Import Question JSON

Current Question (ID: 11009)

Question:

$\text{A pressure } (P) \text{ versus volume } (V) \text{ graph is obtained for an ideal gas from state } A \text{ to state } B \text{ (as shown in the figure below). Which of the following is the correct statement?}$

Options:

1. $\text{The temperature of gas increases continuously.}$
2. $\text{The temperature of gas decreases continuously.}$
3. $\text{The temperature first increases, then decreases.}$ (Correct)
4. $\text{The temperature first decreases, then increases.}$

Solution:

$\text{From the given } P \text{ versus } V \text{ graph, the relationship can be expressed as a linear equation:}$ \n\n $P = -aV+b$ \n\n $\text{where } a \text{ and } b \text{ are positive constants.}$ \n\n $\text{Multiplying both sides by } V \text{ to relate to the ideal gas law:}$ \n\n $PV = -aV^2 + bV$ \n\n $\text{According to the ideal gas law, } PV = nRT \text{, so we can substitute this into the equation:}$ \n\n $nRT = -aV^2 + bV$ \n\n $\text{Solving for } T \text{ (temperature):}$ \n\n $T = \frac{1}{nR}(-aV^2 + bV)$ \n\n $\text{This is the equation of a parabola that opens downward, as the coefficient of the } V^2 \text{ term is negative. The graph of a downward-opening parabola first increases to a maximum value and then decreases.}$ \n\n $\text{Therefore, as the volume } (V) \text{ increases from state } A \text{ to state } B \text{, the temperature } (T) \text{ will first increase and then decrease. This behavior is represented by the given } T \text{ vs } V \text{ graph in the solution figure.}$ \n\n $\text{Hence, the correct statement is that the temperature first increases and then decreases.}$ \n\n $\text{The correct option is 3.}$

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
}