Bohr's Hydrogen Model and spectrum lines

I promise I'll put all equations into MS word mode when doc version is published. At the same time, this is the 100th pieces of note is this blog. ^_^

Quantization of energy in atoms

Assumption of Bohr’s model

-          Validity of Rutherford’s partial model and classical physics, i.e., electron orbiting the nucleus, and the classical laws like circular motion and electric force is valid.

-          The orbit of electron is stationary (stationary orbit). Electron stayed on a stationary orbit is called in a stationary state.

-          Stationary orbits have definite energy levels. Each transition due to emission or absorption of photon must start and end on the energy levels of the atom. i.e., hf = ΔE = E_higher - E_lower for emitting photon and hf = ΔE = E_lower - E_higher for absorbing photon.

-          Bohr’s quantum condition: angular momentum is fixed to integral multiples for h/2π, i.e., Angular momentum = m_evr = nh/2π, where n is a positive integer for energy level E_n and n is called the principal quantum number of that orbit.

Energy of hydrogen atom

-          In Rutherford’s model we know that m_ev²/r = e²/(4πε₀r²)           ---(1)

-          Put m_evr_n = nh/2π, v = nh/2πm_er_n into (1) we will have r_n = (n²h²ε₀)/(e²πm_e)

-          In Rutherford’s model we have E_tot = - e²/(8πε₀r) = -(1/n²)(e⁴m_e/8ε₀²h²)

-          r₁ is called the Bohr’s radius, and E₁ = -13.6eV. For simplicity we have r_n=n²r₁ and E_n = E₁/n² = -13.6eV/n²

-          Note that all energy levels are negative so that the electron is bounded to the atom and energy is required to excite the electron into a higher level.

-          E_∞ refers to 0eV and the state of n = ∞ in which theoretically has infinite radius and both K.E. and P.E. is zero. The electron is just to escape from the atom. Beyond E_∞, the electron becomes free electron and its energy can be in any positive value.

-          n=1 is called the ground state while n=k>1 is called the (k-1)^th excited state.

-          Excitation energy is the energy to excite an electron to a higher energy level (excitation). i.e., excitation energy = ΔE = E_higher - E_lower

-          Ionization energy is the minimal energy to remove an electron from an atom, i.e., ionization energy = E_∞ - E_initial = -E_initial. Note that -E_initial is positive, so work done is required to ionize the electron.

Emission line spectrum

-          Ideal body (black body) with T > 0K emits EM waves at all wavelengths, called continuous spectrum. The higher frequency, the more EM waves of higher frequency emitted.

-          However reality atoms only emit EM waves in specified wavelengths only (spectral lines) when gaseous atoms are heated under low pressure or gas discharge tubes.

-          Atoms must be in gaseous form and excited before it emits photon. It can be done by collisions (lost of energy in inelastic collision excites the electron, elastic collision cannot excite the electron), heating or applying high voltage across the gas.

-          hf_emitted = E_higher - E_lower = (e⁴m_e/8ε₀²h²)(1/n_lower² – 1/n_higher²)

f_emitted = (e⁴m_e/8ε₀²h³)(1/n_lower² – 1/n_higher²)

Rewrite the formula by c=fλ we have 1/λ_emitted = (e⁴m_e/8ε₀²h³c)(1/n_lower² – 1/n_higher²)

-          For hydrogen atom we have 1/λ_emitted = (13.6eV/hc)(1/n_lower² – 1/n_higher²) or       R(1/n_lower² – 1/n_higher²). This is called the Rydberg formula and R = 1.097*10⁷m^-1 is the Rydberg constant.

Absorption spectrum

-          When a continuous spectrum of light is passes through the gas at low pressure, the gas will only absorb EM waves at specified wavelengths only, and produce a spectrum with discrete dark lines called absorption spectrum.

-          Light fringes of emission spectrum = dark fringes of absorption spectrum for the same atom. (Or we can say emission + absorption spectrum = continuous spectrum)

-          Those absorbed EM waves will be emitted by the electrons on the atom again since when the electrons absorbed the photon it is excited and unstable. Emitting the photon away make it back to ground state. However this emission is in random direction. Therefore we see very dark fringes.

-          To produce an absorption spectrum, continuous spectrum like white light must be used.

-          We have 1/λ_absorbed = (13.6eV/hc)(1/n_lower² – 1/n_higher²) = R(1/n_lower² – 1/n_higher²).

Spectrum series

-          The emission spectrum lines from E_n to E₁ or the absorption spectrum lines from E₁ to E_n are called Lyman series which is UV radiation.

-          The emission spectrum lines from E_n to E₂ or the absorption spectrum lines from E₂ to E_n are called Balmer series which is visible light.

-          The emission spectrum lines from E_n to E₃ or the absorption spectrum lines from E₃ to E_n are called Paschen series which is IR radiation.

-          These series do not overlap each other.

-          Spectrum lines are packed closer as n increases. (The lines are packed closer for the higher frequency part for each series)

-          Only one photon is emitted / absorbed for each transition of an atom.

X-ray spectrum

According to classical theory, X-ray is produced when fast moving electrons are decelerated by the target. However we compose the X-ray spectrum (Intensity VS wavelength), there are several spikes where the intensity of a specific wavelength is abnormally high. It is because the electron beam knocked out an electron in the inner orbit, than the electron in the outer orbit will emit a photon and fall to the inner orbit. As a result, the spikes show the characteristic spectrum of the target atom.