Final answer:
The molar mass of a gas that effuses three times faster than radon (222 g/mol) is calculated using Graham's law of effusion, leading to a molar mass of approximately 24.67 g/mol. The gas is likely methane (CH4).
Step-by-step explanation:
We can use Graham's law of effusion to find the molar mass of the unknown gas. This law taps into the inverse relationship between the rate of effusion and the square root of the molar mass of gases. In mathematical terms, the rate of effusion of gas A divided by the rate of effusion of gas B equals the square root of the molar mass of gas B divided by the molar mass of gas A:
rate A / rate B = √(molar mass B / molar mass A)
Given that radon (Rn) has a molar mass of 222 g/mol and the unknown gas effuses three times faster than Rn, we can set up the following equation:
rate unknown / rate Rn = √(molar mass Rn / molar mass unknown)
3 / 1 = √(222 g/mol / molar mass unknown)
Squaring both sides of the equation and solving for the molar mass of the unknown gas gives us:
9 = 222 g/mol / molar mass unknown
molar mass unknown = 222 g/mol / 9
molar mass unknown = 24.67 g/mol
The gas with a molar mass approximately equal to 24.67 g/mol is methane (CH4).