Question

Calculate the rms speed of \(\mathrm{CO}_{2}\) at \(40^{\circ} \mathrm{C}\).

Short answer

The rms speed of CO\(_2\) at 40°C is 412.2 m/s.

Step-by-step solution

Understand the given data

We are asked to calculate the root mean square (rms) speed of carbon dioxide (CO\(_2\)) at a temperature of 40°C. This is an ideal gas problem involving kinetic theory.

Convert Temperature to Kelvin

The given temperature is 40°C. To use the rms speed formula, we need to convert the temperature to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \] Therefore: \[ T = 40 + 273.15 = 313.15 ext{ K} \]

Use the rms speed formula

The formula for the rms speed \( v_{rms} \) is given by: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \] where \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \text{ J/K}) \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of one molecule of the gas. For CO\(_2\), \( m \) in kilograms can be calculated from its molar mass.

Calculate the mass of one molecule of CO2

The molar mass of CO\(_2\) is 44.01 g/mol. To find the mass of one molecule, we convert this to kilograms and divide by Avogadro's number (\(6.022 \times 10^{23} \) mol\(^{-1}\)): \[ m = \frac{44.01 \times 10^{-3}}{6.022 \times 10^{23}} = 7.31 \times 10^{-26} \text{ kg/molecule} \]

Insert values and calculate rms speed

Substitute the known values into the rms speed formula: \[ v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \text{ J/K} \times 313.15 \text{ K}}{7.31 \times 10^{-26} \text{ kg}}} \] Calculating the above expression gives: \[ v_{rms} = 412.2 \text{ m/s} \]

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental concept in chemistry and physics that explains the behavior of gases at the molecular level. It proposes that gas particles, which can be atoms or molecules, are in constant random motion. This theory helps us understand several properties of gases, such as pressure, temperature, and volume. The central idea is that the energy of a gas is related to the temperature. Higher temperatures mean higher kinetic energy and faster-moving particles. The kinetic theory is essential in deriving the root mean square (rms) speed of gas molecules, which is a measure of the speed of particles in a gas. The rms speed formula is used in conjunction with other constants, such as the Boltzmann constant, to calculate the average speed that molecules in a gas would have as they move in a chaotic manner. This concept is pivotal because it links the microscopic behavior of particles with macroscopic properties like pressure and temperature.

Temperature Conversion to Kelvin

Temperature conversion is crucial when dealing with scientific calculations, especially in the context of gases. The Kelvin scale is the preferred unit of measurement in scientific work because it starts at absolute zero, where molecular motion stops. It provides a direct correlation between temperature and molecular kinetic energy. To convert temperatures from Celsius to Kelvin, you simply add 273.15. For example, if a temperature is given in the exercise as 40°C, you can convert it by adding:

• Use the formula: \[ T(K) = T(°C) + 273.15 \]
• For 40°C, the temperature in Kelvin becomes: \[ 40 + 273.15 = 313.15 ext{ K} \]

This conversion ensures accurate calculations in various scientific contexts, such as when using the rms speed formula.

Molar Mass Calculation

Molar mass is the mass of one mole of a given substance and is usually expressed in grams per mole (g/mol). Understanding how to calculate and use molar mass is critical in tasks involving molecular weights or conversions, like determining the mass of individual molecules in grams. Molar mass can be derived from the periodic table by summing the average atomic masses of the elements making up a compound. For carbon dioxide (\( ext{CO}_2 \)), the molar mass is calculated as follows:

• Carbon has an atomic mass of about 12.01 g/mol.
• Oxygen has an atomic mass of about 16.00 g/mol.
• So, molar mass of \( ext{CO}_2 = 12.01 + 2 imes 16.00 = 44.01 \text{ g/mol} \)

This molar mass is crucial for transforming the mass from grams to kilograms, especially when calculating the mass of one molecule, which involves dividing by Avogadro's number.

Avogadro's Number Usage

Avogadro's number, approximately \( 6.022 imes 10^{23} \), is fundamental in chemistry and is used to establish the link between moles and the number of particles. When calculating the mass of a single molecule of a substance, Avogadro's number helps us convert from macroscopic to microscopic scales. Here's the step-by-step process using carbon dioxide as an example:

• We start with the molar mass of \( ext{CO}_2 \), which is 44.01 g/mol.
• Convert this value into kilograms for SI unit compatibility: \( 44.01 ext{ g/mol} = 44.01 imes 10^{-3} ext{ kg/mol} \).
• Divide by Avogadro’s number to find the mass of one molecule: \[ rac{44.01 imes 10^{-3}}{6.022 imes 10^{23}} = 7.31 imes 10^{-26} \text{ kg/molecule} \]

This application of Avogadro’s number is integral to accurately understanding and predicting molecular behavior in different chemical contexts.