The Bohr Model

bohrBohr’s model is essentially restricted to a hydrogen atom, consisting of only 1 proton and 1 electron and the intention of this discussion is only to provide some additional level of description of the rationale behind the development of the atomic model. It is highlighted from the start that this is a description of a model rather than the actual structure of the hydrogen atom as subsequent application of this model to more complex atoms failed to explain all observations and pointed to its limitations. However, the Bohr model is an important step in the transition between classical to quantum physics. In this perspective, the Bohr model is essentially a hybrid model that still depends on classical physics for much of its derivation, but introduces the key element of quantized momentum. However, the Bohr model possibly raises more questions than answers in the sense that the structure of the electron becomes increasingly ambiguous as the wave-particle duality appears to emerge.  Equally, while it defies classical physics by describing a stable electron orbit in terms of quantized angular momentum, it does not explain why the charged electron ceases to radiate energy as it orbits the proton. While quantum theory has gone on to provide an additional mathematical rationale that has proved successful in predicting the outcome of many experiments, for many, this quantum model has still failed to provide an adequate description of the `real` physical structure and processes within the atom.

Model Introduction

In the following text, the energy equation for the ground state orbit of the hydrogen atom is derived and shown to be dependent on the principle quantum number [n]. The common form of this result is given as 13.6/n2 eV, where 1eV = 1.60*10-19 Joules. Often, the general explanation of this quantized effect and the discrete spectral emission lines is often linked directly to Planck’s equation E=hf, where [h=6.63*10-34 Js] and [f] is frequency. However, this provides no real explanation of the ground state energy level, being some 15 orders of magnitude greater than [h] when multiplied by a frequency of 1. Clearly, E=hf does not provide any explanation of quantized energy levels within the hydrogen atom, if the frequency of a photon can range continuously over the entire electromagnetic spectrum from AM radio to Gamma rays, i.e. 104 – 1022Hz. In order to have a quantized orbit, there must be some associated parameter, which is directly proportional to the Planck Constant [h] and the principle quantum number [n]. This parameter turns out to be angular momentum [L] and therefore the derivation will highlight the importance of this step in the process.

Classical Assumptions

The classical requirement for a stable orbit demands that the outward force be matched by an equal and opposite inward force. The outward force is analogous to a centrifugal force and can be described by the following Newtonian equation:

[1]      Fout = mv2/r = 6.79*10-6          when n=1

Where [m] is mass of the electron, [v] is the tangential velocity at a radius [r]. In contrast, the inward force, in this specific case, corresponds to Coulomb’s Law derived in 1785:

[2]      Fin = k(q1*q2)/r2 = 6.79*10-6     when n=1

Note, there is an additional inward force of gravity between the proton and electron, but the magnitude of this force is 2.95*10-45, which is so small in comparison to the electrostatic force that it can be neglected for the purpose of this initial analysis. The parameter [k] is a constant equal to [1/4πe], the parameters [q1*q2] correspond to the attractive charges of the electron and proton can be reduced to [e2] and [r] is the orbital radius. So combining [1] and [2] becomes:

           mv2/r = ke2/r2                          rationalising to

[3]      r = ke2/mv2

However, both [r] and [v] were originally unknown, so an alternative line of reasoning was required. Bohr’s idea was linked to the possibility that the momentum of the electron, within the atom, was quantised along with its energy, in-line with the thinking of Planck and Einstein.  If so, the electron orbits would only be stable for certain values of momentum.  However, the process of derivation can again start with a classical equation for angular momentum [L]:

[4]      L = I*ω

Where [I] is the moment of inertia and [w] is angular velocity. However, we can translate these terms back into linear parameters [v] and [r] just to follow the general logic:

[5]    I = mr2

         ω = 2πf                   where frequency [f] = v/λ (wavelength)

         ω = 2π v/λ               where wavelength [λ] = 2πr

[6]    ω = v/r

[7]     L = I*ω = (mr2) * (v/r) = mvr

Note: today, these equations are typically presented in vector form, but the essential principles can still be seen in algebraic form without having to introduce the complexity of vector maths.

