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Current Question (ID: 11091)

Question:

$\text{An ideal heat engine (Carnot engine) works between temperatures } T_1 \text{ and } T_2 \text{ has an efficiency } \eta\text{. The new efficiency if both the source and sink temperatures are doubled will be:}$

Options:

1. $\frac{\eta}{2}$
2. $\eta$ (Correct)
3. $2\eta$
4. $3\eta$

Solution:

$\text{Hint: } \eta = 1 - \frac{T_2}{T_1}$ $\text{Step: Find the new efficiency if both the source and sink temperatures are doubled.}$ $\text{The efficiency of the heat engine, when the temperature of the source and sink are doubled, then efficiency is given by:}$ $\eta = 1 - \frac{T_2}{T_1}$ $\text{In the first case, } \eta_1 = \frac{T_1 - T_2}{T_1} = \eta$ $\text{In the second case } \eta_2 = \frac{2T_1 - 2T_2}{2T_1}$ $\eta_2 = 1 - \frac{2T_2}{2T_1}$ $\eta_2 = \frac{T_1 - T_2}{T_1} = \eta$ $\text{Hence, option (2) is the correct answer.}$

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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
}