Question

At \(27^{\circ} \mathrm{C}\) the ratio of rms velocities of ozone to oxygen is (1) \(\sqrt{5 / 5}\) (2) \(\sqrt{4 / 3}\) (3) \(\sqrt{2 / 3}\) (4) \(0.25\) (5) \(\sqrt{3 / 4}\)

Short answer

The ratio of rms velocities of ozone to oxygen is \( \sqrt{\frac{2}{3}} \).

Step-by-step solution

Write the formula for rms velocity

The formula for the root mean square (rms) velocity of a gas molecule is given by: \[ v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}} \] where \( k_B \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the molar mass.

Express the ratio of rms velocities

Let's denote the rms velocity of ozone as \(v_{\text{O}_3}\) and the rms velocity of oxygen as \(v_{\text{O}_2}\). The ratio of their rms velocities is: \[ \frac{v_{\text{O}_3}}{v_{\text{O}_2}} = \sqrt{\frac{m_2}{m_3}} \] where \( m_2 \) is the molar mass of oxygen and \( m_3 \) is the molar mass of ozone.

Find molar masses

The molar mass of oxygen (O_2) is \( m_2 = 32 \ g/mol \), and the molar mass of ozone (O_3) is \( m_3 = 48 \ g/mol \).

Compute the ratio

Substitute the molar masses into the equation: \[ \frac{v_{\text{O}_3}}{v_{\text{O}_2}} = \sqrt{\frac{32}{48}} = \sqrt{\frac{2}{3}} \]

Key concepts

rms velocity

The root mean square (rms) velocity is a statistical measure used to define the speed of gas molecules. It helps us understand how fast gas molecules are moving on average. The formula for rms velocity is: \[ v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}} \] Here, \( k_B \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the molar mass of the gas. This equation shows that at a fixed temperature, a gas with a smaller molar mass will have a higher rms velocity. It's important because it ties together temperature, kinetic energy, and the mass of gas particles into one concept.
For example, if you have two different gases at the same temperature, the one with the lighter molecules will move faster on average.

gaseous molecules

Gases are composed of molecules or atoms in rapid, random motion. These particles are constantly colliding with each other and the walls of their container. Because of this behavior, gases exhibit properties like pressure, volume, and temperature. According to the kinetic theory of gases:

• The gas molecules are in constant, rapid motion.
• Most of the volume of a gas is empty space, meaning gas molecules are far apart.
• There are no intermolecular forces between the gas molecules in an ideal gas.
• The average kinetic energy of gas molecules is directly proportional to the temperature of the gas.

These properties help explain why rms velocity depends on temperature and molar mass. When considering an ideal gas, the assumptions make the calculations easier and more manageable.

molar mass ratios

The concept of molar mass ratios is crucial in understanding the properties of different gases, especially their velocities. The molar mass is the mass ofone mole of a substance, typically expressed in grams per mole (g/mol). In exercises like determining rms velocities, you often need to compare the molar masses of different gases. For example, in our exercise, we compared the molar masses of ozone \(O_3\) and oxygen \(O_2\):

• The molar mass of oxygen \(O_2\) is 32 g/mol
• The molar mass of ozone \(O_3\) is 48 g/mol

To find the ratio of their rms velocities, we use the formula: \[ \frac{v_{\text{O}_3}}{v_{\text{O}_2}} = \sqrt{\frac{m_2}{m_3}} = \sqrt{\frac{32}{48}} = \sqrt{\frac{2}{3}}\] Hence, light gases (with lower molar mass) move faster on average than heavier gases. Molar mass ratios are significant in different fields like chemistry and physics, providing insights into gas behavior under various conditions.