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Current Question (ID: 19667)

Question:

$'n'$ \text{ moles of an ideal gas undergo a process } A \rightarrow B \text{ as shown in the figure. The maximum temperature of the gas during the process will be:}$

Options:

1. $\frac{9P_0V_0}{4nR}$
2. $\frac{3P_0V_0}{2nR}$
3. $\frac{9P_0V_0}{2nR}$
4. $\frac{nR}{9P_0V_0}$

Solution:

$\text{Hint: } PV = nRT$ $P = 2P_0 - \frac{P_0}{V_0}V + P_0$ $nRT = -\left(\frac{P_0}{V_0}\right)V + 3P_0$ $T = \frac{1}{nR}\left[-\frac{P_0}{V_0}V^2 + 3P_0V\right]$ $\text{for } T_{\text{max}}, \frac{dT}{dV} = 0$ $\frac{1}{nR}\left[-\frac{2P_0}{V_0}V + 3P_0\right] = 0$ $V = \frac{3}{2}V_0$ $\therefore T_{\text{max}} = \frac{1}{nR}\left[-\frac{P_0}{V_0}\times\left(\frac{3}{2}V_0\right)^2 + 3P_0\times\frac{3}{2}V_0\right]$ $= \frac{1}{nR}\left[-\frac{9}{4}P_0V_0 + \frac{9}{2}P_0V_0\right] = \frac{9P_0V_0}{4nR}$

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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
}