Energy and radius of first Bohr orbit of He⁺ and Li2⁺ are [Given R_H = 2.18×10^–18 J, a₀ = 52.9 pm]

The correct option is:
En (Li²⁺) = –19.62 × 10^–18 J; rn (Li²⁺) = 17.6 pm; En (He⁺) = –8.72 × 10^–18 J; rn (He⁺) = 26.4 pm

To find the energy and radius of the first Bohr orbit ($n=1$) for hydrogen-like species like ${\mathrm{He}}^{+}$ and ${\mathrm{Li}}^{2+}$, we use the standard Bohr formulas:

1. Energy of the n-th Bohr orbit ($E_n$):
$E_n=-R_H\times \frac{Z^2}{n^2}$
where $R_H=2.18\times {10}^{-18} J$ and $Z$ is the atomic number of the element.

2. Radius of the n-th Bohr orbit ($r_n$):
$r_n=a_0\times \frac{n^2}{Z}$
where $a_0=52.9 \mathrm{pm}$.

Step-by-step Calculations:

For Helium ion (${\mathrm{He}}^{+}$):
Atomic number ($Z$) = 2, and first orbit ($n$) = 1.
• Energy calculation:
$E_1\left({\mathrm{He}}^{+}\right)=-2.18\times {10}^{-18} J\times \frac{2^2}{1^2}$
$E_1\left({\mathrm{He}}^{+}\right)=-2.18\times {10}^{-18} J\times 4=-8.72\times {10}^{-18} J$

• Radius calculation:
$r_1\left({\mathrm{He}}^{+}\right)=52.9 \mathrm{pm}\times \frac{1^2}{2}$
$r_1\left({\mathrm{He}}^{+}\right)=26.45 \mathrm{pm}\approx 26.4 \mathrm{pm}$

For Lithium ion (${\mathrm{Li}}^{2+}$):
Atomic number ($Z$) = 3, and first orbit ($n$) = 1.
• Energy calculation:
$E_1\left({\mathrm{Li}}^{2+}\right)=-2.18\times {10}^{-18} J\times \frac{3^2}{1^2}$
$E_1\left({\mathrm{Li}}^{2+}\right)=-2.18\times {10}^{-18} J\times 9=-19.62\times {10}^{-18} J$

• Radius calculation:
$r_1\left({\mathrm{Li}}^{2+}\right)=52.9 \mathrm{pm}\times \frac{1^2}{3}$
$r_1\left({\mathrm{Li}}^{2+}\right)=17.63 \mathrm{pm}\approx 17.6 \mathrm{pm}$