Atomic Physics

The formula for the radius of the nth orbit in a hydrogen-like atom (with atomic number Z) is given by the Bohr model:
, where is the Bohr radius (a constant).
For the hydrogen atom (H), the atomic number is . The radius of the second orbit (n=2) is:
.
For the unknown ion (x) with atomic number Z, the radius of the nth orbit is:
.
We are given that these two radii are the same: .
.
Canceling gives the condition , or .
Now we test the given options to see which pair (n, x) satisfies this condition.
(A) n=4, x=Be . For Beryllium, Z=4. So, . And . So is satisfied. But the ion is Be , which has 2 electrons. The Bohr model applies to single-electron species. This option is physically questionable but mathematically fits if we only consider Z.
(B) n=3, x=Li . For Lithium, Z=3. So, . And . . This is incorrect. (Li is a valid single-electron ion).
(C) n=4, x=Be . For Beryllium, Z=4. So, . And . is satisfied. Be is a single-electron ion, so the Bohr model is applicable. This is a valid option.
(D) n=2, x=He . For Helium, Z=2. So, . And . . This is incorrect. (He is a valid single-electron ion).
Comparing (A) and (C), both satisfy the mathematical relation . However, the Bohr model is strictly valid only for hydrogen-like (single-electron) species. Be has two electrons, while Be has one electron. Therefore, (C) is the physically correct answer.

Step 1: Understanding the Concept:

The radius of the orbit for a hydrogen-like species is given by the Bohr formula: where is the Bohr radius (constant), is the principal quantum number, and is the atomic number.
Step 2: Key Formula or Approach:

1. For Hydrogen atom ( ): , . Given radius . 2. For Helium ion ( ): , . Find radius .
Step 3: Detailed Explanation:

Write the expression for the radius of Hydrogen ( ): We are given , so: Now, write the expression for the radius of ( ): Substitute into this equation:
Step 4: Final Answer:

The radius is .