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Current Question (ID: 11022)

Question:

$\text{One mole of an ideal gas at an initial temperature of } T K \text{ does } 6R \text{ joules of work adiabatically. If the ratio of specific heats of this gas at constant pressure and at constant volume is } \frac{5}{3}, \text{ the final temperature of the gas will be:}$

Options:

1. $(T + 2.4)K$
2. $(T - 2.4)K$
3. $(T + 4)K$
4. $(T - 4)K$ (Correct)

Solution:

$\text{For an adiabatic process, the work done is given by:}$ $W = \frac{R(T_i - T_f)}{\gamma - 1}$ $\text{Given:}$ $\text{- Initial temperature } = T K$ $\text{- Work done } W = 6R \text{ joules}$ $\text{- Ratio of specific heats } \gamma = \frac{5}{3}$ $\text{Substituting the values:}$ $6R = \frac{R(T - T_f)}{\frac{5}{3} - 1}$ $6R = \frac{R(T - T_f)}{\frac{5-3}{3}}$ $6R = \frac{R(T - T_f)}{\frac{2}{3}}$ $6R = \frac{3R(T - T_f)}{2}$ $\text{Solving for } T_f:$ $12R = 3R(T - T_f)$ $4 = T - T_f$ $T_f = T - 4$ $\text{Therefore, the final temperature is } (T - 4)K$

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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
}