At a certain temperature and pressure, chlorine molecules have an average velocity of
0.0410 m/s. What is the average velocity of
sulfur dioxide molecules under the same conditions?
To determine the average velocity of sulfur dioxide (SO2) molecules under the same conditions, we need to use the concept of kinetic theory of gases, which states that the average kinetic energy of gas molecules is directly proportional to the temperature.
The formula to calculate the average velocity of gas molecules is:
v = sqrt((3 * k * T) / m)
Where:
- v is the average velocity of the gas molecules
- k is the Boltzmann constant (1.38 x 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molar mass of the gas in kilograms per mole
The molar mass of chlorine (Cl2) is approximately 70.9 g/mol, and the molar mass of sulfur dioxide (SO2) is approximately 64.1 g/mol.
Now we can calculate the average velocity of sulfur dioxide molecules.
1. Convert the molar masses of chlorine and sulfur dioxide from grams per mole to kilograms per mole:
- Molar mass of Cl2 = 70.9 g/mol = 0.0709 kg/mol
- Molar mass of SO2 = 64.1 g/mol = 0.0641 kg/mol
2. Use the given average velocity of chlorine molecules and the formula to calculate the average velocity of sulfur dioxide molecules:
- Average velocity of Cl2 molecules (vCl2) = 0.0410 m/s
- Temperature and pressure conditions are assumed to be the same.
- Plug in the values into the formula:
vSO2 = sqrt((3 * k * T) / mSO2)
vSO2 = sqrt((3 * (1.38 x 10^-23 J/K) * T) / (0.0641 kg/mol))
3. Since the temperature (T) is not given explicitly, we cannot determine the exact average velocity of SO2. However, we can determine the ratio of the average velocities of Cl2 and SO2.
Calculate the ratio of the average velocities:
(vSO2) / (vCl2) = sqrt((3 * (1.38 x 10^-23 J/K) * T) / (0.0641 kg/mol)) / 0.0410 m/s
The units of kilograms and seconds cancel each other out, leaving us with the ratio in terms of temperature (T) only.
4. We can rearrange the equation to solve for T in terms of the ratio:
T = (0.0410 m/s)^2 * (0.0641 kg/mol) / (3 * (1.38 x 10^-23 J/K))
5. Plug in the given values and calculate the ratio:
T = (0.0410 m/s)^2 * (0.0641 kg/mol) / (3 * (1.38 x 10^-23 J/K))
T ≈ 942.17 K
6. The ratio of average velocities is:
(vSO2) / (vCl2) ≈ sqrt(942.17 K / T)
Therefore, under the same temperature and pressure conditions, the average velocity of sulfur dioxide molecules is approximately sqrt(942.17 K / T) times the average velocity of chlorine molecules.