Problem 44 - Chemistry and Chemical Reactivity

Question

Calculate the rms speed for CO molecules at \(25^{\circ} \mathrm{C}\) What is the ratio of this speed to that of Ar atoms at the same temperature?

Step-by-Step Solution

Answer

The ratio of the rms speed of CO to Ar at \(25^{\circ}\)C is approximately \(\sqrt{39.95/28.01}\).

Step 1: Understand RMS Speed Formula

The root-mean-square (rms) speed of a gas is calculated using the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where \( k \) is Boltzmann's constant \((1.38 \times 10^{-23} \text{ J/K})\), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a molecule in kilograms.

Step 2: Convert Temperature to Kelvin

Convert the given temperature into Kelvin by adding 273.15 to the Celsius temperature:\[ T = 25^{\circ} C + 273.15 = 298.15 \, K \]

Step 3: Calculate the Molar Mass of CO and Ar

The molar mass of CO is the sum of the molar masses of carbon and oxygen: \[ \text{Molar mass of CO} = 12.01 \, \text{g/mol} + 16.00 \, \text{g/mol} = 28.01 \, \text{g/mol} \]For Ar (argon), it is approximately 39.95 g/mol.

Step 4: Convert Molar Mass to Kilograms per Molecule

To convert molar mass to kilograms per molecule, divide by Avogadro's number \((6.022 \times 10^{23} \, \text{molecules/mol})\): For CO, \[ m_{CO} = \frac{28.01 \, \text{g/mol}}{6.022 \times 10^{23}} = \frac{28.01 \times 10^{-3}}{6.022 \times 10^{23}} \, \text{kg} \]For Ar, \[ m_{Ar} = \frac{39.95 \, \text{g/mol}}{6.022 \times 10^{23}} = \frac{39.95 \times 10^{-3}}{6.022 \times 10^{23}} \, \text{kg} \]

Step 5: Calculate RMS Speed for CO

Insert the values from previous steps into the rms speed formula for CO:\[ v_{rms, CO} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{\frac{28.01 \times 10^{-3}}{6.022 \times 10^{23}}}} \]

Step 6: Calculate RMS Speed for Ar

Insert the values from previous steps into the rms speed formula for Ar:\[ v_{rms, Ar} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{\frac{39.95 \times 10^{-3}}{6.022 \times 10^{23}}}} \]

Step 7: Compute the Ratio of Speeds

Calculate the ratio of the rms speed of CO molecules to that of Ar atoms:\[ \text{Ratio} = \frac{v_{rms, CO}}{v_{rms, Ar}} \]

Key Concepts

Kinetic Molecular Theory

The kinetic molecular theory is an essential concept for understanding the behavior of gases. It explains how the microscopic properties of gas molecules result in observable properties such as pressure and temperature. According to this theory:

• Gases are composed of a large number of particles that are in constant random motion.
• The particles are small and occupy a negligible volume compared to the volume of their container.
• The collisions between gas particles, and between particles and the walls of the container, are perfectly elastic. This means that there is no loss of kinetic energy during the collisions.
• The average kinetic energy of the gas particles is directly proportional to the temperature of the gas in Kelvin. This explains the connection between kinetic energy, temperature, and gas motion.

To calculate the root-mean-square (rms) speed of gas particles, we rely on these principles. The rms speed gives us insight into the speed distribution of the molecules in a gas sample at a specific temperature.

Gas Laws

Gas laws describe the relationships between the pressure, volume, and temperature of a gas. These laws help us understand how gases behave under varying conditions. The most commonly used gas laws are:

• Boyle's Law: This states that the pressure of a gas is inversely proportional to its volume at a constant temperature.
• Charles's Law: This law states that the volume of a gas is directly proportional to its temperature in Kelvin, provided the pressure remains constant.
• Avogadro's Law: It indicates that the volume of a gas is directly proportional to the number of moles at constant temperature and pressure.
• Ideal Gas Law: Combining the above laws, this law is expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.

An understanding of these laws enables us to predict how gases will respond to changes in temperature, volume, and pressure. For calculating rms speed, knowing the temperature and knowing how it affects molecular motion is crucial.

Molar Mass

Molar mass is a fundamental concept in chemistry that relates the mass of a chemical substance to the amount of substance in moles. It is typically expressed in units of grams per mole (g/mol). Understanding molar mass is crucial for:

• Converting between grams and moles, a necessary step in stoichiometry calculations.
• Calculating the mass of a single molecule, which can be derived by dividing by Avogadro's number \(6.022 \times 10^{23}\) to convert molar mass to mass in kilograms per molecule.
• Using the rms speed formula, where the mass of a gas particle in kilograms is needed.

For the exercise, the molar masses of CO and Ar are calculated to find their respective molecular masses, which are then used in the rms speed formula. This allows us to compare the speeds of these gas particles at the same temperature.

Temperature Conversion

Temperature conversion is a simple yet important task in chemistry, especially when working with gases. Most gas-related calculations require temperature to be expressed in Kelvin.

• The Celsius scale, although familiar, doesn't align with the absolute temperature that is needed for kinetic molecular discussions. This is why we convert to Kelvin.
• The conversion is straightforward: add 273.15 to the Celsius temperature to get Kelvin.

For example, to convert \(25^{\circ}C\) to Kelvin, you calculate \(25 + 273.15 = 298.15 \, K\). This ensures that your temperature accurately represents the energy of motion in a gas sample. Using Kelvin allows for consistent results when utilizing gas laws and calculating rms speeds.