To determine the molar mass of the unknown gas, we can use Graham's Law of Effusion. Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.
Let's assume the molar mass of oxygen gas (O2) is M1, and the molar mass of the unknown gas is M2.
According to the information given, the oxygen gas is passing through a tiny hole 2.5 times faster than the unknown gas, when they are at the same conditions. This can be expressed mathematically as:
(rate of effusion of oxygen gas) / (rate of effusion of unknown gas) = 2.5
Using Graham's Law, we can rewrite this equation as:
(sqrt(M2) / sqrt(M1)) = 2.5
Taking the square of both sides gives:
(M2 / M1) = (2.5)^2
Simplifying:
M2 = (2.5)^2 * M1
Since we know the molar mass of oxygen gas (M1 = 32 g/mol), we can substitute it into the equation:
M2 = (2.5)^2 * 32
Calculating this expression, we find:
M2 โ 200 g/mol
Therefore, the molar mass of the unknown gas is approximately 200 g/mol.