Question

The rate of diffusion of methane at a given temperature is twice that of a gas \(\mathrm{X}\). The molecular weight of \(X\) is (1) 64 (2) 32 (3) \(4.0\) (4) \(8.0\)

Short answer

The molecular weight of gas X is 64.

Step-by-step solution

Understand the Given Data

The rate of diffusion of methane (CH4) is twice that of gas X. Let the rate of diffusion of methane be represented as \(r_{\text{CH}_4}\) and the rate of diffusion of gas X be represented as \(r_X\). This can be written as: \[r_{\text{CH}_4} = 2r_X\]. Methane's molecular weight (M) is 16 g/mol.

Use Graham's Law of Diffusion

According to Graham's Law, the rate of diffusion of a gas is inversely proportional to the square root of its molecular weight. Mathematically, \[r \propto \frac{1}{\sqrt{M}}.\] For two gases, it can be written as, \[ \frac{r_{\text{CH}_4}}{r_X} = \sqrt{ \frac{M_X}{M_{\text{CH}_4}}} \]

Apply the Given Ratio

From the given data, \[r_{\text{CH}_4} = 2r_X \] Plug this into Graham's Law equation: \[ \frac{2r_X}{r_X} = \sqrt{ \frac{M_X}{16}} \Rightarrow 2 = \sqrt{ \frac{M_X}{16}} \]

Solve for the Molecular Weight of X

Square both sides to get: \[(2)^2 = \frac{M_X}{16} \Rightarrow 4 = \frac{M_X}{16} \Rightarrow M_X = 4 \times 16 = 64 \]

Key concepts

Rate of Diffusion

Diffusion is the process where particles move from an area of high concentration to an area of low concentration. The rate of diffusion is a measure of how quickly this occurs. It depends on several factors, including temperature, the nature of the medium, and the properties of the diffusing substance.
In the context of gases, the rate of diffusion can be significantly influenced by the molecular weight of the gas. Lighter molecules tend to diffuse faster than heavier ones.
For example, in the exercise, methane diffuses at twice the rate of gas X. This statement gives us a comparative measure of how quickly each gas spreads out, useful for deriving further properties, such as molecular weight.

Molecular Weight

Molecular weight, often referred to as molecular mass, is the weight of a single molecule of a substance. It is usually expressed in units of grams per mole (g/mol).
Methane (CH4), for instance, has a molecular weight of 16 g/mol.
Remember, this weight is crucial in determining how gases behave under different conditions, particularly under Graham's Law.
Heavier gases will generally diffuse more slowly due to their greater mass, while lighter gases will diffuse more rapidly.
In the given problem, to find the molecular weight of gas X, we compare it to methane using the rate of diffusion as guided by Graham's Law.

Inverse Proportionality

Inverse proportionality means that as one quantity increases, the other decreases in such a way that their product remains constant. In the context of Graham's Law of Diffusion, the rate of diffusion of a gas is inversely proportional to the square root of its molecular weight.
Mathematically, this is expressed as \[ r \text{ is proportional to } \frac{1}{\text{sqrt}(M)} \]
This implies that if the rate of diffusion doubles, the molecular weight must increase by a factor of four for the inverse square relationship to hold.
From the exercise, we applied this concept to find that methane diffuses twice as fast as gas X, leading us to the equation: \[ 2 = \text{sqrt}(\frac{M_X}{16}) \]
By solving this proportion, we derived that the molecular weight of gas X is 64 g/mol.