Import Question JSON

Current Question (ID: 19864)

Question:

$\text{If the ratio of the number density per cm}^3 \text{ of the two gases is } 5 : 3$ $\text{and the ratio of the diameters of the molecules of the two gases is } 4 : 5,$ $\text{then, the ratio of the mean free path of molecules of two gases is:}$

Options:

1. $\frac{16}{15}$
2. $\frac{15}{16}$
3. $\frac{3}{4}$
4. $\frac{4}{3}$

Solution:

$\text{Hint: } \lambda = \frac{1}{\sqrt{2 \pi n d^2}}$ $\text{Step: Find the ratio of the mean free path of molecules of two gases.}$ $\text{(given } n_1 = 5 \text{ and } n_2 = 3 \quad d_1 = 4 \text{ and } d_2 = 5)$ $\text{we know that } \lambda = \frac{1}{\sqrt{2 \pi n d^2}} \Rightarrow \lambda \propto \frac{1}{n \cdot d^2}$ $\Rightarrow \frac{\lambda_1}{\lambda_2} = \frac{n_2 \cdot d_2^2}{n_1 \cdot d_1^2} = \frac{3 \times (5)^2}{5 \times (4)^2} = \frac{15}{16}$ $\text{Hence, option (2) is the correct answer.}$

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Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}