Import Question JSON

Current Question (ID: 11085)

Question:

$\text{The Carnot cycle (reversible) of gas is represented by a pressure-volume curve}$ $\text{as shown in the figure. Consider the following statements:}$ $\text{I. The area } ABCD = \text{The work done on the gas}$ $\text{II. The area } ABCD = \text{The net heat absorbed}$ $\text{III. The change in the internal energy in the cycle } = 0$ $\text{Which of the statement(s) given above is/are correct?}$

Options:

1. $\text{I only}$
2. $\text{II only}$
3. $\text{II and III}$ (Correct)
4. $\text{I, II, and III}$

Solution:

$\text{Hint: The internal energy, which is a state function, remains unchanged over}$ $\text{a complete cycle.}$ $\text{Step: Identify the correct statements.}$ $\text{The work done by the gas (as the cyclic process is clockwise)}$ $\Delta W = \text{Area } ABCD$ $\text{So from the first law of thermodynamics } \Delta Q$ $\text{(the net heat absorbed) } = \Delta W = \text{Area } ABCD$ $\text{As the change in internal energy in cycle } \Delta U = 0.$ $\text{In a complete Carnot cycle, the area enclosed by the cycle also represents}$ $\text{the net heat absorbed by the gas, because in a cyclic process, the net work}$ $\text{done is equal to the net heat absorbed. Since the Carnot cycle is a cyclic}$ $\text{process, the system returns to its initial state after completing one cycle.}$ $\text{This means that the internal energy, which is a state function, remains}$ $\text{unchanged over a complete cycle.}$ $\text{Therefore, the total change in internal energy for the entire cycle is zero.}$ $\text{Hence option (3) is the correct answer.}$

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
}