The root mean square (rms) speed of a gas molecule can be calculated using the following equation:
v_rms = √(3kT / m)
Where:
- v_rms is the root mean square speed
- k is the Boltzmann constant (1.38 × 10^-23 J/K)
- T is the temperature in Kelvin
- m is the molar mass of the gas molecule
To determine the ratio of the root mean square speed of argon (Ar) and chlorine (Cl2) molecules, we need to compare their respective molar masses.
The molar mass of argon (Ar) is approximately 39.95 g/mol, and the molar mass of chlorine (Cl2) is approximately 70.90 g/mol.
Now, let's calculate the root mean square speeds for both gases.
For argon (Ar):
v_rms_Ar = √(3kT / m_Ar)
= √(3 * 1.38 × 10^-23 J/K * 300 K / 39.95 g/mol)
= √(8.28 × 10^-21 J / 39.95 g/mol)
= √(2.07 × 10^-22 J/g)
≈ 4.54 × 10^2 m/s
For chlorine (Cl2):
v_rms_Cl2 = √(3kT / m_Cl2)
= √(3 * 1.38 × 10^-23 J/K * 300 K / 70.90 g/mol)
= √(1.66 × 10^-21 J / 70.90 g/mol)
= √(2.34 × 10^-23 J/g)
≈ 4.83 × 10^2 m/s
Finally, we can calculate the ratio of the root mean square speeds:
Ratio = v_rms_Cl2 / v_rms_Ar
= (4.83 × 10^2 m/s) / (4.54 × 10^2 m/s)
≈ 1.06
Therefore, the ratio of the root mean square speeds of the chlorine and argon molecules is approximately 1.06.