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Current Question (ID: 7520)

Question:

$\text{Assertion (A): The radius of the second orbit of He}^+ \text{ is equal to that of the first orbit of hydrogen.}$\n\n$\text{Reason (R): The radius of an orbit in hydrogen-like species is directly proportional to n and inversely proportional to Z.}$

Options:

1. $\text{Both (A) and (R) are True and (R) is the correct explanation of the (A).}$
2. $\text{Both (A) and (R) are True and (R) is not the correct explanation of the (A).}$
3. $\text{(A) is True but (R) is False.}$
4. $\text{Both (A) and (R) are False.}$

Solution:

$\text{Hint: } r_n = \frac{52.9 (n^2)}{Z}$\n\n$\text{Step 1:}$\n\n$\text{Calculate the radius of the second orbit of He}^+ \text{ and the first orbit of hydrogen as follows:}$\n\n$r_H = \frac{52.9 (1^2)}{1} = \frac{52.9}{1} = 52.9 \text{ pm}$\n\n$r_{He^+} = \frac{52.9 (2^2)}{2} = \frac{52.9 \times 4}{2} = 105.8 \text{ pm}$\n\n$\text{Step 2:}$\n\n$\text{From the calculations, we can see that the radius of the second orbit of He}^+ \text{ (105.8 pm) is not equal to the radius of the first orbit of hydrogen (52.9 pm). Therefore, Assertion (A) is false.}$\n\n$\text{Examining the Reason (R): According to Bohr's model, the radius of an orbit in hydrogen-like species is given by:}$\n\n$r_n = \frac{52.9 \times n^2}{Z} \text{ pm}$\n\n$\text{This formula shows that the radius is directly proportional to } n^2 \text{ (not } n \text{) and inversely proportional to } Z. \text{ Therefore, Reason (R) is also false because it incorrectly states that the radius is directly proportional to } n \text{ rather than } n^2.$\n\n$\text{Hence, both Assertion (A) and Reason (R) are false statements.}$

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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
}