Import Question JSON

Current Question (ID: 11033)

Question:

$\text{If the ratio of specific heat of a gas at constant pressure to that at constant volume is } \gamma\text{, the change in internal energy of a mass of gas, when the volume changes from V to 2V at constant pressure, P is:}$

Options:

1. $\frac{R}{(\gamma - 1)}$
2. $PV$
3. $\frac{PV}{(\gamma - 1)}$ (Correct)
4. $PV(\gamma - 1)$

Solution:

$\text{Hint: At constant pressure change in internal energy is given as } \Delta U = nC_V \Delta T$ $\text{Step 1: Calculate the change in temperature.}$ $\text{Using ideal gas equation,}$ $PV_1 = nRT_1 \text{ (Isobaric Process [P is constant])}$ $PV_2 = nRT_2$ $nR \Delta T = P[2V - V] = PV$ $\text{Or, } \Delta T = \frac{PV}{nR}$ $\text{Step 2: Calculate the change in internal energy.}$ $\text{According to the question:}$ $\frac{C_p}{C_v} = \gamma$ $C_V = \frac{R}{\gamma - 1}$ $\text{Change in internal energy,}$ $\Delta U = nC_V \Delta T = \frac{nR\Delta T}{\gamma - 1}$ $\Delta U = \frac{PV}{\gamma - 1}$

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
}