Question

If the rate of effusion of helium gas at a pressure of 1000 torr is 10 torr \(\min ^{-1}\). Find the rate of effusion of hydrogen gas at a pressure of 2000 torr at the same temperature. (a) 20 torr \(\mathrm{min}^{-1}\) (b) 10 torr \(\min ^{-1}\) (c) \(30 \sqrt{2}\) torr \(\min ^{-1}\) (d) \(20 \sqrt{2}\) torr \(\mathrm{min}^{-1}\)

Short answer

(d) \(20 \sqrt{2}\) torr \(\mathrm{min}^{-1}\)

Step-by-step solution

Understand the Problem

We need to find the rate of effusion of hydrogen gas given the rate of helium gas effusion and the pressures of both gases.

Apply Graham's Law of Effusion

Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. The formula is given by: \( \frac{\text{rate of effusion of gas 1}}{\text{rate of effusion of gas 2}} = \sqrt{\frac{M_2}{M_1}} \), where \( M \) is the molar mass of the respective gases.

Substitute Known Values for Helium and Hydrogen

For helium, the rate of effusion is 10 torr/min and molar mass is 4 g/mol. For hydrogen, the molar mass is 2 g/mol. Substituting in Graham's Law gives: \( \frac{10}{\text{rate of hydrogen}} = \sqrt{\frac{2}{4}} \).

Solve for Rate of Effusion of Hydrogen

Simplify the expression: \( \frac{10}{\text{rate of hydrogen}} = \sqrt{0.5} = \frac{1}{\sqrt{2}} \). Therefore, the rate of effusion of hydrogen is \(10 \times \sqrt{2} \).

Adjust for Pressure Differences

The rate is directly proportional to pressure, so doubling the pressure for hydrogen from 1000 torr to 2000 torr doubles the rate derived in Step 4. Hence: \(10 \times \sqrt{2} \times 2 = 20 \sqrt{2} \) torr/min.

Key concepts

Rate of Effusion

The rate of effusion is a fascinating concept that relates to how gases travel through small openings or barriers. In simpler terms, effusion is the process by which gas particles escape from a container. The rate at which this happens can depend on many factors, like the kind of gas and pressure. To understand rate of effusion better, think about how perfume spreads quickly around a room when it's sprayed. The speed of this spreading represents the rate of effusion. This rate is important in many scientific and practical applications, like in airbags and respiratory systems. Remember that the rate of effusion is crucially linked to a gas's molar mass and the pressure it is under.

Molar Mass

Molar mass is a key player in understanding how different gases behave, especially when discussing effusion. Molar mass is simply the mass of one mole of a substance, measured in grams per mole. For gases, it influences how quickly they effuse through a small hole. Specifically, Graham's Law of Effusion tells us that lighter gases (those with a smaller molar mass) effuse faster than heavier gases. This is because they have less mass pushing against them, allowing them to move more freely. In the context of our helium and hydrogen gases, hydrogen, with a molar mass of 2 g/mol, effuses faster than helium with a molar mass of 4 g/mol. So, always bear in mind the molar mass when predicting how gases will behave in effusion scenarios.

Helium and Hydrogen Gases

Helium and hydrogen are two commonly discussed gases, especially when studying effusion. Helium is a light, inert gas with a molar mass of 4 g/mol. It is often used in balloons because it is lighter than air, allowing them to float. Hydrogen, on the other hand, is even lighter with a molar mass of 2 g/mol and is the most abundant element in the universe. Its lightweight nature enables hydrogen to effuse faster than helium. This distinction is essential when applying Graham's Law of Effusion, as seen in our exercise, where hydrogen's faster effusion rate, due to its lighter molar mass, plays a vital role in determining its behavior under different pressures. Understanding these differences helps predict and calculate the behavior of these gases in various conditions.

Gas Pressure and Effusion

Gas pressure is a force that the particles of a gas exert on the walls of its container. It's a crucial factor in understanding the rate of effusion. In our exercise, it's noted that when the pressure of a gas is increased, the effusion rate also increases, provided the temperature stays constant. Why does this happen? Because higher pressure means more force pushing gas particles toward a hole, so more particles escape in a given time. This direct proportionality between pressure and effusion rate is an important concept in gas dynamics.
When dealing with hydrogen and helium, remember that doubling the pressure can double the rate of effusion, as our exercise demonstrates with hydrogen gas moving from 1000 torr to 2000 torr. Recognizing how pressure interacts with effusion gives us deeper insight into gas behaviors in various physical and chemical processes.