Solid state practice · Hard

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Analyzing +4 | -1

Under identical conditions of pressure and temperature,

2 \mathrm{L}

of gaseous mixture

\left(\mathrm{H}_{2}\right.

and

\left.\mathrm{CH}_{4}\right)

effuses through a hole in 5 minutes whereas

2 \mathrm{L}

of gas

X

of molecular mass 36 takes 10 minutes to effuse through the same hole. The mole ratio of

\mathrm{H}_{2}: \mathrm{CH}_{4}

in the mixture is

View answer and explanation

Correct option: D

1: 1

Option D (

1: 1

) is correct because Graham's law of effusion relates the rate of effusion to the square root of the molar mass. By comparing the effusion rates of the mixture and gas X, and using the given molecular mass of X, the mole ratio of H2 and CH4 can be determined to be 1:1.

Solution approach

Correct Answer + Core Reason: $1: 1$ . The mole ratio is determined by comparing effusion rates using Graham's law.

Step-by-Step Solution:

Step 1: State Graham's Law of Effusion.
$\frac{{Rate_1}}{{Rate_2}} = \sqrt{\frac{{M_2}}{{M_1}}}$ , where Rate is the rate of effusion and M is the molar mass.

Step 2: Define the rates and molar masses.
$Rate_1$ = Rate of effusion of the mixture (H2 and CH4)
$Rate_2$ = Rate of effusion of gas X
$M_1$ = Molar mass of the mixture
$M_2$ = Molar mass of gas X = 36 g/mol

Step 3: Calculate the rates of effusion.\nSince the volume is the same (2L), the rate is inversely proportional to the time taken.
$Rate_1 = \frac{{2 \text{{ L}}}}{{5 \text{{ min}}}} = 0.4 \text{{ L/min}}$
$Rate_2 = \frac{{2 \text{{ L}}}}{{10 \text{{ min}}}} = 0.2 \text{{ L/min}}$

Step 4: Apply Graham's Law.
$\frac{{0.4}}{{0.2}} = \sqrt{\frac{{36}}{{M_1}}}$
$2 = \sqrt{\frac{{36}}{{M_1}}}$
$4 = \frac{{36}}{{M_1}}$
$M_1 = \frac{{36}}{{4}} = 9 \text{{ g/mol}}$

Step 5: Determine the mole ratio of H2 and CH4 in the mixture.\nLet $x$ be the mole fraction of H2 and $y$ be the mole fraction of CH4.
$M_1 = x \times M_{{H_2}} + y \times M_{{CH_4}}$
$9 = x \times 2 + y \times 16$ \nAlso, $x + y = 1$ , so $x = 1 - y$
$9 = (1 - y) \times 2 + y \times 16$
$9 = 2 - 2y + 16y$
$7 = 14y$
$y = \frac{{7}}{{14}} = 0.5$
$x = 1 - 0.5 = 0.5$

Step 6: Calculate the mole ratio.\nThe mole ratio of H2 : CH4 is $x : y = 0.5 : 0.5 = 1 : 1$ .

Why other options are wrong:
- **Option A ( $1: 2$ ): This is incorrect because it would result in a different average molar mass for the mixture.
- Option B ( $2: 1$ ): This is incorrect because it would result in a different average molar mass for the mixture.
- Option C ( $2: 3$ ): This is incorrect because it would result in a different average molar mass for the mixture.

Key Concepts:
- Graham's Law of Effusion
- Mole fraction
- Average molar mass

Memory Aids:
- Remember Graham's Law: Rate is inversely proportional to the square root of molar mass.

Quick Tips:**
- Pay attention to the units.
- Use mole fractions to calculate the average molar mass of a mixture.

Revision notes and key takeaways

Review SOLID STATE in your textbook or class notes, then use this set to check your understanding.

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