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Current Question (ID: 10998)

Question:

$\text{A monoatomic gas at a pressure } P, \text{ having a volume } V, \text{ expands isothermally to a volume } 2V \text{ and then adiabatically to a volume } 16V. \text{ The final pressure of the gas is:}$ $\left(\text{Take: } \gamma = \frac{5}{3}\right)$

Options:

1. $64P$
2. $32P$
3. $\frac{P}{64}$ (Correct)
4. $16P$

Solution:

$\text{Hint: } PV^\gamma = \text{constant}$ $\text{Step: Find the final pressure of the gas.}$ $\text{In the case of the isothermal process: } PV = \text{constant}$ $\Rightarrow P_1V_1 = P_2V_2$ $\Rightarrow PV = P_2(2V)$ $\Rightarrow P_2 = \frac{P}{2}$ $\text{In adiabatic process: } PV^\gamma = \text{constant}$ $\Rightarrow P_1V_1^\gamma = P_2V_2^\gamma$ $\Rightarrow P_2(2V)^\gamma = P_f(16V)^\gamma$ $\Rightarrow P_f = P_2 \left(\frac{2V}{16V}\right)^\gamma$ $\Rightarrow P_f = \frac{P}{2} \times \left(\frac{1}{8}\right)^{\frac{5}{3}} = \frac{P}{2} \times \left(\frac{1}{2}\right)^5$ $\Rightarrow P_f = \frac{P}{64}$ $\text{Hence, option (3) is the correct answer.}$

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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
}