Question

Which hydrogen-like species will have the same radius as the 1st Bohr orbit of a hydrogen atom?

• A. n = 2, Li²⁺
• B. n = 2, Be³⁺
• C. n = 2, He⁺
• D. n = 3, Li²⁺

Answer 1

The Correct Option is B

The radius of the nth Bohr orbit for a hydrogen-like species is given by:

\( r = \frac{n^2a_0}{Z} \)

where n is the principal quantum number, \(a_0\) is the Bohr radius for hydrogen, and Z is the atomic number. For the 1st Bohr orbit of hydrogen (n = 1, Z = 1), \(r = a_0\).

We need to find a species where \(r = a_0\). Let us check the options:

• Option 1: n = 2, Li²⁺ (Z = 3):

  \( r = \frac{2^2a_0}{3} = \frac{4}{3}a_0 \)

• Option 2: n = 2, Be³⁺ (Z = 4):

  \( r = \frac{2^2a_0}{4} = a_0 \)

• Option 3: n = 2, He⁺ (Z = 2):

  \( r = \frac{2^2a_0}{2} = 2a_0 \)

• Option 4: n = 3, Li²⁺ (Z = 3):

  \( r = \frac{3^2a_0}{3} = 3a_0 \)

Only option 2 gives the same radius as the 1st Bohr orbit of hydrogen.