Energy of S1 State Relative to Hydrogen Ground State

To find the energy of state $\text{S}_1$ in units of the hydrogen atom ground state energy, we analyze the characteristics of state $\text{S}_1$:

1. Identification of Quantum Numbers for State $\text{S}_1$:

   • The state $\text{S}_1$ belongs to the hydrogen-like species $\text{Li}^{2+}$, which has an atomic number $Z = 3$.
   • The passage states that $\text{S}_1$ is spherically symmetric, which implies that the azimuthal quantum number $l = 0$ (an s-orbital).
   • The number of radial nodes is given by the formula:
     $\text{Number of radial nodes} = n - l - 1$
     Given that there is $1$ radial node:
     $n - 0 - 1 = 1 \implies n = 2$
   • Thus, state $\text{S}_1$ corresponds to the $2\text{s}$ state of $\text{Li}^{2+}$ with principal quantum number $n = 2$.

2. Energy Calculation:

   • The energy $E_n$ of a hydrogen-like system is given by Bohr's formula:
     $E_n = E_{\text{H}} \times \frac{Z^2}{n^2}$
     where $E_{\text{H}}$ represents the ground state energy of the hydrogen atom ($Z = 1, n = 1$).
   • For state $\text{S}_1$ of $\text{Li}^{2+}$ ($Z = 3, n = 2$):
     $E(\text{S}_1) = E_{\text{H}} \times \frac{3^2}{2^2} = E_{\text{H}} \times \frac{9}{4} = 2.25 \, E_{\text{H}}$

Therefore, the energy of state $\text{S}_1$ in units of the hydrogen atom ground state energy is $2.25$.

Correct Option: C