Spectrum of Hydrogen Atom and Bohr Model Basics

1. The Electromagnetic Spectrum

Before diving into the hydrogen atom's spectrum, it's crucial to understand the electromagnetic spectrum. This spectrum encompasses all types of electromagnetic radiation, ordered by their frequency or wavelength. It includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. Each type of radiation carries energy and travels at the speed of light.

The relationship between the speed of light (c), frequency (ν), and wavelength (λ) is given by the fundamental equation:

$c = \nu \lambda$

Where:

• $c$ is the speed of light (approximately $3 \times 10^8$ m/s)
• $\nu$ is the frequency (in Hertz, Hz, or s^-1)
• $\lambda$ is the wavelength (in meters, m)

Energy (E) of a photon is directly proportional to its frequency, as described by Planck's equation:

$E = h\nu$

Where:

• $E$ is the energy of the photon
• $h$ is Planck's constant (approximately $6.626 \times 10^{-34}$ J·s)

Combining these equations, we can also express energy in terms of wavelength:

$E = \frac{hc}{\lambda}$

2. Atomic Spectra: Emission and Absorption

When substances are heated or subjected to an electric discharge, they emit light. If this emitted light is passed through a prism, it separates into its constituent colors, forming an emission spectrum. This spectrum appears as a series of bright lines against a dark background.

Conversely, if white light is passed through a substance, the substance absorbs certain wavelengths, and the transmitted light, when analyzed by a prism, shows a continuous spectrum with dark lines at the positions of the absorbed wavelengths. This is called an absorption spectrum.

Crucially, the bright lines in an emission spectrum of an element correspond exactly to the dark lines in its absorption spectrum. This indicates that atoms can absorb and emit light only at specific, discrete wavelengths. This observation was a major puzzle for classical physics.

3. The Hydrogen Atom's Line Spectrum

The simplest atom, hydrogen, has a particularly well-studied line spectrum. When hydrogen gas is excited (e.g., by passing an electric current through it), it emits light that, when passed through a prism, reveals a series of distinct lines in the visible region. These lines were found to be grouped into series, named after their discoverers:

• Lyman Series: In the ultraviolet region (transitions to n=1).
• Balmer Series: In the visible region (transitions to n=2). This was the first series to be observed and studied in detail.
• Paschen Series: In the infrared region (transitions to n=3).
• Brackett Series: In the infrared region (transitions to n=4).
• Pfund Series: In the infrared region (transitions to n=5).

These series suggested that electrons in the hydrogen atom could only exist at specific energy levels.

4. Rutherford's Model and its Limitations

Ernest Rutherford's nuclear model proposed that the atom consists of a small, dense, positively charged nucleus at the center, with electrons orbiting it, much like planets orbiting the sun. While this model explained the scattering of alpha particles, it failed to explain the stability of the atom and its discrete emission spectrum.

According to classical electromagnetic theory, an accelerating charged particle (like an electron orbiting a nucleus) should continuously radiate energy. This would cause the electron to lose energy, spiral inwards, and eventually collapse into the nucleus. This predicted instability contradicted the observed stability of atoms. Furthermore, it predicted a continuous spectrum, not the observed line spectrum.

5. Bohr's Model of the Hydrogen Atom (1913)

Niels Bohr, building upon Rutherford's model and Planck's quantum theory, proposed a revolutionary model for the hydrogen atom that successfully explained its stability and line spectrum. Bohr's model is based on several postulates:

Postulate 1: Quantized Orbits

Electrons do not radiate energy while revolving in certain specific orbits, called stationary orbits or stationary states. In these orbits, the angular momentum of the electron is quantized, meaning it can only take discrete values that are integer multiples of $h/2\pi$.

Mathematically, this is expressed as:

$m_e v r = \frac{nh}{2\pi}$

Where:

• $m_e$ is the mass of the electron
• $v$ is the velocity of the electron
• $r$ is the radius of the stationary orbit
• $n$ is the principal quantum number (a positive integer: 1, 2, 3, ...)
• $h$ is Planck's constant

The integer $n$ is called the principal quantum number, and it determines the energy level and radius of the orbit. $n=1$ corresponds to the lowest energy level (ground state), $n=2$ to the next level (first excited state), and so on.

Postulate 2: Energy Levels

Each stationary orbit corresponds to a definite amount of energy. The electron possesses this energy as long as it remains in that particular orbit. Energy is neither emitted nor absorbed when the electron moves from one stationary orbit to another.

Postulate 3: Energy Transitions and Photon Emission/Absorption

An electron can jump from one stationary orbit to another, but it can only do so by absorbing or emitting energy in the form of a photon. Energy is absorbed when an electron jumps from a lower energy orbit to a higher energy orbit, and energy is emitted when an electron jumps from a higher energy orbit to a lower energy orbit.

The energy of the absorbed or emitted photon ($E_{photon}$) is equal to the difference in energy between the two orbits:

$E_{photon} = E_{final} - E_{initial} = h\nu$

Where $E_{final}$ is the energy of the final orbit and $E_{initial}$ is the energy of the initial orbit.

6. Derivations from Bohr's Model for Hydrogen Atom

Bohr's model allows for the calculation of various properties of the hydrogen atom, such as the radius of the orbits, the energy levels, and the frequencies of the emitted spectral lines.

