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Current Question (ID: 19686)

Question:

$\text{One mole of a rigid diatomic gas performs a work of } \frac{Q}{5} \text{ when heat } Q \text{ is supplied to it. The molar heat capacity of the gas during this transformation is:}$

Options:

1. $\frac{15}{8} R$
2. $\frac{5}{8} R$
3. $\frac{25}{8} R$
4. $\frac{5}{7} R$

Solution:

$\text{Hint: } C = \frac{Q}{n \Delta T}$ $\text{Step 1: Find the change in internal energy.}$ $Q = \Delta U + W$ $\Rightarrow Q = \Delta U + \frac{Q}{5}$ $\text{The change in internal energy is given by;}$ $\Rightarrow \Delta U = \frac{4Q}{5}$ $\text{Step 2: Find the molar heat capacity of the gas.}$ \Rightarrow nC_v \Delta T = \frac{4}{5} nC \Delta T \quad [\Delta U = nC_v \Delta T, \ Q = nC \Delta T]$ $\Rightarrow C = \frac{5}{4} C_v$ $\Rightarrow C = \frac{5}{4} \left( \frac{f}{2} \right) R \quad \left[ C_v = \frac{f}{2} R \right]$ $\Rightarrow C = \frac{5}{4} \left( \frac{5}{2} \right) R \quad [f = 5 \text{ for diatomic gas}]$ $\Rightarrow C = \frac{25}{8} R$ $\text{Therefore, the molar heat capacity of the gas is } \frac{25}{8} R.$ $\text{Hence, option (3) is the correct answer.}$

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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
}