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

 

The Coulomb Electron Speed and 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 defines the electrical force between two charged objects which is directly proportional to the product of the charges on these two objects and inversely proportional to the square of the separation distance between them. In the equation form

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, q ₂ is the charge of another object and R is the distance between these objects.

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 rest mass of electron m_e = 9.11 × 10^-31 kg.

The Coulombic force F_p between these two charges is at a distance R would be F_p = k_ee²/R². 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 orbiting with a speed υ_e.

We know in this orbit the electron is exposed to the Coulombic centripetal force: F_p = k_ee²/R² and the centrifugal force: F_f = m_eυ_n²/R. These two forces are balanced: F_p = F_f or k_ee²/R² = m_eυ_n²/R. An electron to move one-dimensionally along the straight line towards the proton Fc > Fp or, after some algebra, (m_eυ_nR)υ_n < k_ee². Applying Bohr’s quantum condition m_eυ_nR = nħ[1], after a bit of algebra, we get

υ_n < (k_ee²/ħ) × 1/n.

Let us assume that the electron moves along a straight one-dimensional path in the direction of the proton of hydrogen atom with a final speed equal to υ₁ about 2.2 × 10⁶ m sec^-1. When this electron reaches the ground state it would orbit the proton in a circle with this speed as in Bohr’s model.

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 mass of electron m_e. This can be represented by the equation

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

By employing eqn. (2), we find that an electron situated at a distance of Bohr’s radius from the proton would reach an acceleration of about 10³⁰ m sec^-2. Taking it - as an average acceleration - the electron would reach the speed υ₁ for Δt ≈ 2.2 × 10^-24 sec.

To get as close to the proton as possible the electron must move at a relativistic speed much higher than υ₁ or close to the speed of light c (≈ 3 × 10⁸ m sec^-1). The relativistic centrifugal force F_f = m_eυ_e²/R_{min /}[√(1 - υ_e²/c²)] has to be equal to the centripetal force F_p = k_ee²/R_min. This can be expressed as the equation

 m_eυ_e²/R_min√(1 –  υ_e²/c²) = k_ee²/R_min²

or, after some algebra,
Rmin = (kee2/me) × [√(1 –  υe2/c2)] × 1/υe2 
where Rmin is the smallest distance between the electron approaching the proton with a relativistic speed 
υe.

Solving for Rmin and substituting known values, we get
                                                Rmin ≈ 2.6 × 10-7 [√(1 –  υe2/c2)] × 1/υe2.
The electron is traveling at 99.9999992 % of the speed of light[1] (or υe = 0.999999992c) in the electron
beam of energy as high as 4GeV.[2] Plugging this value for υe after some calculation we find that Rmin 
would be about 3.6 × 10-28 m. For comparison, the charge radius of the proton is about 1 × 10-15 m.
Let us denote with ae the average acceleration of the electron approaching to the proton. This acceleration
 would be higher than aC and it can be expressed by the following equation
ae = (υe – υ1)/Δt
or

Δt = (υe – υ1)/ae

where Δt is the corresponding time interval.  As ae > 1030 m sec-2, we write Δt < (υe – υ1)/1030. Solving
for Δt and substituting the known values we find Δt < 3 × 10-22 sec. Of note, it is generally accepted that
the Planck time tp (≈ 10-43 sec) is the smallest time interval that has a physical meaning. This time interval
is about 1022 times smaller than the smallest time interval measured to date: 10-21 sec. 


[1] To avoid confusion in further text, the SI units are given in italics.
[2] We know that meυnR is the angular momentum of an electron in its orbit which is, according to Bohr’s model,
quantized. 
[3] Data from Jefferson Lab.
[4] The relativistic mass of the electron would be about 8000 me.