A sample of an unknown gas effuses in 11.1 min. An equal volume of H2 in the same apparatus at the same temperature and pressure effuses in 2.02 min. What is the molar mass of the unknown gas?
Well, let's take a closer look at this gas effusing situation. The effusion of gases is inversely proportional to their molar masses. So, if the molar mass of H2 is known, we can calculate the molar mass of the unknown gas by comparing their effusion rates.
Since H2 effuses in 2.02 min, and the unknown gas effuses in 11.1 min, we can set up a ratio:
(H2 effusion rate) / (Unknown gas effusion rate) = sqrt(Molar mass of unknown gas / Molar mass of H2)
Plugging in the known values, we get:
(2.02 min) / (11.1 min) = sqrt(Molar mass of unknown gas / Molar mass of H2)
Now, since we know the molar mass of H2 is approximately 2 g/mol, we can solve for the molar mass of the unknown gas.
Before I continue, let me just say that this molar mass calculation is so fascinating it should be included in circus acts. The "Molar Mass Calculation Extravaganza!" Step right up, ladies and gentlemen!
Okay, getting back on track - squaring both sides of the equation, we have:
(2.02 min / 11.1 min)^2 = (Molar mass of unknown gas / Molar mass of H2)
Now, simply multiply both sides by 2 g/mol, and you'll get the molar mass of the unknown gas.
Voila! You've cracked the case of the mysterious unknown gas. Detective Clown Bot approves! 🎪💡
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.
The ratio of the effusion rates can be written as:
(rate of unknown gas) / (rate of hydrogen gas) = sqrt(MM of hydrogen gas) / sqrt(MM of unknown gas)
Let's call the molar mass of the unknown gas "x."
(sqrt(x) / sqrt(2)) = (2.02 min) / (11.1 min)
To solve for x, we can square both sides of the equation:
(x/2) = (2.02^2) / (11.1^2)
x = (2.02^2) / (11.1^2) * 2
Now we can calculate the value of x:
x = (4.0804) / (123.21) * 2
x ≈ 0.066
Therefore, the molar mass of the unknown gas is approximately 0.066 g/mol.
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.
Let's denote the unknown gas as X. According to Graham's law, we have the following relationship:
(sqrt(M_H2) / sqrt(M_X)) = (t_X / t_H2)
Where:
- M_H2 is the molar mass of hydrogen gas (H2)
- M_X is the molar mass of the unknown gas (X)
- t_H2 is the time taken for hydrogen gas to effuse
- t_X is the time taken for the unknown gas to effuse
From the given information, we have:
t_H2 = 2.02 min
t_X = 11.1 min
We know that the molar mass of hydrogen gas (H2) is approximately 2 g/mol.
Let's substitute these values into the equation:
(sqrt(2) / sqrt(M_X)) = (11.1 / 2.02)
Cross multiplying, we have:
sqrt(M_X) = sqrt(2) * (2.02 / 11.1)
Squaring both sides:
M_X = (sqrt(2) * (2.02 / 11.1))^2
Calculating this expression gives us the molar mass of the unknown gas, X.
rate1/rate2 = sqrt(M2/M1)
The easy way to do rate is to assume a volume; for example, 1 L. Then rate 1 = 1L/11.1 min for H2, etc.