Line Spectra of Hydrogen — Revision Notes

⚡ 30-Second Revision

Energy Levels (Hydrogen): — $E_n = -\frac{13.6}{n^2}\,\text{eV}$ ($n=1, 2, 3, \dots$)
Rydberg Formula (Hydrogen): — $\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
Rydberg Formula (Hydrogen-like ions): — $\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
Rydberg Constant (R): — $1.097 \times 10^7\,\text{m}^{-1}$
Lyman Series: — $n_f=1$, $n_i=2,3,4,\dots$, UV region
Balmer Series: — $n_f=2$, $n_i=3,4,5,\dots$, Visible region
Paschen Series: — $n_f=3$, $n_i=4,5,6,\dots$, IR region
Brackett Series: — $n_f=4$, $n_i=5,6,7,\dots$, IR region
Pfund Series: — $n_f=5$, $n_i=6,7,8,\dots$, IR region
Series Limit (shortest $\lambda$): — $n_i = \infty$
First Line (longest $\lambda$): — $n_i = n_f+1$
Photon Energy: — $E = h\nu = hc/\lambda$

2-Minute Revision

The line spectrum of hydrogen is a key concept demonstrating the quantization of atomic energy. Electrons in hydrogen atoms exist in discrete energy levels, $E_n = -13.6/n^2\,\text{eV}$. When an electron transitions from a higher energy level ($n_i$) to a lower one ($n_f$), it emits a photon whose wavelength is given by the Rydberg formula: $1/\lambda = R (1/n_f^2 - 1/n_i^2)$. The Rydberg constant $R$ is approximately $1.097 \times 10^7\,\text{m}^{-1}$.

These transitions form distinct spectral series:

Lyman Series ($n_f=1$): — Ultraviolet region.
Balmer Series ($n_f=2$): — Visible region (e.g., H-alpha line).
Paschen Series ($n_f=3$): — Infrared region.
Brackett Series ($n_f=4$): — Infrared region.
Pfund Series ($n_f=5$): — Infrared region.

For any series, the shortest wavelength (series limit) occurs when $n_i = \infty$, and the longest wavelength (first line) occurs when $n_i = n_f+1$. For hydrogen-like ions (e.g., He$^+$), the Rydberg formula is modified by a $Z^2$ factor: $1/\lambda = R Z^2 (1/n_f^2 - 1/n_i^2)$. Remember that $E = hc/\lambda$ links energy, wavelength, and frequency. This topic is frequently tested in NEET for both conceptual understanding and numerical calculations.

5-Minute Revision

The line spectrum of hydrogen is a direct consequence of the quantized energy levels within the atom, a concept explained by Bohr's model. Electrons can only occupy specific orbits, each with a discrete energy $E_n = -13.

6/n^2\,\text{eV}$, where$n$is the principal quantum number. When an electron jumps from a higher energy level ($n_i$) to a lower one ($n_f$), it emits a photon with energy equal to the energy difference,$E_{photon} = E_{n_i} - E_{n_f}$.

This energy corresponds to a specific wavelength $\lambda = hc/E_{photon}$.

The wavelengths of these spectral lines are precisely predicted by the Rydberg formula: $1/\lambda = R (1/n_f^2 - 1/n_i^2)$, where $R$ is the Rydberg constant ($1.097 \times 10^7\,\text{m}^{-1}$). For hydrogen-like ions (like He$^+$ with $Z=2$), the formula becomes $1/\lambda = R Z^2 (1/n_f^2 - 1/n_i^2)$.

The spectral lines are grouped into series based on the final energy level $n_f$:

1
Lyman Series ($n_f=1$): — Transitions from $n_i=2,3,4,\dots$. All lines are in the ultraviolet (UV) region.
2
Balmer Series ($n_f=2$): — Transitions from $n_i=3,4,5,\dots$. Contains lines in the visible region (e.g., H-alpha at $656\,\text{nm}$ from $n_i=3 \to n_f=2$).
3
Paschen Series ($n_f=3$): — Transitions from $n_i=4,5,6,\dots$. All lines are in the infrared (IR) region.
4
Brackett Series ($n_f=4$): — Transitions from $n_i=5,6,7,\dots$. All lines are in the infrared (IR) region.
5
Pfund Series ($n_f=5$): — Transitions from $n_i=6,7,8,\dots$. All lines are in the infrared (IR) region.

