Question

Two vessels having equal volumes contain \(\mathrm{H}_{2}\) at one atmosphere pressure and \(\mathrm{He}\) at two atmosphere pressure respectively. If both samples are at the same temperature the mean velocity of hydrogen molecule is (1) equal to helium (2) twice that of helium (3) half that of helium (4) \(\sqrt{2}\) times that of helium

Short answer

\(\sqrt{2}\) times that of helium

Step-by-step solution

- Understand the Mean Velocity Formula

Mean velocity of a gas molecule is given by the formula: \( v = \sqrt{\frac{8RT}{\pi M}} \) where R is the gas constant, T is the temperature, and M is the molar mass of the gas.

- Compare Molar Masses

The molar mass of hydrogen ( \( \mathrm{H}_{2} \) ) is 2 g/mol, and the molar mass of helium ( \(\textrm{He}\) ) is 4 g/mol.

- Using the Same Temperature

Since both gases are at the same temperature, T, we can disregard T in the ratio of their velocities.

- Velocity Ratio Calculation

From Step 1 the velocities are: \(\textrm{v}_{\textrm{H}_{2}} = \sqrt{\frac{8RT}{\pi M_{\textrm{H}_{2}}}}\) and \( \textrm{v}_{\textrm{He}} = \sqrt{\frac{8RT}{\pi M_{\textrm{He}}}}\)\ .\ Now ratio : \ \frac{\textrm{v}_{\textrm{H}_{2}}}{\textrm{v}_{\textrm{He}}} = \sqrt{\frac{M_{\text{He}}}{M_{\textrm{H}_{2}}}} = \sqrt{\frac{4}{2}} = \sqrt{2} }

Key concepts

kinetic theory of gases

The kinetic theory of gases is a model that helps explain the behavior of gases. According to this theory, gases consist of tiny particles that are in constant, random motion. These particles collide with each other and with the walls of their container, which causes the gas pressure.

Important points to note:

• Gas particles are in constant, random motion.
• Collisions between gas particles are perfectly elastic, meaning there is no loss of energy.
• The pressure of a gas is due to collisions of its particles with the walls of the container.

Understanding this theory helps explain why gas molecules move at different velocities depending on their temperature and mass.

molar mass of gases

Molar mass is the mass of a given substance (chemical element or chemical compound) divided by its amount of substance. It is usually expressed in grams per mole (g/mol).

Key points to consider:

• Hydrogen (\textrm{H}\(_2\)) has a molar mass of 2 g/mol.
• Helium (He) has a molar mass of 4 g/mol.
• Molar mass is a crucial factor in determining the velocity of gas molecules. Lighter molecules move faster than heavier ones at the same temperature.

In the context of our problem, the gases hydrogen and helium have different molar masses, which will affect their mean velocities.

temperature and velocity relationship

Temperature and velocity of gas molecules are related. According to the kinetic theory, the average kinetic energy of gas molecules is directly proportional to temperature.

Important details:

• The formula for mean velocity of gas molecules is: \( v = \sqrt{\frac{8RT}{\pi M}} \), where R is the gas constant, T is temperature, and M is molar mass.
• At the same temperature, different gases will have different velocities based on their molar masses.
• For instance, in our problem, hydrogen molecules are lighter than helium molecules.

The ratio of velocities, \( \frac{v_{H_2}}{v_{He}} = \sqrt{\frac{M_{He}}{M_{H_2}}} \), will thus be \( \sqrt{\frac{4}{2}} = \sqrt{2} \). This indicates that the mean velocity of hydrogen molecules is \( \sqrt{2} \) times that of helium molecules at the same temperature.