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Current Question (ID: 10725)

Question:

$\text{A copper rod of 88 cm and an aluminium rod of an unknown length have an equal increase in their lengths independent of an increase in temperature. The length of the aluminium rod is:}$ $(\alpha_{\text{Cu}} = 1.7 \times 10^{-5} \text{ K}^{-1} \text{ and } \alpha_{\text{Al}} = 2.2 \times 10^{-5} \text{ K}^{-1})$

Options:

1. $68 \text{ cm}$ (Correct)
2. $6.8 \text{ cm}$
3. $113.9 \text{ cm}$
4. $88 \text{ cm}$

Solution:

$\text{Solution:}$ $\text{When the temperature is increased by } \Delta T\text{, their final lengths are:}$ $L_{\text{Cu}} = L_1[1 + \alpha_1 \Delta T]$ $L_{\text{Al}} = L_2[1 + \alpha_2 \Delta T]$ $\text{As change in lengths are equal:}$ $L_{\text{Cu}} - L_1 = L_{\text{Al}} - L_2$ $\text{This gives us:}$ $\alpha_1 L_1 \Delta T = \alpha_2 L_2 \Delta T$ $\text{Simplifying:}$ $\alpha_1 L_1 = \alpha_2 L_2$ $\text{Substituting the given values:}$ $1.7 \times 10^{-5} \times 0.88 = 2.2 \times 10^{-5} \times L_2$ $\text{Solving for } L_2\text{:}$ $L_2 = \frac{1.7 \times 10^{-5} \times 0.88}{2.2 \times 10^{-5}}$ $L_2 = 0.68 \text{ m} = 68 \text{ cm}$

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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
}