Balmer Series — Revision Notes

⚡ 30-Second Revision

Balmer Series: — Electron transitions to $n_f = 2$.
Initial States ($n_i$): — $3, 4, 5, \dots$
Spectral Region: — Visible and near Ultraviolet.
Rydberg Formula: — $1/\lambda = R_H (1/n_f^2 - 1/n_i^2)$
Rydberg Constant ($R_H$): — $1.097 \times 10^7\,\text{m}^{-1}$
H-alpha ($n_i=3 \to n_f=2$): — Longest wavelength, lowest energy in Balmer series (Red, $\approx 656.3\,\text{nm}$).
Series Limit ($n_i=\infty \to n_f=2$): — Shortest wavelength, highest energy in Balmer series (UV, $\approx 364.6\,\text{nm}$).
Energy of $n$-th level: — $E_n = -13.6/n^2\,\text{eV}$
Photon Energy: — $E = h\nu = hc/\lambda$

2-Minute Revision

The Balmer series is a set of spectral lines in the hydrogen atom's emission spectrum, arising from electron transitions where the final energy level is $n_f=2$. The initial energy levels ($n_i$) can be $3, 4, 5, \dots$.

Its most distinctive feature is that its prominent lines fall within the visible part of the electromagnetic spectrum, making it historically significant. The wavelengths are precisely given by the Rydberg formula: $1/\lambda = R_H (1/2^2 - 1/n_i^2)$, where $R_H$ is the Rydberg constant ($1.

097 \times 10^7\,\text{m}^{-1}$). The H-alpha line ($n_i=3 \to n_f=2$) is the longest wavelength (red,$\approx 656.3\,\text{nm}$) and the lowest energy line in the series. As$n_i$increases, the lines get closer and converge to a series limit ($n_i=\infty \to n_f=2$), which represents the shortest wavelength (UV,$\approx 364.

6\,\text{nm}$) and highest energy. Remember that energy levels are quantized ($E_n = -13.6/n^2\,\text{eV}$), and emitted photon energy equals the difference in energy levels.

5-Minute Revision

The Balmer series is one of the fundamental spectral series of the hydrogen atom, crucial for understanding atomic structure and quantum mechanics. It is defined by electron transitions from any higher energy level ($n_i = 3, 4, 5, \dots$) down to the second principal energy level ($n_f = 2$).

What makes the Balmer series particularly important is that its most prominent lines (H-alpha, H-beta, H-gamma) are visible to the human eye, appearing as distinct red, blue-green, and violet lines, respectively.

This visibility allowed early scientists to study and quantify these lines, leading to the empirical Rydberg formula.

Key Formula and Calculations:

The wavelength ($\lambda$) of any line in the Balmer series can be calculated using the Rydberg formula:

$$\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

For the Balmer series, $n_f = 2$. The Rydberg constant $R_H \approx 1.097 \times 10^7\,\text{m}^{-1}$.

H-alpha line ($n_i=3 \to n_f=2$): — This is the longest wavelength line in the series, corresponding to the smallest energy transition. \$\frac{1}{\lambda_{H\alpha}} = R_H \left(\frac{1}{2^2} - \frac{1}{3^2}\right) = R_H \left(\frac{1}{4} - \frac{1}{9}\right) = R_H \left(\frac{5}{36}\right) \implies \lambda_{H\alpha} \approx 656.3\,\text{nm}$ (Red).
H-beta line ($n_i=4 \to n_f=2$): — \$\frac{1}{\lambda_{H\beta}} = R_H \left(\frac{1}{2^2} - \frac{1}{4^2}\right) = R_H \left(\frac{1}{4} - \frac{1}{16}\right) = R_H \left(\frac{3}{16}\right) \implies \lambda_{H\beta} \approx 486.1\,\text{nm}$ (Blue-Green).
Series Limit ($n_i=\infty \to n_f=2$): — This represents the shortest wavelength and highest energy in the series. \$\frac{1}{\lambda_{limit}} = R_H \left(\frac{1}{2^2} - \frac{1}{\infty^2}\right) = R_H \left(\frac{1}{4}\right) \implies \lambda_{limit} \approx 364.6\,\text{nm}$ (Ultraviolet).

