Import Question JSON

Current Question (ID: 19875)

Question:

$\text{One mole of a monoatomic gas is mixed with three moles of a diatomic gas. If the molecular specific heat of the mixture at constant volume is } \frac{\alpha^2}{4} R \text{ J/mol-K, then the value of } \alpha \text{ will be:}$ $\text{(assume that the given diatomic gas has no vibrational mode)}$

Options:

1. $5$
2. $4$
3. $3$
4. $2$

Solution:

$\text{Hint: } (C_V)_{\text{mix}} = \frac{n_1C_{V_1} + n_2C_{V_2}}{n_1+n_2}$ $\text{Step: Find the specific heat of the mixture.}$ $\text{The specific heat of the mixture is given by:}$ $C_{V(\text{mix})} = \frac{n_1C_{V_1} + n_2C_{V_2}}{n_1+n_2} = \frac{1 \times \frac{3}{2} R + 3 \times \frac{5}{2} R}{1+3} = \frac{\alpha^2}{4}$ $\Rightarrow \frac{9}{4} R = \frac{\alpha^2}{4} R$ $\Rightarrow \alpha = 3$ $\text{Hence, option (3) is the correct answer.}$

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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
}