6.1. Radius of Bohr Orbits

By equating the electrostatic force of attraction between the nucleus and the electron to the centripetal force required for circular motion, and applying the quantization of angular momentum, Bohr derived an expression for the radius ($r_n$) of the $n^{th}$ orbit:

$r_n = \frac{n^2 h^2 \epsilon_0}{\pi m_e e^2}$

Where:

• $n$ is the principal quantum number
• $h$ is Planck's constant
• $\epsilon_0$ is the permittivity of free space
• $m_e$ is the mass of the electron
• $e$ is the magnitude of the electronic charge

The radius of the first orbit ($n=1$), known as the Bohr radius ($a_0$), is approximately $5.29 \times 10^{-11}$ m (or 52.9 pm).

$a_0 = r_1 = \frac{h^2 \epsilon_0}{\pi m_e e^2} \approx 5.29 \times 10^{-11}$ m

The radius of any orbit is proportional to the square of the principal quantum number: $r_n \propto n^2$. This means orbits further from the nucleus are much larger.

6.2. Energy Levels

The total energy of an electron in the $n^{th}$ orbit is the sum of its kinetic and potential energies. After applying the postulates and the force balance, Bohr derived the energy ($E_n$) of the electron in the $n^{th}$ orbit for a hydrogen atom:

$E_n = -\frac{m_e e^4}{8 \epsilon_0^2 h^2} \times \frac{1}{n^2}$

The negative sign indicates that the electron is bound to the nucleus. The energy is lowest (most negative) for $n=1$ (ground state) and becomes less negative (increases) as $n$ increases. For $n \to \infty$, the energy approaches zero, meaning the electron is completely removed from the atom (ionization).

The constant term can be expressed using fundamental constants:

$E_n = -\frac{2.18 \times 10^{-18} \text{ J}}{n^2} = -\frac{13.6 \text{ eV}}{n^2}$

Where eV stands for electron volts, a common unit of energy in atomic physics ($1 \text{ eV} \approx 1.602 \times 10^{-19}$ J).

The energy levels are quantized, meaning only specific energy values are allowed for the electron.

6.3. Spectral Lines (Rydberg Formula)

When an electron transitions from an initial energy level $n_i$ to a final energy level $n_f$, the energy of the emitted or absorbed photon is:

$h\nu = E_{n_i} - E_{n_f} = \left(-\frac{13.6 \text{ eV}}{n_i^2}\right) - \left(-\frac{13.6 \text{ eV}}{n_f^2}\right)$

$h\nu = 13.6 \text{ eV} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

Since $c = \nu \lambda$, we have $\nu = c/\lambda$. Substituting this and converting energy to Joules if needed, we get the wave number ($\bar{\nu} = 1/\lambda$):

$\frac{1}{\lambda} = \bar{\nu} = \frac{13.6 \text{ eV}}{hc} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

The term $\frac{13.6 \text{ eV}}{hc}$ is the Rydberg constant ($R_H$) for hydrogen.

$R_H \approx 1.097 \times 10^7 \text{ m}^{-1}$

So, the Rydberg formula for the hydrogen spectrum is:

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

This formula accurately predicts the wavelengths of all spectral lines in the hydrogen spectrum.

7. Successes of Bohr's Model

Bohr's model was a monumental achievement because it successfully explained:

• The stability of atoms: Electrons in stationary orbits do not radiate energy.
• The discrete line spectra of hydrogen: Transitions between quantized energy levels result in the emission or absorption of photons of specific energies (and thus specific wavelengths).
• The Rydberg formula: It provided a theoretical basis for the empirical formula describing the hydrogen spectrum.
• The radius and energy levels of the hydrogen atom.

8. Limitations of Bohr's Model

Despite its successes, Bohr's model had significant limitations:

• Multi-electron Atoms: It could not accurately predict the spectra of atoms with more than one electron.
• Fine Structure: It could not explain the splitting of spectral lines into finer lines (fine structure) observed with high-resolution spectroscopy.
• Zeeman and Stark Effects: It failed to explain the splitting of spectral lines in the presence of external magnetic fields (Zeeman effect) or electric fields (Stark effect).
• Wave Nature of Matter: It did not incorporate the wave nature of electrons, which was later proposed by de Broglie.
• Chemical Bonding: It offered no explanation for how atoms combine to form molecules.
• Intensity of Lines: It could not explain why some spectral lines are more intense than others.

These limitations paved the way for the development of the more sophisticated quantum mechanical model of the atom. However, Bohr's model remains a crucial stepping stone in understanding atomic structure and spectroscopy, particularly for introductory purposes and for the hydrogen atom.

9. Example Calculation: Balmer Series

Let's calculate the wavelength of the first line in the Balmer series. The Balmer series corresponds to transitions ending at the $n_f = 2$ energy level. The first line corresponds to the transition from the next higher level, $n_i = 3$.

Using the Rydberg formula:

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

$\frac{1}{\lambda} = R_H \left(\frac{1}{2^2} - \frac{1}{3^2}\right)$

$\frac{1}{\lambda} = R_H \left(\frac{1}{4} - \frac{1}{9}\right)$

$\frac{1}{\lambda} = R_H \left(\frac{9 - 4}{36}\right)$

$\frac{1}{\lambda} = R_H \left(\frac{5}{36}\right)$

Now, substitute the value of $R_H \approx 1.097 \times 10^7 \text{ m}^{-1}$:

$\frac{1}{\lambda} = (1.097 \times 10^7 \text{ m}^{-1}) \times \frac{5}{36}$

$\frac{1}{\lambda} \approx 1.524 \times 10^6 \text{ m}^{-1}$

$\lambda = \frac{1}{1.524 \times 10^6 \text{ m}^{-1}} \approx 6.56 \times 10^{-7} \text{ m}$

Converting this to nanometers (1 nm = $10^{-9}$ m):

$\lambda \approx 656 \text{ nm}$

This wavelength corresponds to red light, which is indeed the first observed line in the visible Balmer series of the hydrogen spectrum.