The Quantum Transition

At this point, most derivations seem to simply introduce the concept of quantized angular velocity without necessarily explaining its fundamental importance to the entire proof, i.e. atomic orbits proceed in discrete jumps. It can be seen, in [8] below, that angular momentum [L] is directly proportional to the orbit number [n] and is therefore the parameter that determines the quantized energy levels of the atom:

[8]      L = n*h/2π               where [n] is integer orbit number

While many sources use [8], they give little explanation of its derivation. The following steps are actually based on the later assumptions of Compton (1922) and deBroglie (1923), but hopefully help to highlight a number of key issues. Compton’s wavelength can be derived directly from both Planck’s and Einstein’s energy equations:

[9]      E = hf = mc2 = hc/λ

Note: the relationship between frequency [f] and wavelength [λ] is based on a dispersion relationship, which can be generalised as [f=v/λ] that depends on the media of propagation. However, when the media is a vacuum and the wave is a photon, this relationship becomes [f=c/λ].

The Compton and deBroglie Wavelengths

electronWhile the equivalence of [hf=mc2] is used in this discussion, there is a reservation about the way the energy of a quantum of electromagnetic energy, associated with a discrete wave frequency [hf], is often simply equated to the mass energy of even macroscopic objects. However, this issue is deferred to a later discussion addressing the particle nature of photons.

[10]     λ = h/mc                 Compton’s wavelength

[11]    f = mc2/h                 substitute f = c/λ in vacuum

[12]     m = h/λc                 effective mass of a photon?

Equation [10] is the normal form associated with Compton’s wavelength, but it skips some important steps that may be relevant to any wider discussion concerning the underlying nature of matter. For a start, while we all probably recognise Einstein’s famous equation E=mc2, the scope of the mass [m] has not been clearly defined. Physicists define two distinct types of mass called invariant and relativistic mass, which we can simplify to rest mass [m0 when v=0] and the relativistic or kinetic mass [mk when v≠0].

[13]     13

The term [γ] defines the effect of special relativity and leads to an expanded form of Einstein’s equation:

[14]     E2 = mk2c4 = m02 c4 / (1-v2/c2)

           mk2c 4 - mk2c2 v2 = m0 2 c4

[15]     E2 = m02 c4 + [mk2v2]c 2  

Where [mk] now reflects the relativistic or kinetic mass associated with a velocity [v], rather than [m0], which reflects the rest mass. As such, the [mk2v2] term is normally replaced by momentum [ρ2] based on the argument that the only real form of mass is rest mass. However, we will continue with the expanded form, highlighted by [mk], so as not to lose sight of the specific components of mass when applied to 3 situations:

  • Photon with zero rest mass where v=c
  • Electron with zero velocity
  • Electron with non-zero velocity approaching [c]

Note: before actually discussing each case, it should be highlighted that this aspect of the discussion has implications that go well beyond the scope of the Bohr model. We are discussing what are known as the Compton and deBroglie wavelengths and, as will be shown, both can be derived from [15] when combined with the assumption often referred to as the Planck-Einstein Relationship, i.e. E=mc2=hf. While there appears to be some debate about the validity of this assignment, without it, the following derivations that lead to the definition Compton and deBroglie wavelengths would be undermined. Personally, I think the relationship is valid and is telling us something quite profound about the true nature of fundamental particles. However, this aspect of the discussion will be deferred to the page entitled: Interpretations and Speculations.

Photon with zero rest mass where v=c:

In this case, equation [15] reduces to the following form based on [m0=0] and [v=c]:

[16]    E = mkc2 = hf = hc/λ

          mk = E/c2 = hf/c2 = h/λc

[17]    λ = h/mkc                                        

Note: This compares directly with Compton’s wavelength given in [10]. In this case, we can justify the equality of [mc2=hf] by the assumption that a photon is a quantum of energy with a discrete frequency and wavelength, which has the attribute of a kinetic mass [mk] and velocity [v=c], even though it has no rest mass [m0]. However, it should be highlighted that the structure of a photon is still an issue of debate.

Electron with rest mass [m0] but zero velocity:

In this case, equation [15] reduces to the following form because [v=0]:

[18]    E = m0c2

And if directly equated to hf = hc/λ leads to:

[19]    λ = h/m0c    where f = c/λ

Note: Again, the form of [19] appears to be analogous to Compton’s wavelength as per [10], but we have to now justify the assumption that [m0c2=hf]. By definition, we have said that the electron has no apparent velocity [v=0], but in order to proceed with the wave-particle duality assumption; we must assign a wavelength to the electron mass [m0=me]. If we plug the figures into equation [19], we get a wavelength [λ=2.42*10-12] and on the basis that [f=c/λ], the electron has a frequency [f=1.24*1020, even though there is no obvious associated velocity [v=c] other than via association to the value of [c] in E=mc2. However, on first exposure, many may be left wondering as to whether a vague comment about wave-particle duality is an adequate explanation of the wave structure within the electron?

