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Current Question (ID: 19672)

Question:

$\text{An engine takes in } 5 \text{ moles of air at } 20^\circ \text{ C and } 1 \text{ atm, and compresses it adiabatically to } \frac{1}{10} \text{ th of the original volume.}$ $\text{Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be:}$

Options:

1. $46 \text{ J}$
2. $46 \text{ kJ}$
3. $23 \text{ J}$
4. $23 \text{ kJ}$

Solution:

$\text{Hint: } PV^\gamma = \text{Constant for adiabatic process.}$ $\text{Diatomic:}$ $f=5$ $\gamma = \frac{7}{5}$ $T_i = \frac{T}{V} = 273 + 20 = 293 \text{ K}$ $V_f = \frac{V}{10}$ $\text{Adiabatic } T_1V_1^{\gamma-1} = T_2V_2^{\gamma-1}$ $T \cdot V^{7/5-1} = T_2 \left(\frac{V}{10}\right)^{7/5-1}$ $T_2 = T \cdot 10^{2/5}$ $\Delta U = \frac{nf_R(T_2-T_1)}{2}$ $= \frac{25}{6} T (10^{2/5} - 1)$ $= \frac{5 \times 5 \times \frac{25}{3} \times (T \cdot 10^{2/5} - T)}{2}$ $= \frac{625}{6} \times 293 \times (10^{2/5} - 1)$ $= 4.033 \times 10^3 \approx 4 \text{ kJ}$

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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
}