An unknown gas Q requires 2.17 times as long to effuse under the same conditions as the same amount of nitrogen gas. What is the molar mass (g/mol) of Q?
Graham's law of effusion: 2.17=sqrt(mm/28) solve for molmass mm.
To solve this problem, 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 nitrogen gas (N2) is M, and the molar mass of the unknown gas (Q) is M' (in g/mol).
According to Graham's Law of Effusion:
Rate of effusion of N2/Rate of effusion of Q = √(M'/M)
Given that the unknown gas Q requires 2.17 times as long to effuse as the same amount of nitrogen gas, we can write:
Rate of effusion of N2/Rate of effusion of Q = 1/2.17
Substituting the known values into the equation and solving for M':
1/2.17 = √(M'/M)
To isolate M', let's square both sides of the equation:
(1/2.17)^2 = (M'/M)
1/4.68 = (M'/M)
Cross-multiplying:
M' = M/4.68
Therefore, the molar mass of Q is 1/4.68 times the molar mass of N2.
Note: The molar mass of nitrogen gas (N2) is approximately 28 g/mol. So, the molar mass of Q is (28 g/mol)/4.68 ≈ 5.98 g/mol.
To find the molar mass of gas Q, we need to compare the rates of effusion of gas Q and nitrogen gas. The rate of effusion is inversely proportional to the square root of the molar mass of the gas.
Let's assume the molar mass of nitrogen gas (N₂) is M and the molar mass of gas Q is M(Q).
According to Graham's law of effusion, the ratio of the rates of effusion of two gases is equal to the square root of the ratio of their molar masses:
Rate(Q) / Rate(N₂) = sqrt(M(N₂) / M(Q))
Given that gas Q takes 2.17 times as long to effuse compared to nitrogen gas:
Rate(Q) / Rate(N₂) = 1 / 2.17
Simplifying the equation:
sqrt(M(N₂) / M(Q)) = 1 / 2.17
Now, square both sides of the equation to eliminate the square root:
M(N₂) / M(Q) = (1 / 2.17)²
M(N₂) / M(Q) = 1 / 4.7089
Cross multiply the equation:
M(N₂) = M(Q) × 4.7089
We know the molar mass of nitrogen gas (N₂) is approximately 28 g/mol. Substituting this value into the equation:
28 g/mol = M(Q) × 4.7089
Now, solve for M(Q):
M(Q) = 28 g/mol / 4.7089
Calculating the value:
M(Q) ≈ 5.95 g/mol
Therefore, the molar mass of gas Q is approximately 5.95 g/mol.