Question

Diatomic carbon dioxide gas (CO2(CO2 has molar mass of 44.0 g/molg/mol ) is at a temperature of 296 KK

A)Calculate the most probable speed vmp
B)Calculate the average speed vavvav
C)Calculate the root-mean-square speed vrmsvrmsv_rms.

Answer 1

The speed (vmp) of CO2 gas at 296 K is 385 m/s, while both the average speed (vav) and root-mean-square speed (vrms) are also 385 m/s for diatomic carbon dioxide molecules at this temperature and molar mass of 44.0 g/mol.

Explanation:

The most probable speed (vmp) of CO2 gas at 296 K, we can use the equation vmp = √(2 * Boltzmann constant * temperature / molar mass). Given the temperature as 296 K and the molar mass of CO2 as 44.0 g/mol, plugging these values into the equation yields vmp = √(2 * 1.38 ×
`10^{-23` J/K * 296 K / (44.0 g/mol)), resulting in vmp = 385 m/s.

The average speed (vav) of gas molecules can be calculated using vav = √(8 * Boltzmann constant * temperature / (π * molar mass)). Substituting the known values gives vav = √(8 * 1.38 ×
`10^{-23` J/K * 296 K / (π * 44.0 g/mol)), which simplifies to vav = 385 m/s.

The root-mean-square speed (vrms) formula is vrms = √(3 * Boltzmann constant * temperature / molar mass). Applying the given temperature and molar mass, vrms = √(3 * 1.38 ×
`10^{-23` J/K * 296 K / 44.0 g/mol), resulting in vrms = 429 m/s.

In a gas at a given temperature, the most probable speed refers to the speed most likely for the particles to have, while the average speed considers all the speeds and the root-mean-square speed accounts for the square of the speeds, giving a higher weight to faster particles. These calculations help understand the distribution of speeds among CO2 molecules at 296 K.

Answer 2

The most probable speed, average speed, and root-mean-square speed of CO2 gas at a temperature of 296 K can be calculated using the relevant formulas. The most probable speed is 263 m/s, the average speed is 352 m/s, and the root-mean-square speed is 408 m/s.

Step-by-step explanation:

A) The most probable speed (vmp) of a gas can be calculated using the formula:

vmp = √(2kT/m)

Where vmp is the most probable speed, k is the Boltzmann constant (1.38 x 10-23 J/K), T is the temperature in Kelvin, and m is the molar mass of the gas.

In this case, for carbon dioxide gas (CO2) at 296 K, the molar mass is 44.0 g/mol. Plugging these values into the formula, we get:

vmp(CO2) = √(2 x 1.38 x 10-23 J/K x 296 K / 44.0 g/mol) = 263 m/s

B) The average speed (vav) of a gas can be calculated using the formula:

vav = √(8kT/πm)

Where vav is the average speed, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of the gas.

Using the same values as before, we get:

vav(CO2) = √(8 x 1.38 x 10-23 J/K x 296 K / (π x 44.0 g/mol)) = 352 m/s

C) The root-mean-square speed (vrms) of a gas can be calculated using the formula:

vrms = √(3kT/m)

Where vrms is the root-mean-square speed, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the molar mass of the gas.

Using the same values as before, we get:

vrms(CO2) = √(3 x 1.38 x 10-23 J/K x 296 K / 44.0 g/mol) = 408 m/s