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Current Question (ID: 11097)

Question:

$\text{Two Carnot engines A and B are operated in succession. The first one, A receives heat from a source at } T_1 = 800 \text{ K and rejects to sink at } T_2 \text{ K. The second engine, B, receives heat rejected by the first engine and rejects to another sink at } T_3 = 300 \text{ K. If the work outputs of the two engines are equal, then the value of } T_2 \text{ will be:}$

Options:

1. $100 \text{ K}$
2. $300 \text{ K}$
3. $550 \text{ K}$ (Correct)
4. $700 \text{ K}$

Solution:

$\text{For Carnot engines, the efficiency is given by:}$ $\eta_A = \frac{T_1 - T_2}{T_1} = \frac{W_A}{Q_1} \Rightarrow \eta_B = \frac{T_2 - T_3}{T_2} = \frac{W_B}{Q_2}$ $\text{Since the work outputs are equal: } W_A = W_B$ $\text{From the efficiency equations:}$ $\frac{Q_1}{Q_2} = \frac{T_1}{T_2} \times \frac{T_2 - T_3}{T_1 - T_2} = \frac{T_1}{T_2}$ $\text{Therefore: } W_A = W_B$ $\text{Solving for } T_2\text{:}$ $T_2 = \frac{T_1 + T_3}{2} = \frac{800 + 300}{2} = 550 \text{ K}$

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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
}