A sample of an unknown gas effuses in 11.5 min. An equal volume of H2 in the same apparatus at the same temperature and pressure effuses in 2.77 min. What is the molar mass of the unknown gas?
To determine the molar mass of the unknown gas, we can use Graham's law of effusion, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.
According to Graham's law, the rate of effusion (R) is given by:
R = (1 / √M)
where M represents the molar mass of the gas.
Now, let's denote the rate of effusion of the unknown gas as Runknown and the rate of effusion of H2 as RH2. Given that the time taken for the unknown gas to effuse is 11.5 min, and the time taken for H2 to effuse is 2.77 min, we can set up the following equation using the ratio of the rates:
Runknown / RH2 = √(MH2 / Munknown)
Substituting the given values:
Runknown / 2.77 = √(2 / Munknown)
To isolate Munknown, we square both sides of the equation:
(Runknown / 2.77)^2 = 2 / Munknown
Munknown = (2 * 2.77^2) / Runknown^2
To find the value of Munknown, you need to know the values of Runknown and RH2. The rate of effusion can be calculated using the time taken for effusion. Since the volume and temperature are constant, we can simplify the equation:
R = Volume / time
Substituting the given values:
RH2 = Volume / 2.77
Runknown = Volume / 11.5
Once you have the values for RH2 and Runknown, you can substitute them into the equation to find the molar mass of the unknown gas (Munknown).
To find the molar mass of the unknown gas, we can use Graham's Law of Effusion, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.
Let's denote the molar mass of the unknown gas as M and the molar mass of H2 as MH2.
According to Graham's Law of Effusion, we have the following relationship:
(rate of effusion of unknown gas) / (rate of effusion of H2) = sqrt(MH2 / M)
We can rearrange this equation to solve for M:
M = (MH2 / ((rate of effusion of unknown gas) / (rate of effusion of H2)))^2
Now, let's plug in the given values:
MH2 = 2 g/mol (molar mass of H2)
(rate of effusion of unknown gas) = 11.5 min
(rate of effusion of H2) = 2.77 min
M = (2 g/mol / (11.5 min / 2.77 min))^2
M = (2 g/mol / 4.149)) ^ 2
M ≈ 0.963 g/mol
Therefore, the molar mass of the unknown gas is approximately 0.963 g/mol.