Problem 102 - Master Resource Book in JEE Main Physics

Question

At a certain temperature, the ratio of the rms velocity of \(\mathrm{H}_{2}\) molecules to \(\mathrm{O}_{2}\) molecule is (a) \(1: 1\) (b) \(1: 4\) (c) \(4: 1\) (d) \(16: 1\)

Step-by-Step Solution

Answer

The ratio of rms velocities is 4:1.

Step 1: Understanding rms velocity formula

The root mean square (rms) velocity of a gas molecule is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( v_{rms} \) is the rms velocity, \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the molar mass of the gas molecule.

Step 2: Setting up the equation for velocity ratio

Since both gases are at the same temperature, the ratio of their rms velocities depends inversely on the square root of their molar masses: \( \frac{v_{rms,\,H_2}}{v_{rms,\,O_2}} = \sqrt{\frac{m_{O_2}}{m_{H_2}}} \).

Step 3: Substituting molar masses

The molar mass of \( \mathrm{H}_2 \) is approximately 2 g/mol, and the molar mass of \( \mathrm{O}_2 \) is approximately 32 g/mol. Substitute these values in the ratio: \( \frac{v_{rms,\,H_2}}{v_{rms,\,O_2}} = \sqrt{\frac{32}{2}} \).

Step 4: Calculating the ratio

Simplify \( \sqrt{\frac{32}{2}} \) to get \( \sqrt{16} \), which equals 4. This gives the ratio of \( v_{rms,\,H_2} \) to \( v_{rms,\,O_2} \) as 4:1.

Key Concepts

Molar Mass

Molar mass is an essential concept when discussing rms velocity of gas molecules. Every chemical element or compound has a molar mass, which tells us how much one mole of that substance weighs. It's typically expressed in g/mol. For example, the molar mass of hydrogen (\(\mathrm{H}_2\)) is about 2 g/mol, while oxygen (\(\mathrm{O}_2\)) has a molar mass of about 32 g/mol. These values are crucial because they directly affect the rms velocity. This is because the rms velocity formula involves the molar mass in its denominator.
In simple terms, the larger the molar mass, the slower the particle moves, assuming constant temperature. This is because a heavier particle has more inertia, requiring more energy to achieve the same velocity as a lighter particle. This understanding allows us to compute velocity ratios between different gases, making molar mass a fundamental factor to consider.

Temperature Effects

Temperature plays a vital role in determining the rms velocity of gas molecules. Temperature, measured in Kelvin, is directly proportional to the average kinetic energy of molecules in a gas. According to the kinetic molecular theory, when temperature increases, the kinetic energy increases as well. This is represented in the rms velocity formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( T \) is the temperature. You can see how the rms speed increases with higher temperatures and vice versa.
A higher temperature means molecules will generally move faster. This is because kinetic energy is a function of velocity, and more energy translates into higher speeds. Understanding this relationship helps us grasp how changing temperature conditions can affect gas behavior, including velocity magnitude and pressure exerted by gas molecules. For calculations at a constant temperature, however, we focus on differences resulting from varying molecular properties, like molar mass.

Gas Molecular Properties

Gas molecular properties include attributes such as molar mass, molecular structure, and the overall interaction of molecules with each other. These properties extensively influence how gases behave under various conditions.

• **Molecular Mass:** As discussed, the molar mass of gas molecules affects their rms velocity.
• **Structure & Size:** The structure can impact how molecules interact, determining collision frequency and energy exchange.
• **Interactions:** The degree of attraction or repulsion between molecules affects velocity and pressure, due to space occupied by gases.

Considering these factors helps to predict and explain the motion and speed of gas molecules. Especially in the case of the rms velocity, knowing how molecular properties come into play adds a layer of understanding. For instance, lighter gases like hydrogen will naturally have higher rms velocities than heavier gases like oxygen, assuming the same conditions, because their molecular properties allow more rapid movement.