Import Question JSON

Current Question (ID: 11102)

Question:

$\text{Two Carnot engines x and y are working between the same source temperature } T_1 \text{ and the same sink temperature } T_2\text{. If the temperature of the source in Carnot engine x is increased by } \Delta T\text{, and in the Carnot engine y, the temperature of the sink is increased by } \Delta T\text{, then the efficiency of x and y becomes } \eta_x \text{ and } \eta_y\text{. Then:}$

Options:

1. $\eta_x = \eta_y$
2. $\eta_x < \eta_y$
3. $\eta_x > \eta_y$ (Correct)
4. $\text{The relation between } \eta_x \text{ and } \eta_y \text{ depends on the nature of the working substance}$

Solution:

$\text{The correct option is 3: } \eta_x > \eta_y$ $\text{Explanation}$ $\text{The efficiency } (\eta) \text{ of a Carnot engine is given by the formula:}$ $\eta = 1 - \frac{T_{\text{sink}}}{T_{\text{source}}}$ $\text{For engine x, the source temperature } (T_{\text{source}}) \text{ increases from } T_1 \text{ to } T_1 + \Delta T\text{. This makes the fraction } \frac{T_{\text{sink}}}{T_{\text{source}}} \text{ smaller, which increases the overall efficiency.}$ $\eta_x = 1 - \frac{T_2}{T_1 + \Delta T} > 1 - \frac{T_2}{T_1} = \eta_{\text{initial}}$ $\text{For engine y, the sink temperature } (T_{\text{sink}}) \text{ increases from } T_2 \text{ to } T_2 + \Delta T\text{. This makes the fraction } \frac{T_{\text{sink}}}{T_{\text{source}}} \text{ larger, which decreases the overall efficiency.}$ $\eta_y = 1 - \frac{T_2 + \Delta T}{T_1} < 1 - \frac{T_2}{T_1} = \eta_{\text{initial}}$ $\text{By comparing the two outcomes, we can conclude that } \eta_x > \eta_y\text{. The efficiency of a Carnot engine is independent of the working substance, making option 4 incorrect.}$

Import JSON File

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
}