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Current Question (ID: 11035)

Question:

$\text{The pressure in a monoatomic gas increases linearly from 4 atm to 8 atm when its volume increases from 0.2 m}^3 \text{ to 0.5 m}^3\text{. The increase in internal energy will be:}$

Options:

1. $480 \text{ kJ}$ (Correct)
2. $550 \text{ kJ}$
3. $200 \text{ kJ}$
4. $100 \text{ kJ}$

Solution:

$\text{Hint: Internal energy depends only on temperature}$ $\text{Step 1: Find } C_V$ $C_V = \frac{R}{\gamma - 1} = \frac{R}{\frac{5}{3} - 1} = \frac{3}{2}R$ $\text{Step 2: Find difference of PV}$ $P_2V_2 - P_1V_1 = 0.5 \times 8 \times 10^5 - 0.2 \times 4 \times 10^5 = 3.2 \times 10^5 \text{J}$ $\text{Step 3: Use relation for change in internal energy}$ $\Delta U = nC_V \Delta T = \frac{3}{2}nR \Delta T$ $\text{From } PV = nRT \text{, we get } nR \Delta T = \Delta(PV)$ $\text{Therefore, } \Delta U = \frac{3}{2} \Delta PV = \frac{3}{2} \times 3.2 \times 10^5 = 480 \text{ kJ}$

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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
}