The rate of effusion of a particular gas was measured and found to be 23.0 mL/min. Under the same conditions, the rate of effusion of pure methane (CH4) gas is 46.2 mL/min. What is the molar mass of the unknown gas?
(rateunk/rateCH4) = sqrt(molar mass CH4/molar mass unknown)
To find 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 define the variables:
- r1 is the rate of effusion of the unknown gas (23.0 mL/min)
- r2 is the rate of effusion of pure methane gas (46.2 mL/min)
- M1 is the molar mass of the unknown gas (what we want to find)
- M2 is the molar mass of methane gas (16.04 g/mol)
Graham's law can be written as:
(r1 / r2) = √(M2 / M1)
Rearranging the equation, we can solve for M1:
M1 = M2 * (r2 / r1)^2
Plugging in the values we know:
M1 = 16.04 g/mol * (46.2 mL/min / 23.0 mL/min)^2
Let's calculate that:
M1 = 16.04 g/mol * (2)^2
M1 = 16.04 g/mol * 4
M1 = 64.16 g/mol
Therefore, the molar mass of the unknown gas is approximately 64.16 g/mol.
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 assume the molar mass of the unknown gas is M.
Now, we can set up a proportion using the rates of effusion:
(rate of unknown gas) / (rate of CH4) = sqrt(Molar mass of CH4) / sqrt(M)
23.0 mL/min / 46.2 mL/min = sqrt(16.04 g/mol) / sqrt(M)
Simplifying the left side of the equation:
0.4989 = sqrt(16.04) / sqrt(M)
Now, let's square both sides of the equation:
0.4989^2 = (sqrt(16.04))^2 / M
0.2489 = 16.04 / M
Cross-multiply:
M * 0.2489 = 16.04
Now, isolate M:
M = 16.04 / 0.2489
M ≈ 64.44 g/mol
Therefore, the molar mass of the unknown gas is approximately 64.44 g/mol.