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 rms speed of CO is about 515 m/s, and its ratio to the rms speed of Ar is 1.2.

Step 1: Formula for RMS Speed

The root-mean-square (rms) speed of gas molecules is given by 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 absolute temperature, and \( m \) is the mass of one molecule.

Step 2: Convert Temperature to Kelvin

First, we need to convert the temperature from Celsius to Kelvin by using the formula: \[ T(K) = T(^{\circ}C) + 273.15 \] Hence, for \(25^{\circ} \text{C}\), the temperature is \(298.15 \text{ K}\).

Step 3: Calculate the Molar Mass and Molecular Mass

The molecular mass of CO (carbon monoxide) is \(28.01 \text{ g/mol}\). Convert this to kg to use in the formula: \[ m_{CO} = \frac{28.01}{6.022 \times 10^{23}} \, \text{kg/mol} = \frac{28.01 \times 10^{-3}}{6.022 \times 10^{23}} \text{ kg} \]

Step 4: Calculate RMS Speed for CO

Plug the values of the constants and calculated mass into the rms speed formula: \[ v_{rms, CO} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{4.66 \times 10^{-26}}} \approx 515 \, \text{m/s} \]

Step 5: Calculate Atomic Mass and Molecular Mass for Ar

The atomic mass of Ar (argon) is \(39.95 \text{ g/mol}\). Convert it to kg: \[ m_{Ar} = \frac{39.95 \times 10^{-3}}{6.022 \times 10^{23}} \text{ kg} \]

Step 6: Calculate RMS Speed for Ar

Use the same rms formula for argon: \[ v_{rms, Ar} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{6.63 \times 10^{-26}}} \approx 429 \, \text{m/s} \]

Step 7: Calculate the Speed Ratio

Find the ratio of the rms speed of CO to that of Ar: \[ \text{Ratio} = \frac{v_{rms, CO}}{v_{rms, Ar}} = \frac{515}{429} \approx 1.2 \]

Key Concepts

Gas Molecules

Gas molecules are the tiny particles that make up gases and are in constant, random motion. This incessant movement is what causes gases to fill any container they are in. When studying gases, scientists often consider the behavior of a large number of gas molecules, as this helps in understanding properties like pressure and temperature.

Gas molecules are incredibly small and move at varying speeds depending on the temperature. The speed and motion patterns of gas molecules are crucial in various calculations, such as determining the root-mean-square speed, which gives us insights into the average kinetic energy of the molecules in a gas.

• Gases consist of numerous small molecules that move randomly.
• The temperature is a pivotal factor in altering the speed of these molecules.
• The behavior of gas molecules can significantly impact different properties in a given system.

Root-Mean-Square Speed

Root-mean-square speed ( v_{rms}) is an important concept in understanding how fast molecules in a gas are moving on average. It provides a measure of the average speed of gas molecules, taking into account the distribution of speeds within the gas.

The formula for root-mean-square speed is: \[ v_{rms} = \sqrt{\frac{3kT}{m}}\]where

• \(k\) is Boltzmann's constant,
• \(T\) is the absolute temperature in Kelvin, and
• \(m\) is the mass of a single molecule in kilograms.

This formula helps in calculating the speed of gas molecules based on temperature and molecular mass, explaining why lighter gases move faster than heavier ones at the same temperature.

• The root-mean-square speed helps understand molecular movement better.
• It considers both temperature and molecular mass.

Molecular Mass

Molecular mass is the sum of the masses of all the atoms in a molecule. It is often measured in grams per mole (g/mol) and is pivotal in chemical calculations, including determining the rms speed of gas molecules.

For instance, the molecular mass of carbon monoxide (CO) is 28.01 g/mol. In order to utilize this in calculations that involve the root-mean-square speed formula, it needs to be converted into kilograms:\[m = \frac{28.01 \times 10^{-3}}{6.022 \times 10^{23}} \text{ kg}\]This conversion is essential because the formula for rms speed requires the mass of a single molecule in kilograms. Understanding molecular mass aids in various calculations particularly when comparing different gases such as in the given exercise.

• Molecular mass is essential in chemical reactions and calculations.
• It must be converted to kilograms for use in certain formulas like rms speed.

Temperature Conversion

Temperature conversion is crucial when calculating the root-mean-square speed of gas molecules. Temperature directly affects the speed of molecular movement, and scientific calculations require the temperature to be in Kelvin rather than Celsius.

To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:\[T(\text{K}) = T(^{\circ}\text{C}) + 273.15\]In our example, to find the rms speed at 25°C, we first convert it to Kelvin:\[T(\text{K}) = 25 + 273.15 = 298.15\text{ K}\]This conversion ensures that the calculations for molecular speed are consistent with other scientific measurements, as Kelvin is the standard temperature unit in these contexts.

• Temperature affects molecular speeds, requiring conversion to Kelvin for calculations.
• Adding 273.15 to Celsius gives the Kelvin temperature needed for formulas.