Problem 27 - Arihant AIEEE Physics

Question

The temperature of \(\mathrm{H}_{2}\) at which the rms velocity of its molecules is seven times the rms velocity of the molecules of nitrogen at \(300 \mathrm{~K}\), is (a) \(2100 \mathrm{~K}\) (b) \(1700 \mathrm{~K}\) (c) \(1350 \mathrm{~K}\) (d) \(1050 \mathrm{~K}\)

Step-by-Step Solution

Answer

The temperature is 1050 K, option (d).

Step 1: Identify the formula for rms velocity

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

Step 2: Establish relation between rms velocities of H2 and N2

Given that the rms velocity of \(\mathrm{H}_{2}\) molecules is seven times the rms velocity of \(\mathrm{N}_{2}\) molecules, we have the equation:\[v_{rms, H_{2}} = 7 \times v_{rms, N_{2}}\]Substituting the formula for rms velocity for each gas, the equation becomes:\[\sqrt{\frac{3kT_{H_{2}}}{m_{H_{2}}}} = 7 \times \sqrt{\frac{3k \times 300}{m_{N_{2}}}}\]Cancel out the constants and simplify:

Step 3: Simplify and solve for temperature

Squaring both sides of the equation we get:\[\frac{3kT_{H_{2}}}{m_{H_{2}}} = 49 \times \frac{3k \times 300}{m_{N_{2}}}\]Cancel the \(3k\) on both sides:\[\frac{T_{H_{2}}}{m_{H_{2}}} = 49 \times \frac{300}{m_{N_{2}}}\]Substitute molecular masses \(m_{H_{2}} = 2\) (since the molar mass of \(\mathrm{H}_{2}\) is 2 g/mol) and \(m_{N_{2}} = 28\) (since the molar mass of \(\mathrm{N}_{2}\) is 28 g/mol):\[\frac{T_{H_{2}}}{2} = 49 \times \frac{300}{28}\]Solve for \(T_{H_{2}}\):\[T_{H_{2}} = 2 \times 49 \times \frac{300}{28} = 2 \times 525 = 1050 \text{ K}\]

Step 4: Determine the correct answer

With \(T_{H_{2}} = 1050 \text{ K}\), we identify the matching choice from the options provided:(a) 2100 K(b) 1700 K(c) 1350 K**(d) 1050 K**Thus, the correct answer is (d) 1050 K.

Key Concepts

Molecular Mass

Molecular mass, also known as molar mass, refers to the mass of one mole of a substance (usually expressed in grams per mole). It is the weight of the combined atoms in a molecule. For gases, understanding molecular mass is crucial when calculating properties like the root mean square (rms) velocity, as it depicts how massive each gas particle is.

• The molecular mass of hydrogen (\(\mathrm{H}_{2}\)) is 2 g/mol.
• The molecular mass of nitrogen (\(\mathrm{N}_{2}\)) is 28 g/mol.

These values are used to determine how speed or kinetic activity will vary in molecules of different gases at given temperatures, with lighter molecules generally moving faster than heavier ones when at the same kinetic energy.

Boltzmann Constant

The Boltzmann constant (\(k\)) is a fundamental constant in physics that links the average kinetic energy of particles in a gas with the temperature of the gas. It is a bridge between macroscopic and microscopic properties, translating the temperature of a system into energy units.

• Its value is approximately \(1.38 \times 10^{-23}\) J/K.
• It appears in the formula for rms velocity and is crucial for calculations in thermodynamics.

By understanding Boltzmann's constant, we appreciate how temperature, a macroscopic property we observe, relates to the motion and energy at the microscopic molecular level. It allows us to compute the distribution of molecular speeds in a gas and understand the noise of molecular dynamics.

Temperature of Gas

The temperature of gas is a measure of the average kinetic energy of the gas molecules. It is a core factor in determining properties like pressure and volume in a gaseous system. When the temperature increases, molecules move faster, often leading to higher velocities.

• In rms velocity calculations, temperature (\(T\)) influences the molecule's speed in the formula \(v_{rms} = \sqrt{\frac{3kT}{m}} \).
• A higher temperature indicates higher molecular speed and higher kinetic energy.

For instance, in this exercise, the challenge was to determine the temperature at which the rms velocity of hydrogen molecules becomes a certain multiple of that of nitrogen molecules, highlighting the intricate relation between temperature and molecular velocity.