At what temperature (∘C) will xenon atoms have the same average speed that Cl2 molecules have at 35∘C?
To determine the temperature at which xenon atoms have the same average speed as Cl2 molecules at 35°C, we can use the kinetic theory of gases. The average speed of gas molecules can be determined using the root mean square (rms) speed formula:
urms = √(3kT/m)
Where:
- urms is the root mean square speed of the gas molecules
- k is the Boltzmann constant (1.38 × 10^-23 J/K)
- T is the temperature in Kelvin (K)
- m is the molar mass of the gas molecules in kilograms (kg)
First, we need to convert the Cl2 temperature from Celsius to Kelvin:
T(Cl2) = 35°C + 273.15 = 308.15 K
Next, we need to determine the molar mass of Cl2. Chlorine (Cl) has an atomic mass of 35.45 amu, and since there are two chlorine atoms in one Cl2 molecule, the molar mass of Cl2 is 2 * 35.45 = 70.90 g/mol = 0.0709 kg/mol.
Now we can calculate the root mean square speed of Cl2 at 35°C:
urms(Cl2) = √(3 * (1.38 × 10^-23 J/K) * (308.15 K) / (0.0709 kg/mol))
Next, we need to determine the molar mass of xenon (Xe). The atomic mass of xenon is 131.29 amu, so the molar mass is 131.29 g/mol = 0.13129 kg/mol.
Now we can rearrange the rms speed formula to solve for temperature:
T(Xe) = urms(Xe)^2 * m(Xe) / (3k)
Substituting the known values into the formula, we can solve for T(Xe):
T(Xe) = urms(Cl2)^2 * m(Cl2) / (urms(Xe)^2 * m(Xe) / (3k))
Therefore, by plugging in the values, we can determine the temperature (in Celsius) at which xenon atoms have the same average speed as Cl2 molecules at 35°C.