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Current Question (ID: 11061)

Question:

$\text{Find out the total heat given to diatomic gas in the process } A \rightarrow B \rightarrow C\text{:}$ $(B \rightarrow C \text{ is isothermal})$ $\text{Given the P-V diagram with points: A}(V_0, P_0)\text{, B}(V_0, 2P_0)\text{, C}(2V_0, P_0)$

Options:

1. $P_0 V_0 + 2P_0 V_0 \ln 2$
2. $\frac{1}{2} P_0 V_0 + 2P_0 V_0 \ln 2$
3. $\frac{5}{2} P_0 V_0 + 2P_0 V_0 \ln 2$ (Correct)
4. $3P_0 V_0 + 2P_0 V_0 \ln 2$

Solution:

$\text{Hint: } \Delta Q = \Delta U + \Delta W$ $\text{Step 1: Find the change in internal energy.}$ $\Delta U = nC_V \Delta T$ $C_V = \frac{R}{\gamma - 1} = \frac{5R}{2}$ $\Delta U = \frac{5nR(T_2 - T_1)}{2}$ $= \frac{5nR(P_2 V_2 - P_1 V_1)}{2}$ $= \frac{5P_0 V_0}{2}$ $\text{Step 2: Find the total work done.}$ $W = W_{AB} + W_{BC}$ $= 0 + nRT \ln \frac{V_2}{V_1}$ $= 2P_0 V_0 \ln 2$ $\text{Step 3: Use the first law of thermodynamics.}$ $\Delta Q = \Delta U + \Delta W$ $= \frac{5}{2} P_0 V_0 + 2P_0 V_0 \ln 2$

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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
}