Import Question JSON

Current Question (ID: 11043)

Question:

$\text{The molar heat capacity in the case of a diatomic gas if it does work of } \frac{Q}{4} \text{ when heat } Q \text{ is supplied to it is:}$

Options:

1. $\frac{2}{5}R$
2. $\frac{5}{2}R$
3. $\frac{10}{3}R$ (Correct)
4. $\frac{6}{7}R$

Solution:

$\text{Hint: } dU = nC_V dT$ $\text{Step: Find the molar heat capacity of the gas.}$ $\text{The internal energy of the diatomic gas is given by;}$ $dU = nC_V dT$ $\Rightarrow dU = \frac{5}{2}RdT$ $\Rightarrow dT = \frac{2}{5} \frac{dU}{R} \ldots (1)$ $\text{From the first law of thermodynamics;}$ $dU = dQ - dW = Q - \frac{Q}{4} = \frac{3Q}{4}$ $\Rightarrow dU = \frac{3}{4}Q \ldots (2)$ $\text{The molar-specific heat of the gas is given by;}$ $\Rightarrow C = \frac{dQ}{dT} = \frac{Q}{2\left(\frac{dU}{5R}\right)} = \frac{5RQ}{2\left[\frac{3Q}{4}\right]} = \frac{10}{3}R$ $\text{Hence, option (3) is the correct answer.}$

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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
}