Energy Considerations:

Each energy level of hydrogen is given by $E_n = -13.6/n^2\,\text{eV}$. The energy of an emitted photon is $E_{photon} = |E_{n_i} - E_{n_f}|$. This energy is also related to frequency ($\nu$) and wavelength ($\lambda$) by $E = h\nu = hc/\lambda$. For example, the energy of the H-alpha photon is $E_{H\alpha} = E_3 - E_2 = (-13.6/3^2) - (-13.6/2^2) = -1.51\,\text{eV} - (-3.40\,\text{eV}) = 1.89\,\text{eV}$.

Comparison with other series:

Lyman Series ($n_f=1$): — All lines in the Ultraviolet region, highest energy transitions.
Paschen Series ($n_f=3$): — All lines in the Infrared region, lower energy transitions than Balmer.

NEET Focus: Be ready to calculate wavelengths, frequencies, and energies for specific transitions. Understand the order of lines (H-alpha is longest wavelength, series limit is shortest). Differentiate between the spectral regions of different series.

Prelims Revision Notes

The Balmer series is a crucial topic for NEET, focusing on the hydrogen atom's emission spectrum. Its defining characteristic is that electrons transition to the **second principal energy level ($n_f = 2$)** from higher energy levels ($n_i = 3, 4, 5, \dots$).

Key Formulas and Constants:

1
Rydberg Formula: — $1/\lambda = R_H (1/n_f^2 - 1/n_i^2)$
2
Rydberg Constant for Hydrogen ($R_H$): — $1.097 \times 10^7\,\text{m}^{-1}$
3
Energy of $n$-th orbit: — $E_n = -13.6/n^2\,\text{eV}$
4
Photon Energy: — $E = h\nu = hc/\lambda$
5
Speed of Light ($c$): — $3 \times 10^8\,\text{m/s}$
6
Planck's Constant ($h$): — $6.626 \times 10^{-34}\,\text{J\cdot s}$

Important Lines and Properties:

H-alpha line: — Transition $n_i=3 \to n_f=2$. This is the longest wavelength and lowest energy line in the Balmer series. It appears red (approx. $656.3\,\text{nm}$). It's the 'first' line.
H-beta line: — Transition $n_i=4 \to n_f=2$. Appears blue-green (approx. $486.1\,\text{nm}$). Its energy is higher and wavelength shorter than H-alpha.
H-gamma line: — Transition $n_i=5 \to n_f=2$. Appears violet (approx. $434.0\,\text{nm}$). Even higher energy, shorter wavelength.
Series Limit: — Transition $n_i=\infty \to n_f=2$. This represents the shortest wavelength and highest energy in the Balmer series. It falls in the ultraviolet region (approx. $364.6\,\text{nm}$). This energy corresponds to the ionization energy from the $n=2$ state.

Spectral Region: The Balmer series lines are predominantly in the visible region of the electromagnetic spectrum, with the series limit extending into the near ultraviolet.

Common Traps:

Confusing $n_i$ and $n_f$ in the Rydberg formula.
Incorrectly identifying the longest vs. shortest wavelength (longest is H-alpha, shortest is series limit).
Mixing up the final energy levels for different series (Lyman $n_f=1$, Balmer $n_f=2$, Paschen $n_f=3$).
Errors in unit conversion (nm to m) or handling powers of ten in calculations.

Strategy: Practice numerical problems involving wavelength, frequency, and energy calculations. Understand the energy level diagram of hydrogen and how transitions relate to emitted photons. Be able to compare the Balmer series with other series based on their characteristics.

Vyyuha Quick Recall

To remember the final energy levels for the first three hydrogen spectral series: Look Before Passing. Lyman ($n_f=1$), Balmer ($n_f=2$), Paschen ($n_f=3$). For Balmer, remember it's the Visible series, like a Vision of 2 (n=2).