Question

The densities of two gases are in the ratio of \(1: 16\). The ratio of their rates of diffusion is (a) \(16: 1\) (b) \(4: 1\) (c) \(1: 4\) (d) \(1: 16\)

Short answer

The ratio of their rates of diffusion is (b) \(4: 1\).

Step-by-step solution

Understanding Graham's Law of Effusion

Graham's Law states that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its density. In formula form, it can be written as: \( \frac{r_1}{r_2} = \sqrt{\frac{d_2}{d_1}} \), where \( r_1 \) and \( r_2 \) are the rates of diffusion for gases 1 and 2, and \( d_1 \) and \( d_2 \) are their respective densities.

Substitute the Densities into the Formula

Given the density ratio between the two gases is \( 1:16 \), identify \( d_1 = 1 \) and \( d_2 = 16 \). Substitute these values into the formula: \( \frac{r_1}{r_2} = \sqrt{\frac{16}{1}} \).

Calculate the Square Root

Calculate the square root of the density ratio: \( \sqrt{16} = 4 \). So, \( \frac{r_1}{r_2} = 4 \).

Express the Result as a Ratio

The ratio of the rates of diffusion \( r_1 : r_2 \) comes out to be \( 4 : 1 \). This means that the first gas diffuses four times faster than the second gas.

Key concepts

Rate of Diffusion

Diffusion is the movement of particles from an area of higher concentration to an area of lower concentration. This process happens until a uniform concentration is achieved throughout. In the context of gases, the rate of diffusion refers to how quickly gas molecules spread out. According to Graham's Law, the rate of diffusion is inversely related to the square root of the gas's density. This means that lighter gases, which have lower densities, will diffuse faster compared to heavier ones.

To visualize this, imagine a perfume bottle opened in a room. The perfume molecules spread out, moving from where they are most concentrated (near the bottle) to less concentrated areas (the rest of the room). If you have two different perfumes, one made of lighter molecules and the other made of heavier molecules, the lighter perfume will spread through the room more quickly due to its faster rate of diffusion.

Density of Gases

The density of a gas is its mass per unit volume, commonly measured in grams per liter (g/L). Density is an essential factor in determining how a gas behaves under different conditions, including its rate of diffusion. For gases, density can vary with changes in temperature and pressure, making it a critical parameter in gas calculations. It is also what contributes to the weight of a gas.

In relation to Graham's Law, the density of a gas plays a crucial role in diffusion and effusion processes. Gas molecules that are denser tend to move more slowly because they have more mass that needs to be moved. Consequently, understanding the density of different gases can help predict their behavior in various scenarios. For example, hydrogen gas, being very light and thus less dense, will diffuse quickly when compared to a heavier gas like sulfur hexafluoride.

Effusion

Effusion is a process where gas molecules escape through a tiny opening or hole. This differs slightly from diffusion in that effusion specifically involves moving through a restricted space. According to Graham’s Law, similar to diffusion, the rate of effusion of a gas is inversely proportional to the square root of its density.

Imagine a scenario where you have a balloon filled with gas and a pinhole made in it. The lighter gas molecules will effuse much faster through the pinhole as compared to heavier gas molecules. This means that if we had two balloons, one filled with a lighter gas like helium and another with a denser gas like nitrogen, the helium would escape more quickly through the pinhole.

• The size of the hole is crucial; it must be small so that only a few molecules can pass through at a time.
• Effusion helps explain why balloons filled with helium deflate faster than those filled with air.

Understanding effusion is fundamental in applications such as gas purging and leak testing.