Import Question JSON

Current Question (ID: 19742)

Question:

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

Options:

1. $RT[1 - 2\sqrt{2}]$
2. $RT[2\sqrt{2} - 1]$
3. $RT[\sqrt{2} - 2]$
4. $RT[2 - \sqrt{2}]$

Solution:

$\text{For an adiabatic process, } TV^{\gamma-1} = \text{constant.}$ $\text{Given } \gamma = \frac{3}{2}, \text{ so } T_2V_2^{\frac{1}{2}} = T_1V_1^{\frac{1}{2}}.$ $\text{Since } V_2 = 2V_1, \text{ we have } T_2 = T_1 \left(\frac{V_1}{2V_1}\right)^{\frac{1}{2}} = \frac{T_1}{\sqrt{2}}.$ $\text{The work done } W = \frac{RT}{\gamma - 1} \left(1 - \left(\frac{1}{2}\right)^{\frac{\gamma - 1}{\gamma}}\right).$ $\text{Substituting } \gamma = \frac{3}{2}, \text{ we get } W = RT[2 - \sqrt{2}].$

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
}