Import Question JSON

Current Question (ID: 19746)

Question:

$\text{A sample of 1 mole gas at temperature } T \text{ is adiabatically expanded to double its volume.}$ $\text{If adiabatic constant for the gas is } \gamma = \frac{3}{2}, \text{ then the work done by the gas in the process is:}$

Options:

1. $RT \left[ 2 - \sqrt{2} \right]$
2. $RT \left[ 2 + \sqrt{2} \right]$
3. $\frac{R}{T} \left[ 2 + \sqrt{2} \right]$
4. $\frac{R}{T} \left[ 2 - \sqrt{2} \right]$

Solution:

$\text{For adiabatic process: } TV^{\gamma - 1} = \text{constant}$ $\text{Given } \gamma = \frac{3}{2}, \text{ so } TV^{\frac{1}{2}} = \text{constant}$ $\text{Work done } W = \frac{RT}{\gamma - 1} \left[ 1 - \left( \frac{V_f}{V_i} \right)^{\gamma - 1} \right]$ $\text{Since } V_f = 2V_i, \text{ we have } W = \frac{RT}{\frac{1}{2}} \left[ 1 - (2)^{-\frac{1}{2}} \right]$ $= 2RT \left[ 1 - \frac{1}{\sqrt{2}} \right] = RT \left[ 2 - \sqrt{2} \right]$

Import JSON File

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
}