The Coulomb Acceleration in Bohr’s Model of the Hydrogen Atom

 

The Coulomb Acceleration in Bohr’s Model of the Hydrogen Atom

Pavle I. Premović, 

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry, 

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

Coulomb's law states that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the separation distance between the two objects. Mathematically, this law can be stated as

F_C = k_eq₁q₂/R²    … (1)

 where k_e is the Coulomb constant (= 8,988 × 10⁹ N m² C^-2)[1], q₁ is the charge of the first object, q₂ is the charge of another object and R is the distance between these objects. F_E is the electrostatic force.

We will apply the expression (1) in the case of the proton and electron of the hydrogen atom. The proton has a positive charge equal in magnitude to a unit of electron charge e (= 1.60 × 10^-19 C). In the ground state of hydrogen atom, its Bohr radius a₀ = 5.3 × 10^-11 m.

The Coulomb force of hydrogen atom is attractive and can cause the electron to deflect from a straight-line path to a circular path (Bohr’s orbit) around the proton. We assume that the electron moves one-dimensionally along the line joining them and that it directly reaches Bohr’s radius without any deflection from a straight-line path.

The Coulombic force FC between these two charges is at a distance R would be

FC = kee2/R2.

According to Newton’s second law of motion, the acceleration of the electron at R would be equal to the Coulombic force divided by the rest mass of electron m_e (= 9.11 × 10^-31 kg). This can be represented by the equation

 a_C = (F_C/m_e) = k_ee²/m_eR²   … (2).

An electron at a distance from the proton equal to Bohr’s radius a₀ (= 5.3 × 10^-11 m)[1] would reach an acceleration of about 10³⁰ m sec^-2 [by employing eqn. (2)]. With this average acceleration the electron would reach the speed of light c (= 299792 km sec^-1 ≈ 3 × 10⁵ km sec^-1) for Δt ≈ 3 × 10^-22 sec.

The quantum mechanical expression for the energy-time uncertainty is

ΔEΔt ≈ h    … (3).

This relates the minimum uncertainty in energy ΔE of a particle and its change during a time interval Δt. Solving for ΔE and substituting Δt ≈ 3 × 10^-22 sec gives ΔE ≈ 2 × 10^-12 J. This uncertainty in energy is about 10⁶ times higher than the ionization energy of the hydrogen atom about 2.2 ×10^-18 J (or 13.6 eV). For the electron of this atom, it is reasonable to assume that ΔE is about its ionization energy then the time interval Δt has to be 3 × 10^-16 sec [eqn. (3)]. By that time the electron speed would be 3 × 10^-16 sec × 10³⁰ m sec^-2 ≈ 3 × 10¹⁴ m sec^-1 or about 10⁶ times higher than the speed of light.

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[1] To avoid confusion in further text, the SI units are given in italics.

[2] According to Bohr’s model, at this distance the electron has a non-relativistic speed of about 2.2 × 10⁶ m sec^-1.