Answer:
[tex]\boxed{3}[/tex]
Explanation:
The Rydberg equation gives the wavelength λ for the transitions:
[tex]\dfrac{1}{\lambda} = R \left ( \dfrac{1}{n_{i}^{2}} - \dfrac{1}{n_{f}^{2}} \right )[/tex]
where
R= the Rydberg constant (1.0974 ×10⁷ m⁻¹) and
[tex]\text{$n_{i}$ and $n_{f}$ are the numbers of the energy levels}[/tex]
Data:
[tex]n_{f} = 2[/tex]
λ = 657 nm
Calculation:
[tex]\begin{array}{rcl}\dfrac{1}{657 \times 10^{-9}} & = & 1.0974 \times 10^{7}\left ( \dfrac{1}{2^{2}} - \dfrac{1}{n_{f}^{2}} \right )\\\\1.522 \times 10^{6} &= &1.0974\times10^{7}\left(\dfrac{1}{4} - \dfrac{1}{n_{f}^{2}} \right )\\\\0.1387 & = &\dfrac{1}{4} - \dfrac{1}{n_{f}^{2}} \\\\-0.1113 & = & -\dfrac{1}{n_{f}^{2}} \\\\n_{f}^{2} & = & \dfrac{1}{0.1113}\\\\n_{f}^{2} & = & 8.98\\n_{f} & = & 2.997 \approx \mathbf{3}\\\end{array}\\\text{The value of $n_{i}$ is }\boxed{\mathbf{3}}[/tex]
