A : The radius of second orbit of ` he^(+)` is equal to that of first orbit of hydrogen
R : The radius of an orbit in hydrogen like species is directly proportional to n and inversely proportional to Z .

Options

A. IF both assertion & reason are true and the reason is the correct explanation of the assertion then mark
B. IF both assertion & reason are true but the reason is not the correct explanation of the assertion , then mark
C. IF assertion is true statement but reason is false , then mark
D. If both assertion and reason are false statements then mark

Answer

To solve the question, we will analyze both the assertion (A) and the reason (R) step by step.

### Step 1: Understanding the Assertion (A)
The assertion states that the radius of the second orbit of \( \text{He}^+ \) is equal to that of the first orbit of hydrogen.

### Step 2: Calculate the Radius of the Second Orbit of \( \text{He}^+ \)
According to Bohr's model, the radius of the nth orbit is given by the formula:

\[ R_n = \frac{0.53 \, n^2}{Z} \, \text{Å} \]

For \( \text{He}^+ \):
- \( n = 2 \) (second orbit)
- \( Z = 2 \) (atomic number of helium)

Substituting these values into the formula:

\[ R_2 = \frac{0.53 \times 2^2}{2} = \frac{0.53 \times 4}{2} = \frac{2.12}{2} = 1.06 \, \text{Å} \]

### Step 3: Calculate the Radius of the First Orbit of Hydrogen
For hydrogen (\( \text{H} \)):
- \( n = 1 \) (first orbit)
- \( Z = 1 \) (atomic number of hydrogen)

Substituting these values into the formula:

\[ R_1 = \frac{0.53 \times 1^2}{1} = 0.53 \, \text{Å} \]

### Step 4: Compare the Two Radii
Now we compare the two calculated radii:
- Radius of the second orbit of \( \text{He}^+ \) = \( 1.06 \, \text{Å} \)
- Radius of the first orbit of hydrogen = \( 0.53 \, \text{Å} \)

Since \( 1.06 \, \text{Å} \neq 0.53 \, \text{Å} \), the assertion is **false**.

### Step 5: Understanding the Reason (R)
The reason states that the radius of an orbit in hydrogen-like species is directly proportional to \( n \) and inversely proportional to \( Z \).

### Step 6: Analyze the Proportionality
From the formula \( R_n = \frac{0.53 \, n^2}{Z} \):
- The radius is **directly proportional to \( n^2 \)**, not \( n \).
- The radius is **inversely proportional to \( Z \)**.

Thus, the reason is also **false**.

### Conclusion
Both the assertion and reason are false. Therefore, the correct option is:

**Option D: Both assertion and reason are false.**

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