A gas of unknown molecular mass was allowed to effuse through a small opening under constant-pressure conditions. It required 52 s for 1.0 L of the gas to effuse. Under identical experimental conditions it required 24 s for 1.0 L of O2 gas to effuse. Calculate the molar mass of the unknown gas. (Remember that the faster the rate of effusion, the shorter the time required for effusion of 1.0 L; that is, rate and time are inversely proportional.)
(rateO2/rateunk) = sqrt(Munk/MO2)
M = molar mass
unk = unknown
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To calculate the molar mass of the unknown gas, we can use Graham's law of effusion. According to Graham's law, the rate of effusion of a gas is inversely proportional to the square root of its molar mass.
First, let's determine the ratio of the rates of effusion between the unknown gas and O2 gas.
Rate of effusion of unknown gas / Rate of effusion of O2 gas = sqrt (Molar mass of O2 gas / Molar mass of unknown gas)
We are given that it took 52 s for 1.0 L of the unknown gas to effuse, and 24 s for 1.0 L of O2 gas to effuse.
Using the ratio of the rates of effusion:
Rate of effusion of unknown gas / 24 s = Rate of effusion of O2 gas / 52 s
Now, we need to solve for the molar mass of the unknown gas. Rearranging the equation:
Rate of effusion of unknown gas = (24 s / 52 s) * Rate of effusion of O2 gas
The rate of effusion of O2 gas is inverse proportion to the square root of its molar mass, so:
(24 s / 52 s) * Rate of effusion of O2 gas = sqrt (Molar mass of O2 gas / Molar mass of unknown gas)
Squaring both sides:
((24 s / 52 s) * Rate of effusion of O2 gas)^2 = Molar mass of O2 gas / Molar mass of unknown gas
After calculating the value on the left-hand side, we can rearrange the equation to solve for the molar mass of the unknown gas:
Molar mass of unknown gas = (Molar mass of O2 gas * ((24 s / 52 s) * Rate of effusion of O2 gas)^2)
Now, plug in the known values:
Molar mass of O2 gas = 32 g/mol (molar mass of O2 gas)
Rate of effusion of O2 gas = 1.0 (as we are using 1.0 L of O2 gas)
Rate of effusion of unknown gas = 1.0 (as we are using 1.0 L of the unknown gas)
Molar mass of unknown gas = (32 g/mol * ((24 s / 52 s) * 1.0)^2)
After calculating the expression on the right-hand side, you will get the molar mass of the unknown gas in grams per mole.