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Current Question (ID: 19682)

Question:

$\text{A diatomic gas, having } C_P = \frac{7}{2} R \text{ and } C_V = \frac{5}{2} R \text{ is heated at constant pressure. The ratio of } dU : dQ : dW \text{ is:}$

Options:

1. $5 : 7 : 3$
2. $5 : 7 : 2$
3. $3 : 7 : 2$
4. $3 : 5 : 2$

Solution:

$\text{Hint: } dQ = dU + dW$ $\text{Step 1: Find the change in internal energy and heat.}$ $dU = nC_V dT \Rightarrow n \left( \frac{5}{2} R \right) dT \quad \ldots (1)$ $dQ = nC_P dT \Rightarrow n \left( \frac{7}{2} R \right) dT \quad \ldots (2)$ $\text{Step 2: Find the work done.}$ $\text{For a constant pressure process: } dW = PdV \Rightarrow nRdT \quad \ldots (3)$ $\text{Step 3: Find the ratio of change in internal energy } dU, \text{ heat supplied } dQ \text{ and work done } dW.$ $dU : dQ : dW = n \left( \frac{5}{2} R \right) dT : n \left( \frac{7}{2} R \right) dT : nRdT$ $\text{Now, divide all terms by the common factor } nRdT \text{ we get;}$ $\Rightarrow dU : dQ : dW = \frac{5}{2} : \frac{7}{2} : 1$ $\Rightarrow dU : dQ : dW = 5 : 7 : 2$ $\text{Hence, option (2) is the correct answer.}$

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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
}