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What would be the radius of the second orbit of $H{e^ + }$ ion?
A. $1.058{A^o}$
B. $3.023{A^o}$
C. $2.068{A^o}$
D. $4.458{A^o}$

Last updated date: 18th Jun 2024
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Hint: The He+ ion is similar to the Hydrogen atom in electron configuration. From the formula for radius given by Bohr’s model of the Hydrogen atom, we can find the radius of the second orbit of the Helium ion. The atomic number of the Helium ion and the principal quantum number for the second orbit must be known.
Formula used:
$r = \dfrac{{{n^2}}}{Z} \times 0.529\mathop {\rm A}\limits^o $

Complete answer:
According to Bohr’s model for the Hydrogen atom, the radius of any ion which has a similar number of electrons as Hydrogen, i.e. one, is given by
$r = \dfrac{{{n^2}}}{Z} \times 0.529\mathop {\rm A}\limits^o $
r is the radius of the orbit
n is the principal quantum number of the orbit
Z is the atomic number
For He+ ion, n = 2 for the second orbit, and the atomic number is 2. Substituting the values in the formula, we get
  & r = \dfrac{{{n^2}}}{Z} \times 0.529\mathop {\rm A}\limits^o \cr
  & \Rightarrow r = \dfrac{{{{\left( 2 \right)}^2}}}{2} \times 0.529\mathop {\rm A}\limits^o \cr
  & \Rightarrow r = 1.058\mathop {\rm A}\limits^o \cr} $
Therefore, the radius of the second orbit of the He+ ion is $1.058\mathop {\rm A}\limits^o $.

Thus, the correct option is A.

Additional Information:
In case any of you can’t remember the formula or need to know the formula in terms of constants, we can derive the equation at hand from the postulates of Bohr’s model of the Hydrogen atom, from the angular momentum, the centripetal and electrostatic forces acting on the electron.

The Bohr’s model of Hydrogen applies only to the atoms which have a single electron in their orbit. So, the formula is only applicable for ions like He$^+$, Li$^{2+}$, Be$^{3+}$, etc., One should put the effort in learning the formula and remembering the constant value, as deriving the equation and finding the constant is a time taking task.