Hydrogen Spectrum — Revision Notes

⚡ 30-Second Revision

Energy Levels: — $E_n = -\frac{13.6}{n^2}\,\text{eV}$ (for hydrogen)
Rydberg Formula: — $\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$
Rydberg Constant: — $R \approx 1.097 \times 10^7 \text{ m}^{-1}$
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 & near-UV region
Paschen Series: — $n_f=3$, $n_i=4,5,6,\dots$, IR region
Brackett Series: — $n_f=4$, $n_i=5,6,7,\dots$, Far IR region
Pfund Series: — $n_f=5$, $n_i=6,7,8,\dots$, Far IR region
Series Limit: — $n_i=\infty$ (shortest wavelength, highest energy)
First Line: — $n_i=n_f+1$ (longest wavelength, lowest energy within a series)

2-Minute Revision

The hydrogen spectrum is a set of discrete lines, not continuous, proving electron energy levels are quantized. Bohr's model explains this: electrons transition between fixed energy levels ($E_n = -13.

6/n^2\,\text{eV}$), emitting or absorbing photons. The Rydberg formula,$\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$, calculates the wavelength of these photons. Remember the five main series: Lyman ($n_f=1$, UV), Balmer ($n_f=2$, Visible/UV), Paschen ($n_f=3$, IR), Brackett ($n_f=4$, Far IR), and Pfund ($n_f=5$, Far IR).

The 'series limit' corresponds to $n_i=\infty$, giving the shortest wavelength and highest energy for that series. The 'first line' corresponds to $n_i=n_f+1$, giving the longest wavelength and lowest energy.

Higher energy transitions mean shorter wavelengths and higher frequencies. Be ready to apply these concepts to calculate wavelengths, energies, and identify spectral regions.

5-Minute Revision

The hydrogen spectrum is a critical topic, showcasing the quantum nature of atoms. It's characterized by discrete lines, explained by Bohr's model where electrons occupy quantized energy levels $E_n = -13.

6/n^2\,\text{eV}$. When an electron drops from a higher energy level ($n_i$) to a lower one ($n_f$), it emits a photon. The wavelength of this photon is given by the Rydberg formula:$\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$, where$R$is the Rydberg constant ($1.

097 \times 10^7 \text{ m}^{-1}$). Remember$n_i > n_f$ for emission.

Key Series to Master:

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 $n_i=3,4,5,\dots$. Lines are in the Visible and near-UV region. This is the only series with visible lines ($H_\alpha, H_\beta, H_\gamma, H_\delta$).
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 Far Infrared (IR) region.
5
Pfund Series ($n_f=5$): — Transitions from $n_i=6,7,8,\dots$. All lines are in the Far Infrared (IR) region.

Important Concepts:

Series Limit: — The shortest wavelength (highest energy) line in a series, occurring when $n_i = \infty$. For example, for Lyman series limit, $\frac{1}{\lambda} = R\left(\frac{1}{1^2} - \frac{1}{\infty^2}\right) = R$.
First Line: — The longest wavelength (lowest energy) line in a series, occurring when $n_i = n_f+1$. For example, for Balmer's first line ($H_\alpha$), $n_i=3, n_f=2$.
Excitation Energy: — Energy needed to move an electron from a lower to a higher state, e.g., from $n=1$ to $n=2$ is $E_2 - E_1 = (-3.4) - (-13.6) = 10.2\,\text{eV}$.
Ionization Energy: — Energy needed to remove an electron completely from an atom (from $n$ to $n=\infty$). From ground state ($n=1$), it's $13.6\,\text{eV}$. From an excited state $n$, it's $|E_n| = 13.6/n^2\,\text{eV}$.

Example: Calculate the wavelength of the first line of the Paschen series. Here, $n_f=3$ and $n_i=4$. $\frac{1}{\lambda} = R\left(\frac{1}{3^2} - \frac{1}{4^2}\right) = R\left(\frac{1}{9} - \frac{1}{16}\right) = R\left(\frac{16-9}{144}\right) = \frac{7R}{144}$. $\lambda = \frac{144}{7R} = \frac{144}{7 \times 1.097 \times 10^7} \approx 1.875 \times 10^{-6}\,\text{m} = 1875\,\text{nm}$ (IR region).

Prelims Revision Notes

The Hydrogen Spectrum is a crucial topic for NEET, primarily tested through the application of Bohr's model and the Rydberg formula. Key points for quick recall:

1. Bohr's Energy Levels:

* Energy of electron in $n$-th orbit: $E_n = -\frac{13.6}{n^2}\,\text{eV}$. * Ground state: $n=1$, $E_1 = -13.6\,\text{eV}$. * First excited state: $n=2$, $E_2 = -3.4\,\text{eV}$. * Second excited state: $n=3$, $E_3 = -1.51\,\text{eV}$. * Energy levels get closer as $n$ increases.

2. Rydberg Formula for Wavelength:

* $\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$, where $n_i > n_f$ for emission. * Rydberg constant $R \approx 1.097 \times 10^7 \text{ m}^{-1}$. * For hydrogen-like ions (e.g., $He^+, Li^{2+}$), use $R Z^2$ instead of $R$.

3. Spectral Series (Memorize $n_f$ and EM Region):

* Lyman Series: $n_f=1$. Transitions from $n_i=2,3,4,\dots$. Region: Ultraviolet (UV). * Balmer Series: $n_f=2$. Transitions from $n_i=3,4,5,\dots$. Region: Visible (partially) and near-UV. * Paschen Series: $n_f=3$. Transitions from $n_i=4,5,6,\dots$. Region: Infrared (IR). * Brackett Series: $n_f=4$. Transitions from $n_i=5,6,7,\dots$. Region: Far Infrared (IR). * Pfund Series: $n_f=5$. Transitions from $n_i=6,7,8,\dots$. Region: Far Infrared (IR).

4. Key Terms:

* First Line of a Series: Corresponds to the smallest energy transition within that series, i.e., $n_i = n_f+1$. This gives the longest wavelength. * Series Limit: Corresponds to the largest energy transition within that series, i.

e., $n_i = \infty$. This gives the shortest wavelength. * Ionization Energy: Energy to remove electron from $n$ to $\infty$. From ground state ($n=1$), it's $13.6\,\text{eV}$. From state $n$, it's $|E_n| = 13.

6/n^2\,\text{eV}$. * **Excitation Energy:** Energy to move electron from$n_f$to$n_i$.$\Delta E = E_{n_i} - E_{n_f}$.

5. Energy-Wavelength-Frequency Relationship:

* $E = h\nu = hc/\lambda$. Higher energy means higher frequency and shorter wavelength.

6. Common Traps:

* Confusing $n_i$ and $n_f$. * Mixing up series (e.g., calculating for Lyman when Balmer is asked). * Confusing 'first line' with 'series limit'. * Arithmetic errors, especially with squares and fractions.

Vyyuha Quick Recall

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

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

(For regions, think: UV, Visible, IR, IR, IR)