Key calculations for NEET:

Longest wavelength in a series: — Corresponds to the smallest energy jump, i.e., $n_i = n_f+1$.

* Example: Longest wavelength of Lyman series ($n_f=1, n_i=2$): $1/\lambda_{max} = R(1/1^2 - 1/2^2) = 3R/4 \implies \lambda_{max} = 4/(3R)$.

Shortest wavelength (series limit) in a series: — Corresponds to the largest energy jump, i.e., $n_i = \infty$.

* Example: Shortest wavelength of Balmer series ($n_f=2, n_i=\infty$): $1/\lambda_{min} = R(1/2^2 - 1/\infty^2) = R/4 \implies \lambda_{min} = 4/R$.

Practice applying these formulas and understanding the characteristics of each series to confidently solve NEET problems.

Prelims Revision Notes

1
Bohr's Model Foundation: — Hydrogen's line spectrum confirms Bohr's postulate of quantized energy levels. Electrons exist in discrete orbits ($n=1, 2, 3, \dots$) with energies $E_n = -13.6/n^2\,\text{eV}$. $n=1$ is the ground state, $n=2$ is the first excited state, etc.
2
Photon Emission/Absorption: — Light is emitted when an electron jumps from a higher energy level ($n_i$) to a lower one ($n_f$). Light is absorbed for the reverse transition. The energy of the photon is $\Delta E = E_{n_i} - E_{n_f}$.
3
Rydberg Formula: — The fundamental equation for calculating wavelength $\lambda$ is $1/\lambda = R (1/n_f^2 - 1/n_i^2)$, where $R = 1.097 \times 10^7\,\text{m}^{-1}$ is the Rydberg constant. Remember $n_i > n_f$ for emission.
4
Hydrogen-like Ions: — For single-electron ions with atomic number $Z$ (e.g., He$^+$, Li$^{2+}$), the formula becomes $1/\lambda = R Z^2 (1/n_f^2 - 1/n_i^2)$. The energy levels are $E_n = -13.6 Z^2/n^2\,\text{eV}$.
5
Spectral Series and Regions:

* Lyman Series: $n_f=1$. Transitions from $n_i=2,3,4,\dots$. All lines are in the Ultraviolet (UV) region. * Balmer Series: $n_f=2$. Transitions from $n_i=3,4,5,\dots$. Lines are in the Visible region (e.

g., H-alpha, H-beta). * Paschen Series: $n_f=3$. Transitions from $n_i=4,5,6,\dots$. All lines are in the Infrared (IR) region. * Brackett Series: $n_f=4$. Transitions from $n_i=5,6,7,\dots$.

All lines are in the Infrared (IR) region. * Pfund Series: $n_f=5$. Transitions from $n_i=6,7,8,\dots$. All lines are in the Infrared (IR) region.

1
Longest and Shortest Wavelengths:

* Longest wavelength (first line): Occurs for the smallest energy jump, i.e., $n_i = n_f+1$. * Shortest wavelength (series limit): Occurs for the largest energy jump, i.e., $n_i = \infty$.

1
Ionization Energy: — Energy required to remove an electron from the ground state ($n=1$) to $n=\infty$. For hydrogen, this is $0 - (-13.6\,\text{eV}) = 13.6\,\text{eV}$.
2
Relationship between E, $\nu$, $\lambda$: — $E = h\nu = hc/\lambda$. Use appropriate constants: $h = 6.626 \times 10^{-34}\,\text{J\cdot s}$, $c = 3 \times 10^8\,\text{m/s}$. For energy in eV, $hc \approx 1240\,\text{eV\cdot nm}$.

Vyyuha Quick Recall

To remember the order of spectral series and their regions: Lovely Boys Play Baseball Professionally.

Lyman ($n_f=1$) - Ultraviolet (UV)
Balmer ($n_f=2$) - Visible (V)
Paschen ($n_f=3$) - Infrared (IR)
Brackett ($n_f=4$) - Infrared (IR)
Pfund ($n_f=5$) - Infrared (IR)

(Note: The regions UV, Visible, IR can be remembered as 'U V I I I' for the first letters of the series.)