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Current Question (ID: 10993)

Question:

$\text{An ideal gas at } 27^\circ\text{C is compressed adiabatically to } \frac{8}{27} \text{ of its original volume. If } \gamma = \frac{5}{3}\text{, then the rise in temperature will be:}$

Options:

1. $450 \text{ K}$
2. $375 \text{ K}$ (Correct)
3. $225 \text{ K}$
4. $405 \text{ K}$

Solution:

$\text{Hint: } TV^{\gamma-1} = \text{constant}$ $\text{Step: Find the rise in the temperature.}$ $\text{Given: } T_1 = 300 \text{ K}, V_1 = V, V_2 = \frac{8}{27}V, \gamma = \frac{5}{3}$ $\text{In adiabatic compression; } TV^{\gamma-1} = \text{constant}$ $\Rightarrow \frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\gamma-1}$ $\Rightarrow \frac{T_2}{300} = \left(\frac{V}{\frac{8V}{27}}\right)^{\frac{5}{3}-1}$ $\Rightarrow T_2 = 300 \left(\frac{27}{8}\right)^{\frac{2}{3}}$ $\Rightarrow T_2 = 300 \times \left(\frac{3}{2}\right)^2 = 300 \times \frac{9}{4}$ $\Rightarrow T_2 = 75 \times 9 = 675 \text{ K}$ $\text{The rise in temperature is given by;}$ $\Delta T = T_2 - T_1$ $\Rightarrow \Delta T = 675 - 300 = 375 \text{ K}$ $\text{Hence, option (2) is the correct answer.}$

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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
}