Answer: The ratio of the average kinetic energy of two gases can be determined using the concept of the equipartition of energy, which states that in a mixture of gases at the same temperature, each molecule has an equal amount of kinetic energy associated with each degree of freedom.
For a monatomic gas, like oxygen (O2), there are three translational degrees of freedom (motion in three directions), and for a diatomic gas, like sulfur dioxide (SO2), there are five degrees of freedom (three translational and two rotational).
The average kinetic energy of a gas is directly proportional to its temperature (in Kelvin) and can be represented by the following equation:
KE = (3/2) * k * T
Where:
KE = Average kinetic energy
k = Boltzmann constant (approximately 1.38 × 10^-23 J/K)
T = Temperature in Kelvin
Since we are comparing the two gases at the same temperature, we can ignore the temperature term in the ratio. Now, let's calculate the ratio of the average kinetic energy of sulfur dioxide (SO2) to oxygen (O2):
For sulfur dioxide (SO2):
Number of degrees of freedom (n) = 5
Average kinetic energy (KE_SO2) = (n/2) * k * T = (5/2) * k * T
For oxygen gas (O2):
Number of degrees of freedom (n) = 3
Average kinetic energy (KE_O2) = (n/2) * k * T = (3/2) * k * T
Now, calculate the ratio:
Ratio of average kinetic energy (SO2/O2) = KE_SO2 / KE_O2
Ratio = [(5/2) * k * T] / [(3/2) * k * T]
The temperature (T) cancels out, leaving us with:
Ratio = (5/2) / (3/2) = 5/3
Therefore, the ratio of the average kinetic energy of a sulfur dioxide molecule to that of oxygen gas in a mixture of the two gases is 5:3.