To determine which hydrogen-like species will have the same radius as the 1 st Bohr orbit of the hydrogen atom, we first need to recall the formula for the radius of the nth orbit in a hydrogen-like atom. The radius for the
nth orbit in a hydrogen-like atom is given by:
rn=‌⋅‌where:
rn is the radius of the
nth orbit.
n is the principal quantum number.
h is Planck's constant.
m is the mass of the electron.
e is the charge of the electron.
Z is the atomic number of the nucleus.
For a hydrogen atom in the 1 st orbit (
n=1,Z=1 ), the radius is:
r1=‌Now, for a hydrogen-like species with atomic number
Z and in the
nth orbit, the radius is:
rn=‌ We need to find the species and quantum number
n such that its radius equals the radius of the 1 st Bohr orbit of hydrogen, i.e.,
r1 of the hydrogen atom. This can be written as:
‌=‌ Simplifying, we get:
‌=1This gives us:
n2=Z So, the species must satisfy this condition:
For
n=2,n2=22=4. So,
Z=4.
For
n=3,n2=32=9. So,
Z≠4 (none of the options align with this condition).
Let's check if any species listed in the options have
Z=4 corresponding to
n=2 :
Option B:
n=2,Be3+( Beryllium has an atomic number of
4(Z=4)).
This option satisfies the condition.
Therefore, the correct answer is:
Option B
n=2,Be3+