Question

The ratio between the root mean square velocity of \(\mathrm{H}_{2}\) at \(50 \mathrm{~K}\) and that of \(\mathrm{O}_{2}\) at \(800 \mathrm{~K}\) is (a) 4 (b) 2 (c) 1 (d) \(\frac{1}{4}\)

Short answer

The ratio is 1.

Step-by-step solution

Understand the Root Mean Square Velocity Formula

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 absolute temperature in Kelvin, and \( M \) is the molar mass of the gas in kg/mol.

Calculate RMS Velocity for Hydrogen Gas

Using the formula \( v_{rms} = \sqrt{\frac{3RT}{M}} \), calculate the RMS velocity for \( \mathrm{H}_{2} \) at \( 50 \mathrm{~K} \). The molar mass \( M_{H_2} \) is \( 2 \times 10^{-3} \) kg/mol. Substitute \( T = 50 \mathrm{~K} \), we have:\[v_{rms, H_2} = \sqrt{\frac{3R \times 50}{2 \times 10^{-3}}}\]

Calculate RMS Velocity for Oxygen Gas

Using the formula again, calculate the RMS velocity for \( \mathrm{O}_{2} \) at \( 800 \mathrm{~K} \). The molar mass \( M_{O_2} \) is \( 32 \times 10^{-3} \) kg/mol. Substitute \( T = 800 \mathrm{~K} \), we have:\[v_{rms, O_2} = \sqrt{\frac{3R \times 800}{32 \times 10^{-3}}}\]

Calculate Ratio of RMS Velocities

To find the ratio of RMS velocities of \( \mathrm{H}_{2} \) to \( \mathrm{O}_{2} \), take the ratio of the two expressions obtained:\[\text{Ratio} = \frac{v_{rms, H_2}}{v_{rms, O_2}} = \frac{\sqrt{\frac{3R \times 50}{2 \times 10^{-3}}}}{\sqrt{\frac{3R \times 800}{32 \times 10^{-3}}}}\]Simplifying this, use the properties of square roots:\[\text{Ratio} = \sqrt{\frac{50 \times 32}{800 \times 2}}\]

Simplify the Ratio Expression

Perform arithmetic to simplify the ratio:\[\text{Ratio} = \sqrt{\frac{1600}{1600}}\]This simplifies to \(\text{Ratio} = \sqrt{1} = 1\).Therefore, the ratio is 1.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental theory that explains the behavior of gases at a molecular level. This theory assumes that gas particles are in constant, random motion and that they collide with the walls of their container, exerting pressure. Let's break down some key points:

• The particles are considered to be small spheres with a perfectly elastic collision. This means they do not lose energy upon impact.
• The average kinetic energy of these gas particles is directly proportional to the temperature of the gas. Hence, higher temperatures lead to faster particle movement.
• Pressure arises from the momentum transfer of these particles hitting the walls of their container.

In essence, the kinetic theory provides the foundation for understanding gas laws like Boyle's and Charles's Law, and it helps us comprehend macroscopic properties such as pressure and temperature through a microscopic lens.

Universal Gas Constant

The universal gas constant, often denoted as \(R\), is a crucial part of many equations involving gases, including the ideal gas law and the formula for root mean square velocity.

• Its value is \(8.314 \, \text{J}\,\text{mol}^{-1}\,\text{K}^{-1}\), which provides the connection between the energy scale and temperature scale.
• In the context of the root mean square velocity (RMS), \(R\) allows us to relate the speed of gas molecules to the temperature of the system.

Using \(R\) helps simplify complex molecular motions into understandable mathematical expressions. It essentially bridges the microscopic world of atoms and molecules with macroscopic observations like temperature and pressure.

Molar Mass

Molar mass is the mass of one mole of a substance, usually measured in grams per mole (g/mol). It's a fundamental concept when dealing with chemical reactions and calculations.

• It essentially tells us how much a given amount of a substance weighs, enabling chemists to balance equations and compute yields.
• In the formula for root mean square velocity, \(M\) represents molar mass, reflecting how molecular mass affects the speed of particles.
• Light molecules like \(\text{H}_2\) move faster than heavier molecules like \(\text{O}_2\) at the same temperature. This is due to their lower molar mass, illustrating the inverse relationship between mass and velocity in kinetic energy.

Molar mass is a cornerstone in understanding chemical reactions quantitatively, as it sets the basis for stoichiometry and enables the comparison of different substances in a common format.