavior of individual molecules microscopically. Therefore, it plays a crucial role in linking the microscopic world of atoms and molecules with the macroscopic properties of matter.

Helium vs Hydrogen Molecules

Even though helium atoms and hydrogen molecules differ in mass, with helium being heavier, their average kinetic energy is the same at a given temperature when considering them as ideal gases. This is because the formula for average kinetic energy depends only on temperature, not mass

• Both helium and hydrogen are monoatomic and diatomic gases, respectively, which means helium is single atoms while hydrogen exists as pairs of atoms (\(H_2\)).
• When both gases are at 298 K, they exhibit identical average kinetic energy due to the temperature dependence rule.

This equivalence in kinetic energy per particle has significant implications for understanding gas behaviors, particularly in mixtures and under various atmospheric conditions. Recognizing that kinetic energy is independent of mass at a constant temperature can simplify many thermodynamic problems.