Problem 103 - A Complete Resource Book in Chemistry for JEE Main

Question

The rms velocity of hydrogen is \(\sqrt{7}\) times the rms velocity of nitrogen. If \(\mathrm{T}\) is the temperature of the gas (a) \(\mathrm{T}\left(\mathrm{H}_{2}\right)=\mathrm{T}\left(\mathrm{N}_{2}\right)\) (b) \(\mathrm{T}\left(\mathrm{H}_{2}\right)>\mathrm{T}\left(\mathrm{N}_{2}\right)\) (c) \(\mathrm{T}\left(\mathrm{H}_{2}\right)<\mathrm{T}\left(\mathrm{N}_{2}\right)\) (d) \(\mathrm{T}\left(\mathrm{H}_{2}\right)=\sqrt{7} \mathrm{~T}\left(\mathrm{~N}_{2}\right)\)

Step-by-Step Solution

Answer

The correct answer is (c) \( T(H_2) < T(N_2) \).

Step 1: Understanding rms velocity

The root mean square (rms) velocity of a gas is given by the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \) where \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molar mass of the gas.

Step 2: Given relationship between velocities

It is given that the rms velocity of hydrogen, \( v_{rms}(H_2) = \sqrt{7} \times v_{rms}(N_2) \). This implies: \( \sqrt{\frac{3RT(H_2)}{M(H_2)}} = \sqrt{7} \times \sqrt{\frac{3RT(N_2)}{M(N_2)}} \).

Step 3: Cancel similar terms and simplify

On cancelling the factor \( \sqrt{3R} \) from both sides, we get: \( \sqrt{\frac{T(H_2)}{M(H_2)}} = \sqrt{7} \times \sqrt{\frac{T(N_2)}{M(N_2)}} \).

Step 4: Substitute known values for molar masses

We know the approximate molar masses: \( M(H_2) = 2 \) g/mol and \( M(N_2) = 28 \) g/mol. Substitute these values into the equation: \( \sqrt{\frac{T(H_2)}{2}} = \sqrt{7} \times \sqrt{\frac{T(N_2)}{28}} \).

Step 5: Solve for temperature relationship

Square both sides to remove the square roots: \( \frac{T(H_2)}{2} = 7 \times \frac{T(N_2)}{28} \). Simplify this to: \( T(H_2) = \frac{7}{14} \times T(N_2) = \frac{1}{2} \times T(N_2) \).

Step 6: Determine the correct answer

From \( T(H_2) = \frac{1}{2} \times T(N_2) \), it can be concluded that \( T(H_2) < T(N_2) \). Therefore, the correct answer is option (c).

Key Concepts

Root Mean Square Velocity

The root mean square velocity, or rms velocity, is an important concept in understanding the behavior of gas particles. This velocity provides an average speed for particles in a gas sample, taking into account the distribution of various speeds among the particles.
In the world of gases, particle speeds are not uniform; instead, they exhibit a wide range of speeds due to constant collisions and motion.
To accommodate this variety, rms velocity is calculated using the formula: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

• \( R \) is the universal gas constant.
• \( T \) represents the absolute temperature of the gas.
• \( M \) is the molar mass of the gas.

This equation shows that rms velocity increases with temperature since particles move faster as the temperature rises.Additionally, gases with lower molar masses have higher rms velocities at the same temperature because they are lighter and thus move quicker.

Hydrogen Gas

Hydrogen gas, denoted as \( H_2 \), is the lightest and most abundant element in the universe.
Due to its low molar mass of approximately 2 g/mol, hydrogen molecules move rapidly compared to heavier gas molecules when at equivalent temperatures.
This is particularly relevant when considering rms velocity, as shown in the formula:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]The low molar mass of hydrogen implies a higher velocity, given the same temperature conditions.Moreover, hydrogen's high rms velocity also means that it diffuses quickly, making it effective in processes like hydrogenation and as a fuel source in energy applications.
In contrast to heavier gases, hydrogen's lighter nature means it escapes Earth's atmosphere more readily, which is an interesting phenomenon in atmospheric science.

Nitrogen Gas

Nitrogen gas, \( N_2 \), makes up about 78% of the Earth's atmosphere.
It boasts a higher molar mass of roughly 28 g/mol, which is a significant factor when calculating its rms velocity.
Because the rms velocity is inversely proportional to the square root of the molar mass, nitrogen particles move slower relative to lighter gases like hydrogen at the same temperature.Using the formula:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]We can observe that while nitrogen has a higher molar mass, it's stability and abundance make it an essential component of the atmosphere.Its relatively slower rms velocity contributes to its stable distribution across the Earth, a fact utilized in industries for inert environments and as a diluting agent in various processes.

Temperature Relation in Gases

Temperature is a fundamental factor influencing the behavior and speed of gas particles.
According to the rms velocity formula:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]We see that rms velocity is directly proportional to the square root of temperature.This relationship implies that as temperature rises, the speed of gas molecules increases significantly.

• Higher temperatures impart more kinetic energy to gas particles, causing faster motion.
• This increase in kinetic energy leads to higher rms velocities across different gases.

Consider the relationship between hydrogen and nitrogen gases: hydrogen, due to its lower molar mass, requires less temperature to achieve the same rms velocity as nitrogen.The exercise explored this relationship, highlighting that for hydrogen to maintain the same rms velocity level as nitrogen, it often resides at a lower temperature, reflecting its distinct molecular characteristics.