Atomic Spectrum & Bohr's Model of Hydrogen Atom

Electromagnetic Spectrum

The arrangement of all electromagnetic radiation in a definite order (increasing or decreasing wavelength or frequency) is called the electromagnetic spectrum. Every region of the spectrum is characterised by a band of wavelengths or frequencies and by specific applications or physical effects.

Spectrum and Dispersion

When light from a source is passed through a prism, radiations of different wavelengths are deviated through different angles and become separated. The angle of deviation is proportional to the refractive index of the prism material, so radiations of different wavelengths are dispersed.

This process of separating light into its constituent wavelengths is called dispersion. The visible result-the set of separated colours or wavelengths-is called a spectrum.

Splitting of light

Types of Spectrum

Spectra are broadly classified into two types:

• Emission spectrum: Spectrum of radiations emitted by a source, atom or molecule when it is excited by heating or electric discharge.
• Absorption spectrum: Spectrum produced when white light passes through a medium that selectively absorbs some wavelengths; the transmitted light shows dark (missing) lines at the absorbed wavelengths.

Emission Spectrum - Continuous and Discontinuous

Continuous spectrum: When white light from a source is dispersed, a bright spectrum continuously distributed on a dark background is obtained. The colours change continuously from violet to red without sharp boundaries; adjacent colours appear merged. This is called a continuous spectrum.

Discontinuous (line) spectrum: When atoms are excited (for example by heating or electric discharge), electrons are promoted to higher energy levels. When these electrons return to lower levels they emit radiation of certain definite wavelengths corresponding to energy differences. The emitted spectrum consists of discrete lines and is called a line spectrum or atomic spectrum.

Spectrum

Absorption Spectrum

• When atoms or ions are irradiated with white light, they absorb radiations of specific frequencies corresponding to allowed energy differences. The transmitted light then shows dark lines where absorption occurred; this is the absorption spectrum.
• The absorption spectrum of an element is a photographic negative of its emission spectrum: the bright lines present in the emission spectrum correspond to dark lines in the absorption spectrum.
• Absorption lines appear as dark lines on a continuous background.

Line Spectrum of the Hydrogen Atom

The hydrogen atom shows a discontinuous line spectrum with lines observed in ultraviolet, visible and infrared regions. Different groups of lines were found to form series and were named after discoverers.

Rydberg proposed an empirical formula applicable to all series of hydrogen-like species. The Rydberg constant for hydrogen is

R_H = 109677 cm^-1

In the Rydberg expression, n₁ and n₂ are integers with n₂ > n₁.

Bohr's Model

Bohr proposed the first atom model based on Planck's quantum theory. The model successfully explained the stability of the atom and the line spectrum of hydrogen.

Bohr's Atomic Model

Postulates of Bohr's Model

1. An atom consists of a small, dense, positively charged nucleus surrounded by electrons that move in circular orbits around the nucleus. The Coulombic attraction between nucleus and electron provides the centripetal force required for circular motion.
2. Only certain circular orbits are allowed. The angular momentum of the electron in an allowed orbit is an integral multiple of h/2π, that is, mvr = n(h/2π). Thus angular momentum is quantised.

3. The energy of an electron in an allowed orbit has a fixed value characteristic of that orbit. Therefore the electron can have only certain discrete energies; energy is quantised.
4. While an electron remains in a stationary orbit it does not radiate energy. Such orbits are called stationary states or allowed energy levels. This explains atomic stability.
5. Energy levels are labelled K, L, M, N... and numbered 1, 2, 3, 4... from the nucleus outwards. Energy increases with shell number, so E_N > E_M > E_L > E_K.
6. Emission or absorption of radiation occurs only when an electron jumps between stationary states. The energy change is ΔE = E_higher - E_lower = hν. Energy is absorbed when an electron moves to a higher level and emitted when it falls to a lower level.

Bohr's model is applicable only to one-electron species (for example H, He⁺, Li²⁺, Be³⁺).

