Problem 22 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The rate of diffusion of methane at a given temperature is twice that of a gas \(X\). The molecular weight of \(X\) is (a) \(64.0\) (b) \(32.0\) (c) \(4.0\) (d) \(8.0\)

Step-by-Step Solution

Answer

The molecular weight of gas X is 64 g/mol.

Step 1: Understanding Rate of Diffusion

According to Graham's Law of Diffusion, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. This can be mathematically expressed as \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \), where \( r \) is the rate of diffusion and \( M \) is the molar mass.

Step 2: Applying Given Conditions

We know that the rate of diffusion of methane is twice that of gas \( X \). Hence, we write \( \frac{r_{\text{methane}}}{r_X} = 2 \). Also, the molar mass of methane \( M_{\text{methane}} = 16 \).

Step 3: Substitute into Graham's Law

Substitute the values into Graham’s Law: \( 2 = \sqrt{\frac{M_X}{16}} \).

Step 4: Square Both Sides

To eliminate the square root, square both sides of the equation: \( 2^2 = \frac{M_X}{16} \), which simplifies to \( 4 = \frac{M_X}{16} \).

Step 5: Solve for Molecular Weight of X

Multiply both sides of the equation by 16 to solve for \( M_X \): \( M_X = 4 \times 16 = 64 \).

Key Concepts

Graham's Law of Diffusion

Graham's Law of Diffusion is a method used to determine how gases will diffuse. It states that the rate at which gases diffuse is inversely proportional to the square root of their molar masses. In simpler terms, lighter gases diffuse faster than heavier gases. We can write this as \( \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \), where \( r \) represents the diffusion rates and \( M \) represents the molar masses of the two gases.
Understanding this concept is key when comparing the diffusion rates of different gases under identical conditions.
When the rate of diffusion of a gas is given, Graham’s Law allows us to rearrange the equation, to solve for unknown values like the molar mass of another gas, making it a powerful tool in chemistry.

Molar Mass

Molar mass is a fundamental concept in chemistry. It is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). Knowing the molar mass is essential for understanding the quantity and proportion of atoms in a given sample of an element or compound.
Molar mass allows us to convert from the number of moles of a substance to the mass in grams and vice versa, offering a bridge between molecular and macroscopic scales.
It plays a crucial role in stoichiometry, chemical reactions and calculations involving gases use, such as in Graham's Law of Diffusion, where it's important to have accurate molar masses for comparing and solving diffusion problems.

Methane

Methane is a simple molecule with the chemical formula \( CH_4 \). It is the main component of natural gas and is a significant fuel source.
Methane has a molar mass of \( 16 \, \text{g/mol} \), calculated from one carbon atom (\( 12 \, \text{g/mol} \)) and four hydrogen atoms (each \( 1 \, \text{g/mol} \)).
In the context of diffusion, methane is often used as a benchmark because of its relatively low molar mass, which leads it to diffuse quickly compared to larger, heavier molecules.
By understanding its properties, we can compare methane's diffusion rate against other gases, leading us to deeper insight into Graham's Law and diffusion rates.

Molecular Weight Calculation

Calculating molecular weight involves adding up the atomic masses of the elements in a molecule. Each element's atomic mass can be found on the periodic table. For instance, the molecular weight of methane is calculated by adding the weight of one carbon atom (roughly \( 12 \, \text{g/mol} \)) and four hydrogen atoms (each \( 1 \, \text{g/mol} \)). This results in a molecular weight of \( 16 \, \text{g/mol} \).
This process is vital for determining the molar mass, which is essential for calculating diffusion rates using Graham's Law.
Accurate determination of molecular weights allows chemists to precisely predict how different gases will interact, react, and diffuse, thus applying theoretical principles to real-world scenarios effectively.