Electron with rest mass [m0] and kinetic mass [mk] as [v→c]:

In this case, equation [15] does not immediately reduce, but as [v] approaches [c], the second term becomes dominant, where it might appear that the situation becomes analogous to [17], but let us do the substitutions for clarity:

           E → mkvc        only as [v] approaches [c]

However, [mk] corresponds to kinetic mass, where [mk=γm0], plus we can also substitute for E=hc/λ :

           hc/λ = γm0vc

[20]    λ = hγ/m0 v = h/mkv

The form of equation [20] does differ from [17] and [19] because unlike a photon, there is a rest mass [m0] being subjected to relativistic effects. So while the substitution starts with the assumption that [v] is approaching [c] to the extent that it has become the dominant term in [15] due to [mk>>m0], by virtue of [13], the variable [v] has to be retained. Just as a reminder, we started this detour at equation [8] in order to justify the quantization of angular momentum [L]. This is based on the hypothesis that the deBroglie wavelength can be assigned to a particle with velocity [v]:

[21]    λ = h/mv                 deBroglie’s wavelength

However, we needed to justify the equality of [20] and [21] for the case of an electron particle with velocity [v], which hopefully the previous rationale has gone someway to explaining. The caveat on this equality being that the particle velocity [v] must be sufficiently relativistic so as to justify the assumption that [m>>m0]. We might also wish to rearrange [21] to provide an expression for the velocity [v]:

[22]    v = h/ mk*λ  

Note: While we have provided some rationale for the derivation for a wavelength to be associated with an electron particle, we have not really explain how the apparent wave properties of an electron are associated with two different wavelengths, i.e. when [v=0] and [v≠0]. Again, see `speculations & assumptions` for some further thoughts on this issue.

Electron Radius and Velocity

bohrWe still have some work to do to justify equation [8], but based on equation [7]

[23]     L = mvr        

 substituting equation [22] into [23]

[24]     L =hr/λ                            

However, we might now make a general substitution that links wavelength to circular rotation, i.e. λ=2πr. As such:

           L = hr/2πr = h/2π

[25]    L = n * h/2π             where n = atomic orbit number

On this basis, [25] now aligns with the assumption given in [8], which underpins the quantization of atomic orbits based on angular momentum [L] and we can now return to the main thrust of Bohr’s model by combining equations [7] and [8]:

          L = n * h/2π = mvr

[26]    v = nh/2π*mr

Remembering equation [3] as a starting point followed by some messy substituting for [v] as given in [22]:

          r = ke2/mv2

         rv2 = ke2/m

         r(nh / 2π*mr)2 = ke2/m

         rn2h2 /( 2π)2m2*r2 = ke2/m

         n2h2 / (2π)2m2*r = ke2/m

Finally, we arrive at the Bohr radius for each atomic orbit [n]:

[27]    r = n2  ( h2 / 4mk*(πe)2 )

First, it might be sensible to clarify that the atomic orbit number [n] is equivalent to what is now called the 'principle quantum number'. Also, given its unwieldy nature, we can rationalise equation [28] by substituting for the known values of the various constants:

Term Value Unit  Description
h 6.62*10-34 Js  Planck constant
e 1.602*10-19 C  electric charge
m 9.11*10-31 Kg  electron mass
ε 8.85*10-12 C2 N-1 m-2  electric permittivity
k 8.99*10+9 N m2 C-2  1/(4πε)

The rationalised form of equation [27] is:

[28]    r = n2 * 5.28*10-11 metres

Of course, we can now substitute the value of [r] back into equation [26] that allows the calculation of both radius [r] and velocity [v] to be expressed independently of each other:

            v = (nh/2π*m) * 1/r                     

            v = (nh/2π*m) * (4mk*( πe)2/n2 h2 )

[29]     v = (1/n)*(2π ke2/h)

Again, all the values in the term on the right can be substituted, which allows the orbital velocity [v] within the Bohr model to also be expressed as a function of the principle quantum number [n]. In line with our classical understanding of angular velocity, the speed of rotation falls as a function of radius:

[30]    v = (1/n)*2.19*106 m/s                 Note: this is ~1% of [c]

In the next page, we will consider the energy levels associated with this model.