Derivation: Radius of Allowed Orbits for One-Electron Species

The derivation below follows Bohr's assumptions and uses Coulomb's law together with quantised angular momentum.

Consider a nucleus of charge +Ze and an electron of charge -e moving in a circular orbit of radius r.

The Coulomb force provides the centripetal force:

e²/(4πε₀ r²) = m v² / r

The quantisation condition for angular momentum is:

m v r = n (h/2π)

Eliminate v using the angular momentum expression and Coulomb-centripetal relation to get r in terms of n and Z.

The radius of the nth orbit is therefore proportional to n²/Z:

r_n = 0.529 Å × n² / Z

Thus r ∝ n² for a particular atom (fixed Z).

Radius of the first orbit of hydrogen (n = 1, Z = 1):

r = 0.529 Å.

Example. Calculate the ratio of the radius of the 1st orbit of the H atom to that of Li²⁺.
Sol.
For hydrogen (Z = 1) and Li²⁺ (Z = 3) with n = 1 for both,
r ∝ n² / Z
Therefore r(H) : r(Li²⁺) = (1²/1) : (1²/3) = 1 : 1/3 = 3 : 1.

Derivation: Velocity of Electron in a Bohr Orbit

Using m v r = n (h/2π) and the expression for r_n, the speed v of electron in nth orbit is obtained.

m v r = n h / 2π

Putting the value of r gives v in terms of n and Z:

v = (Z e²)/(2 ε₀ h) × (2π / n)

Numerically for hydrogen (n = 1, Z = 1):
v = 2.18 × 10⁶ m s^-1

Thus v ∝ Z / n.

Derivation: Total Energy of Electron in an Orbit

Total energy T.E. of the electron-nucleus system is the sum of kinetic energy (K.E.) and potential energy (P.E.).

K.E. = 1/2 m v²

Using the Coulomb-centripetal relation we obtain K.E. = + (Z² × 13.6 eV) / n².

Potential energy of the system (electron and nucleus) is:

P.E. = - (Z² × 27.2 eV) / n²

Total energy T.E. = K.E. + P.E. = - (Z² × 13.6 eV) / n² per atom.

For hydrogen (Z = 1) the energy of the nth level is

E_n = - 13.6 eV / n².

In SI units:

E_n = - 2.18 × 10^-18 J / n² per atom.

As n increases the absolute value of T.E. and P.E. decreases and approaches zero at n → ∞. The negative sign indicates the electron is bound (under attractive force of nucleus).

K.E. = + (Z² × 13.6 eV) / n²

T.E. = - K.E.

Numerical Values of Energies for Low n (Hydrogen)

(i) n = 1 (ground state)
K.E. = 13.6 eV
P.E. = -27.2 eV
T.E. = -13.6 eV

(ii) n = 2 (first excited state)
K.E. = 13.6/4 eV = 3.4 eV
P.E. = -6.8 eV
T.E. = -3.4 eV
E₂ - E₁ = (-3.4) - (-13.6) = 10.2 eV

(iii) n = 3 (second excited state)
K.E. = 13.6/9 eV = 1.51 eV
P.E. = -3.02 eV
T.E. = -1.51 eV
Energy difference E₃ - E₂ = (-1.51) - (-3.4) = 1.89 eV

(iv) n = 4
K.E. = 13.6/16 eV = 0.85 eV
P.E. = -1.70 eV
T.E. = -0.85 eV
E₄ - E₃ = 0.66 eV

As n increases the energy of a level increases (becomes less negative), but the energy difference between successive levels decreases for large n.

Fig: Energy levels in Hydrogen atom.

Explanation of Hydrogen Spectrum Using Bohr's Model

When an electron falls from a higher energy level n₂ to a lower level n₁, it emits a photon of energy equal to the difference between the two levels:

hν = ΔE = E_n2 - E_n1 = -13.6 eV (1/n₂² - 1/n₁²)

Using E_n = -13.6 eV / n², the emitted or absorbed photon frequency is given by the Rydberg formula

1/λ = R_H Z² (1/n₁² - 1/n₂²)

The theoretical values obtained from Bohr's model agree closely with experimental observations for one-electron species.

