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Current Question (ID: 19679)

Question:

$n$ \text{ moles of a perfect gas undergoes a cyclic process } ABCA \text{ (see figure) consisting of the following processes.} $A \rightarrow B$ \text{ Isothermal expansion at temperature } T \text{ so that the volume is doubled from } V_1 \text{ to } V_2 = 2V_1 \text{ and pressure changes from } P_1 \text{ to } P_2. $B \rightarrow C$ \text{ Isobaric compression at pressure } P_2 \text{ to initial volume } V_1. $C \rightarrow A$ \text{ Isochoric change leading to change of pressure from } P_2 \text{ to } P_1. \text{Total work done in the complete cycle } ABCA \text{ is:}

Options:

2. nRT \left( \ln 2 - \frac{1}{2} \right)
3. nRT \ln 2
4. nRT \left( \ln 2 + \frac{1}{2} \right)

Solution:

$\text{Hint: } W_{\text{isothermal}} = nRT \ln \left( \frac{V_2}{V_1} \right)$ $W_{\text{isothermal}} = nRT \ln \left( \frac{V_2}{V_1} \right)$ $W_{\text{isobaric}} = P \Delta V = n \Delta T$ $W_{\text{isochoric}} = 0$ $W_1 = nRT \ln 2$ $W_2 = nR \left( \frac{T}{2} - T \right) = -nR \frac{T}{2}$ $W_3 = 0$ $W_{\text{net}} = W_1 + W_2 + W_3$ $W_{\text{net}} = nRT \left( \ln 2 - \frac{1}{2} \right)$

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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
}