Problem 12 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

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

Step-by-Step Solution

Answer

(c) \(T(H_2) < T(N_2)\)

Step 1: Understand the basic concept

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

Step 2: Define the relation in terms of rms velocities

According to the problem, the rms velocity of hydrogen is \( \sqrt{7} \) times the rms velocity of nitrogen. Thus, we can write: \( v_{rms, H_2} = \sqrt{7} \cdot v_{rms, N_2} \).

Step 3: Substitute the formula for rms velocity

Substitute the rms velocity formula for each gas: \( \sqrt{\frac{3RT_{H_2}}{M_{H_2}}} = \sqrt{7} \cdot \sqrt{\frac{3RT_{N_2}}{M_{N_2}}} \).

Step 4: Simplify the equation

Square both sides to eliminate the square roots: \( \frac{3RT_{H_2}}{M_{H_2}} = 7 \cdot \frac{3RT_{N_2}}{M_{N_2}} \). Simplify by canceling \(3R\): \( \frac{T_{H_2}}{2} = \frac{7T_{N_2}}{28} \).

Step 5: Solve for the relation between temperatures

Since \( M_{H_2} = 2 \) and \( M_{N_2} = 28 \), substitute back to solve for \( T_{H_2} \) in terms of \( T_{N_2} \): \( T_{H_2} = \frac{7}{2} \cdot T_{N_2} \).

Step 6: Conclusion based on the calculation

The conclusion from the calculations shows: \( T_{H_2} < T_{N_2} \). Thus, option 'c' is correct.

Key Concepts

Root Mean Square Velocity

The root mean square (rms) velocity is a concept used to describe the motion of gas molecules. It is defined as the square root of the average of the squares of the individual velocities of the gas molecules. The formula to calculate rms velocity is given by: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] where

• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin, and
• \( M \) is the molar mass of the gas.

This formula shows that rms velocity is tied directly to the temperature and inversely related to the molar mass of the gas. Therefore, for lighter gases like hydrogen, the rms velocity tends to be higher when compared to heavier gases such as nitrogen. The relationship seen in the formula also indicates that rms velocity provides a useful insight into the kinetic energy of gas particles, which correlates with the temperature and molar mass.

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry and physics that describes the behavior of an ideal gas. The law is typically expressed in the equation: \[ PV = nRT \] where

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

The ideal gas law ties together several state properties of gases—pressure, volume, temperature, and number of moles. It assumes gases are composed of a large number of small particles that are in constant, random motion and that collisions between the molecules are perfectly elastic without energy loss. In practice, real gases exhibit slight deviations from ideal behavior, especially at high pressures and low temperatures. Learning how to apply the ideal gas law is instrumental for solving problems related to gas properties and predicting how gases will react to changes in their environment.

Temperature Dependence

Temperature is a critical factor in the behavior of gases, as described by kinetic molecular theory. It directly influences the speed and energy of gas molecules; according to the root mean square velocity formula, when the temperature increases, the rms velocity also increases. This is because the molecules gain more kinetic energy, resulting in faster movement. In gases, temperature and kinetic energy are directly proportional, meaning:

• Higher temperatures lead to higher average velocities of gas molecules.
• When temperature decreases, the kinetic energy and rms velocity decrease as well.

The problem from the original exercise highlights how different gases at the same temperature can have differing speeds, with lighter gases moving faster than heavier ones. This is because their lower molar mass compensates for the energy, allowing them to achieve greater velocities at the same temperature.