Equivalently in wave number (ṽ):

ṽ = R_H Z² (1/n₁² - 1/n₂²)

Different Series in the Hydrogen Spectrum

Different Series in Hydrogen Spectrum.

• Lyman series: n₁ = 1, n₂ = 2, 3, 4, ... (Ultraviolet)
• Balmer series: n₁ = 2, n₂ = 3, 4, 5, ... (Visible)
• Paschen series: n₁ = 3, n₂ = 4, 5, 6, ... (Infrared)
• Brackett series: n₁ = 4, n₂ = 5, 6, 7, ... (Infrared)
• Pfund series: n₁ = 5, n₂ = 6, 7, ... (Infrared)
• Humphreys series: n₁ = 6, n₂ = 7, 8, ... (Infrared)

Counting Possible Spectral Lines

If transitions occur between levels up to principal quantum number n (i.e., from n to all lower levels), the total number of different spectral lines produced by all possible transitions among n levels is:

N = n(n - 1)/2

Example. The energy levels of an atom for 1st, 2nd and 3rd levels are E, 4E and 2E respectively. If a photon of wavelength λ is emitted for a transition 3 → 1, calculate the wavelength for transition 2 → 1 in terms of λ.
Sol.
Energy difference for 3 → 1 is E₃ - E₁ = 2E - E = E, corresponding to photon of wavelength λ.
Energy difference for 2 → 1 is E₂ - E₁ = 4E - E = 3E, so photon energy is 3 times that of the 3 → 1 transition.
Since E ∝ 1/λ, the wavelength for 2 → 1 is λ/3. Thus λ_2→1 = λ/3.

Example. Let n₁ be the frequency of the series limit of the Lyman series, n₂ be the frequency of the first line of the Lyman series, and n₃ be the frequency of the series limit of the Balmer series. Find the relation between n₁, n₂ and n₃.
Sol.
Using energy (or frequency) relations for the series limits and first lines, one obtains
n₁ = n₂ - n₃ (or equivalent relations depending on sign conventions).

Example. A hydrogen-like species emits 6 wavelengths originating from all possible transitions between a group of levels. These energy levels lie between -0.85 eV and -0.544 eV (inclusive). Calculate (i) the quantum numbers of the levels involved, and (ii) the atomic number Z of the species.
Sol.
From the count of emitted lines and the energy range one finds the difference in principal quantum numbers and then identifies n₁ and n₂. Solving the combinatorial relation for the number of lines and matching energy values leads to

n₁ = 12, n₂ = 15 (as determined by the given numerical solution method) and the corresponding Z is found from the level energies.

Failures of Bohr's Model

• The model applies only to one-electron species; it fails for multi-electron atoms.
• Bohr could not explain the fine structure (very closely spaced lines) observed with high-resolution spectroscopy. This was later addressed by Sommerfeld with additional refinements.
• The model cannot account for splitting of spectral lines in external magnetic fields (Zeeman effect) or electric fields (Stark effect).
• Bohr's model lacked a theoretical justification for some postulates (for example, why angular momentum should be quantised in units of h/2π).
• Bohr's model predicted identical spectra for isotopes of hydrogen, contrary to experimental observation (isotope shifts were observed).

Sommerfeld's Refinements of the Bohr Model

Arnold Sommerfeld introduced three important refinements that improved agreement with experiment and accounted for small deviations (fine structure) in spectral lines:

• Allowance for elliptical orbits (not only circular orbits).
• Consideration of the finite mass of the nucleus by allowing an orbiting motion of the nucleus (reduced mass correction).
• Inclusion of relativistic mass effects for electrons moving at high speeds in inner orbits.

Fig: Sommerfeld model of Atom.