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Current Question (ID: 19981)

Question:

$\text{Sound travels in a mixture of two moles of helium and } n \text{ moles of hydrogen.}$ $\text{If RMS speed of gas molecules in the mixture is } \sqrt{2} \text{ times}$ $\text{the speed of sound, then the value of } n \text{ will be:}$ $1.\ 1$ $2.\ 2$ $3.\ 3$ $4.\ 4$

Options:

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

Solution:

$\text{Hint: } v = \sqrt{\frac{\gamma RT}{M}}$ $\text{Step 1: Find the ratio of } v_s \text{ and } v_{\text{rms}}.$ $\text{The speed of sound in a gas is given by:}$ $\Rightarrow v_s = \sqrt{\frac{\gamma k_B T}{m}}$ $\text{The RMS speed of gas molecules is given by:}$ $\Rightarrow v_r = \sqrt{\frac{3k_B T}{m}}$ $\Rightarrow \frac{v_s}{v_{\text{rms}}} = \sqrt{\frac{\gamma}{3}}$ $\Rightarrow \sqrt{\frac{\gamma}{3}} = \frac{1}{\sqrt{2}}$ $\Rightarrow \gamma = \frac{3}{2}$ $\text{Step 2: Find the value of } n.$ $\Rightarrow \gamma = 1 + \frac{2}{f_{\text{mix}} + 2}$ $\Rightarrow \gamma = 1 + \frac{6 + n \times \frac{5}{2}}{7n + 10}$ $\Rightarrow \frac{3}{2} = \frac{6 + 5n}{6 + 5n}$ $\Rightarrow 18 + 15n = 14n + 20$ $\Rightarrow n = 2$ $\text{Hence, option (2) is the correct answer.}$

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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
}