Import Question JSON

Current Question (ID: 19691)

Question:

$\text{Given below are two statements:}$ $\text{Statement I:}$ \text{When } \mu \text{ amount of an ideal gas undergoes adiabatic}$ $\text{change from state } (P_1, V_1, T_1) \text{ to state } (P_2, V_2, T_2),$ $\text{the work done is } W = \frac{\mu R (T_2 - T_1)}{1 - \gamma}, \text{ where } \gamma = \frac{C_P}{C_V} \text{ and } R =$ $\text{universal gas constant,}$ $\text{Statement II:}$ \text{In the above case, when work is done on the gas, the}$ $\text{temperature of the gas would rise.}$

Options:

1. $\text{Both Statement I and Statement II are correct.}$
2. $\text{Both Statement I and Statement II are incorrect.}$
3. $\text{Statement I is correct, but statement II is incorrect.}$
4. $\text{Statement I is incorrect, but statement II is correct.}$

Solution:

$\text{Hint: } \Delta U = \Delta Q - \Delta W$ $\text{Step: Analyse each statement one by one.}$ $\text{In an adiabatic process, no heat is exchanged } (Q = 0), \text{ so the first}$ $\text{law of thermodynamics becomes:}$ $W = -\Delta U = -nC_V (T_2 - T_1)$ $W = -\mu \left( \frac{f}{2} \right) R(T_2 - T_1)$ $\text{We know that: } \frac{2}{f} + 1 = \gamma$ $\frac{f}{2} = \frac{1}{\gamma - 1}$ $W = -\mu \left( \frac{f}{2} \right) R(T_2 - T_1)$ $W = -\mu \left( \frac{1}{\gamma - 1} \right) R(T_2 - T_1) = \frac{\mu R(T_2 - T_1)}{1 - \gamma}$ $Q = W + \Delta U$ $0 = W + \Delta U$ $\Delta U = -W$ $\text{If work is done on the gas, i.e., work is negative so } \Delta U > 0$ $\text{In an adiabatic process, if work is done on the gas (i.e., the gas is}$ $\text{compressed), the internal energy of the gas increases, which raises}$ $\text{its temperature. Since there is no heat exchange } (Q = 0), \text{ the}$ $\text{increase in internal energy comes entirely from the work done on}$ $\text{the gas.}$ $\text{Therefore, both Statement I and Statement II are correct}$ $\text{and consistent with the behavior of an ideal gas undergoing an}$ $\text{adiabatic process.}$ $\text{Hence, option (1) is the correct answer.}$

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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
}