If He(g) has an average kinetic energy of 4410 J/mol under certain conditions, what is the root mean square speed of O2(g) molecules under the same conditions?
See your other rms post.
To find the root mean square (RMS) speed of O2(g) molecules, we need to use the relationship between kinetic energy and RMS speed. The formula for the RMS speed (u) of a gas molecule is given as:
u = √(3 * k * T / M)
Where:
- 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 kg/mol
In this case, we are given the average kinetic energy of He(g) molecules, which is 4410 J/mol. However, we want to find the RMS speed of O2(g) molecules. To do this, we'll need to use the principle of equipartition of energy, which states that each degree of freedom contributes an average kinetic energy of (1/2) * k * T to a molecule.
The O2 molecule has five degrees of freedom, which consist of three translational motions and two rotational motions (since oxygen is diatomic). Therefore, each O2 molecule has an average kinetic energy of (5/2) * k * T.
Now, let's relate the average kinetic energy of He(g) to the average kinetic energy of O2(g) using the molar ratios and the concept of stoichiometry.
The molar mass of He(g) is approximately 4 g/mol, while the molar mass of O2(g) is approximately 32 g/mol. Therefore, the molar ratio of He(g) to O2(g) is 1:8 (32 g/mol ÷ 4 g/mol).
Since the average kinetic energies are proportional to the molar ratios, we have:
(1/8) * (5/2) * k * T(O2) = (4410 J/mol)
Now, we can solve for T(O2) using the given information.
Multiply both sides of the equation by 8/5:
T(O2) = (4410 J/mol) * (8/5) = 7056 J/mol
Now that we have determined T(O2), we can use this value in the RMS speed formula to find the RMS speed of O2 molecules.
Substituting the known values into the RMS speed formula:
u(O2) = √(3 * k * T(O2) / M(O2))
The molar mass of O2 is approximately 32 g/mol, so it can be converted to kg/mol.
M(O2) = 32 g/mol * (1 kg / 1000 g) = 0.032 kg/mol
Substituting the values into the formula:
u(O2) = √(3 * (1.38 x 10^-23 J/K) * (7056 J/mol) / (0.032 kg/mol))
Calculating this expression will give you the RMS speed of O2(g) molecules under the given conditions.