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Current Question (ID: 10759)

Question:
$\text{Three rods made of the same material, having the same cross-sectional area but different lengths 10 cm, 20 cm and 30 cm are joined as shown. The temperature of the junction will be:}$
Options:
  • 1. $10.8^\circ\text{C}$
  • 2. $14.6^\circ\text{C}$
  • 3. $16.4^\circ\text{C}$ (Correct)
  • 4. $18.2^\circ\text{C}$
Solution:
\text{Hint: Apply junction law.} \text{Explanation: At steady state condition: Incoming heat = outgoing heat} \text{Let } T_J \text{ be the junction temperature and heat current } i_1, i_2 \text{ and } i_3 \text{ in the three rods as shown in the figure.} \text{Heat current is given as } \frac{\Delta T}{R} \text{ where } \Delta T \text{ is the change in temperature across the junction and } R \text{ is the thermal resistance given by } \frac{L}{kA} \text{Here } k \text{ is thermal conductivity and } A \text{ is cross-sectional area.} \text{The thermal conductivity and cross-sectional area are the same for the three rods, which means } R \propto L \text{According to the junction rule:} i_1 + i_2 = i_3 \frac{(30 - T_J)}{3R} + \frac{(20 - T_J)}{2R} = \frac{(T_J - 10)}{R} 2(30 - T_J) + 3(20 - T_J) = 6(T_J - 10) T_J = 16.4°\text{C} \text{Option (3) is correct.}

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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
}