In a hydrogen like ion, the energy difference between the 2nd$2^{\mathrm{nd}}$ excitation energy state and ground is 108.8eV$108.8\mathrm{eV}$ . The atomic number of the ion is

Core Logic

Identify the states:

• Ground state:$

  Core Logic

  Identify the states:

  • Ground state: $n_1 = 1
  • $
  • $2^{\mathrm{nd}}excitation energy state:$ excitation energy state: $n_2 = 3

  Substitute values into the energy equation:

  $

Substitute values into the energy equation:

Step 1: Solve for Z

Rearrange the equation to isolate$

Step 1: Solve for Z

Rearrange the equation to isolate $Z^2:

Calculate the numerical value:

Pattern Recognition

Sees: "$

Pattern Recognition

Sees: "$2^{\mathrm{nd}}excitation"$ excitation" $\implies n = 3. Ground state$. Ground state $\implies n = 1. Shortcut: Energy transition ratio for$. Shortcut: Energy transition ratio for $1 \to 3is always$ is always $\frac{8}{9}of$ of $13.6 Z^2 \approx 12.1 Z^2. Since$. Since $108.8 / 12.1 \approx 9,$, $Z^2 = 9 \implies Z = 3$.

Chapter Mix

Class 12 Physics: Atoms