Question

The rates of diffusion of \(\mathrm{SO}_{2}, \mathrm{CO}_{2}, \mathrm{PCl}_{3}\) and \(\mathrm{SO}_{3}\) are in the following (1) \(\mathrm{PCl}_{3}>\mathrm{SO}_{3}>\mathrm{SO}_{2}>\mathrm{CO}_{2}\) (2) \(\mathrm{CO}_{2}>\mathrm{SO}_{2}>\mathrm{PCl}_{3}>\mathrm{SO}_{3}\) (3) \(\mathrm{SO}_{2}>\mathrm{SO}_{3}>\mathrm{PCl}_{3}>\mathrm{CO}_{2}\) (4) \(\mathrm{CO}_{2}>\mathrm{SO}_{2}>\mathrm{SO}_{3}>\mathrm{PCl}_{3}\)

Short answer

(4) \( \mathrm{CO}_{2} > \mathrm{SO}_{2} > \mathrm{SO}_{3} > \mathrm{PCl}_{3} \)

Step-by-step solution

- Understand diffusion rate

The rate of diffusion of gases is inversely proportional to the square root of their molar masses. This relationship is described by Graham's Law of Effusion: \[ r \propto \frac{1}{\sqrt{M}} \] Where \(r\) is the rate of diffusion and \(M\) is the molar mass of the gas.

- Determine molar masses

Calculate the molar masses of the gases involved: \( \text{Molar mass of} \ \mathrm{SO}_{2} = 32 + (2 \times 16) = 64 \text{g/mol} \) \( \text{Molar mass of} \ \mathrm{CO}_{2} = 12 + (2 \times 16) = 44 \text{g/mol} \) \( \text{Molar mass of} \ \mathrm{PCl}_{3} = 31 + (3 \times 35.5) = 136.5 \text{g/mol} \) \( \text{Molar mass of} \ \mathrm{SO}_{3} = 32 + (3 \times 16) = 80 \text{g/mol} \)

- Rank gases by molar mass

Arrange the gases in order of increasing molar mass: \( \mathrm{CO}_{2} (44) < \mathrm{SO}_{2} (64) < \mathrm{SO}_{3} (80) < \mathrm{PCl}_{3} (136.5) \)

- Determine the rate of diffusion

According to Graham's Law, gases with lower molar mass diffuse faster. Therefore, the order of rates of diffusion from fastest to slowest is: \( \mathrm{CO}_{2} > \mathrm{SO}_{2} > \mathrm{SO}_{3} > \mathrm{PCl}_{3} \)

- Choose the correct option

From the given options, the correct order of rates of diffusion is: (4) \( \mathrm{CO}_{2} > \mathrm{SO}_{2} > \mathrm{SO}_{3} > \mathrm{PCl}_{3} \)

Key concepts

Rate of Diffusion

Diffusion is the process where molecules move from an area of higher concentration to an area of lower concentration. For gases, the rate of diffusion can be different based on several factors—one of the most important being the molar mass of the gas. According to Graham's Law of Effusion, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. To put it simply, lighter gases diffuse faster than heavier gases.
You can calculate this using the formula: \( r \propto \frac{1}{\sqrt{M}} \), where \( r \) is the rate of diffusion and \( M \) is the molar mass.
This means that if you know the molar masses of two gases, you can compare their rates of diffusion. For example, if Gas A has a lower molar mass than Gas B, Gas A will diffuse faster.

Molar Mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It's calculated by summing the atomic masses of all atoms in a molecule. For example:
- The molar mass of \( \text{SO}_{2} \) is 64 g/mol, calculated as 32 (sulfur) + 2 × 16 (oxygen).
- The molar mass of \( \text{CO}_{2} \) is 44 g/mol, calculated as 12 (carbon) + 2 × 16 (oxygen).
- The molar mass of \( \text{PCl}_{3} \) is 136.5 g/mol, calculated as 31 (phosphorus) + 3 × 35.5 (chlorine).
- The molar mass of \( \text{SO}_{3} \) is 80 g/mol, calculated as 32 (sulfur) + 3 × 16 (oxygen). This concept is crucial in determining the rates of diffusion, as heavier gases will diffuse more slowly according to Graham's Law.

Inverse Proportionality

Inverse proportionality means that as one value increases, the other decreases. For Graham's Law, the rate of diffusion \( r \) and the molar mass \( M \) are inversely proportional. This is expressed in the equation \( r \propto \frac{1}{\sqrt{M}} \).
Let’s break it down:
- If the molar mass increases, the rate of diffusion decreases.
- If the molar mass decreases, the rate of diffusion increases.
For instance, in the given exercise, \( \text{CO}_{2} \) with the lowest molar mass (44 g/mol) diffuses the fastest, while \( \text{PCl}_{3} \) with the highest molar mass (136.5 g/mol) diffuses the slowest. So, when you are given a set of gases and asked to rank them by their rates of diffusion, simply determine their molar masses and use the principle of inverse proportionality to find the order.