Import Question JSON

Current Question (ID: 19687)

Question:

$\text{A Carnot engine whose heat sinks at } 27^\circ\text{C, has an efficiency of } 25\%. \text{ By how many degrees should the temperature of the source be changed to increase the efficiency by } 100\% \text{ of the original efficiency?}$

Options:

1. $\text{Increase by } 18^\circ\text{C}$
2. $\text{Increase by } 200^\circ\text{C}$
3. $\text{Increase by } 120^\circ\text{C}$
4. $\text{Increase by } 73^\circ\text{C}$

Solution:

$\text{Hint: } \eta = 1 - \frac{T_2}{T_1}$ $\text{Step 1: Find the temperature of the sink.}$ $\text{The efficiency of a Carnot engine is given by:}$ $\eta = 1 - \frac{T_2}{T_1}$ $\text{by substituting the known values we get:}$ $1 - \frac{300}{T} = 0.25$ $\Rightarrow T = 400 \text{ K}$ $\text{Step 2: Find the change in temperature of the sink.}$ $\text{If efficiency increased by } 100\% \text{ then new efficiency, } \eta' = 50\%$ $\text{Now,}$ $\eta = 1 - \frac{T_2}{T_1}$ $\text{by substituting the known values we get:}$ $1 - \frac{300}{T'} = 0.5$ $\Rightarrow T' = 600 \text{ K}$ $\text{Step 3: Find the increase in temperature of the source.}$ $\text{The increase in temperature of the engine is given by:}$ $\Delta T = T' - T$ $\Rightarrow \Delta T = 600 - 400 = 200^\circ\text{C}$ $\text{Hence, option (2) is the correct answer.}$

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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
}