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Current Question (ID: 10788)

Question:

$\text{A body cools from a temperature of } 3T \text{ to } 2T \text{ in 10 minutes. The room temperature is } T\text{. Assuming that Newton's law of cooling is applicable, the temperature of the body at the end of the next 10 minutes will be:}$

Options:

1. $\frac{7}{4}T$
2. $\frac{3}{2}T$ (Correct)
3. $\frac{4}{3}T$
4. $T$

Solution:

$\text{According to Newton's law of cooling, the rate of cooling is proportional to the temperature difference between the body and surroundings.}$ $\frac{dT}{dt} = -k(T_{body} - T_{room})$ $\text{For the first 10 minutes:}$ $\frac{3T - 2T}{10} = k[2.5T - T]$ $\frac{T}{10} = k[1.5T]$ $\text{This gives us: } k = \frac{1}{15}$ $\text{For the next 10 minutes, starting from } 2T\text{:}$ $\frac{2T - T_f}{10} = k\left[\frac{2T + T_f}{2} - T\right]$ $\frac{2T - T_f}{10} = \frac{1}{15}\left[\frac{2T + T_f - 2T}{2}\right]$ $\frac{2T - T_f}{10} = \frac{1}{15} \cdot \frac{T_f}{2}$ $\frac{T}{2T - T_f} = \frac{1.5T}{0.5T_f}$ $T_f = 1.5T$

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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
}