Question

At a given temperature the root mean square velocities of Oxygen and hydrogen molecules are in the ratio (A) \(1: 4\) (B) \(1: 16\) (C) \(16: 1\) (D) \(4: 1\)

Short answer

The root mean square velocities of Oxygen and Hydrogen molecules are in the ratio \(1:4\). The correct answer is (A) \(1:4\).

Step-by-step solution

Write down the formula for root mean square velocity

The root mean square velocity (Vrms) 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, and M is the molar mass of the gas.

Assign values to the constants and variables

For this exercise, let's assign the following values: R = 8.314 J/mol∙K (universal gas constant) T = temperature (constant for both Oxygen and Hydrogen) M_oxygen = M_o (molar mass of Oxygen) = 32 g/mol M_hydrogen = M_h (molar mass of Hydrogen) = 2 g/mol

Write down the Vrms equations for Oxygen and Hydrogen

Using the formula for root mean square velocity, we can write down the Vrms equations for Oxygen and Hydrogen: \(V_{rms,oxygen} = \sqrt{\frac{3RT}{M_o}}\) \(V_{rms,hydrogen} = \sqrt{\frac{3RT}{M_h}}\)

Write the velocity ratio

We want to find the ratio of the root mean square velocities of Oxygen and Hydrogen. We can write it as: \(\frac{V_{rms,oxygen}}{V_{rms,hydrogen}} = \frac{\sqrt{\frac{3RT}{M_o}}}{\sqrt{\frac{3RT}{M_h}}}\)

Simplify the velocity ratio

Now, let's simplify the ratio equation and solve for the velocity ratio: \(\frac{V_{rms,oxygen}}{V_{rms,hydrogen}} = \frac{\sqrt{\frac{3RT}{M_o}}}{\sqrt{\frac{3RT}{M_h}}} = \sqrt{\frac{M_h}{M_o}}\) Considering the molar masses of Oxygen and Hydrogen: \(\frac{V_{rms,oxygen}}{V_{rms,hydrogen}} = \sqrt{\frac{2}{32}} = \sqrt{\frac{1}{16}}\)

Calculate the velocity ratio

Now, let's calculate the root mean square velocity ratio: \(\frac{V_{rms,oxygen}}{V_{rms,hydrogen}} = \sqrt{\frac{1}{16}} = \frac{1}{4}\) So the correct ratio is \(1 : 4\). The correct answer is (A) \(1 : 4\).

Key concepts

Root Mean Square Velocity

The concept of root mean square velocity (Vrms) is fundamental in understanding the kinetic theory of gases. It provides a way to measure the average speed of gas molecules at a given temperature. To calculate the Vrms, we use the formula: \[ V_{rms} = \sqrt{\frac{3RT}{M}} \] Where:

• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.
• \( M \) is the molar mass of the gas in kg/mol.

This equation shows that the Vrms is inversely proportional to the square root of the gas's molar mass. Therefore, lighter molecules move faster at the same temperature. This relationship helps us compare the speeds of different gases, such as oxygen and hydrogen, illustrating that hydrogen molecules will typically have a higher root mean square velocity than oxygen molecules at the same temperature. Understanding this concept allows us to better grasp how gases behave in different conditions.

Molar Mass

Molar mass is the weight of one mole of a given substance, usually measured in g/mol. It signifies the mass of the substance's molecules and plays a critical role in various calculations in chemistry. When discussing gases, molar mass affects the velocity of the molecules significantly. A molecule with a smaller molar mass will tend to move faster compared to a molecule with a larger molar mass when both are at the same temperature. For example, in our exercise:

• Oxygen has a molar mass of 32 g/mol.
• Hydrogen has a molar mass of 2 g/mol.

As hydrogen has a much smaller molar mass relative to oxygen, its molecules have a higher root mean square velocity, explaining why lighter gases diffuse faster. This concept is pivotal when working with gaseous substances and can help explain phenomena such as effusion and diffusion rates, and why lighter gases travel faster through space.

Universal Gas Constant

The universal gas constant \( R \) is a key component in the ideal gas law and in equations relating to the behavior of gases. It makes the connection between energy and temperature possible, playing an essential part in the root mean square velocity formula. The value of the universal gas constant is commonly used in calculations involving the behavior of ideal gases. Its value is approximately:

• \( R = 8.314 \) J/mol·K

This constant is important as it links together various physical properties of gases, such as pressure, volume, and temperature, in the equation: \[ PV = nRT \] where \( P \) is the pressure, \( V \) is the volume, \( n \) is the amount of substance in moles, and \( T \) is the temperature in Kelvin.By understanding and applying \( R \), it becomes easier to solve problems regarding the properties of gases, calculate changes in these properties, and predict how gases will behave under different conditions.