# The Bohr Model

Bohr’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/n^{2}
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.
10^{4} – 10^{22}Hz. 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]
F_{out} = mv^{2}/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]
F_{in} = k(q_{1}*q_{2})/r^{2}
= 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 [q_{1}*q_{2}]
correspond to the attractive charges of the electron and proton can
be reduced to [e^{2}] and [r] is the orbital radius. So combining
[1] and [2] becomes:

mv^{2}/r = ke^{2}/r^{2
}rationalising to

[3] r = ke^{2}/mv^{2}

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 = mr^{2}

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

ω = 2π v/λ where wavelength [λ] = 2πr

[6] ω = v/r

[7] L = I*ω = (mr^{2})
* (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 = mc^{2} = 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

While
the equivalence of [hf=mc^{2}] 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 = mc^{2}/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=mc^{2},
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 [m_{0} when v=0] and the relativistic
or kinetic mass [m_{k} when v≠0].

[13]

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

[14] E^{2} = m_{k}^{2}c^{4
}= m_{0}^{2 }c^{4 }/ (1-v^{2}/c^{2})

m_{k}^{2}c^{
4 }- m_{k}^{2}c^{2 }v^{2} = m_{0}^{
2 }c^{4}

[15] E^{2 =} m_{0}^{2
}c^{4 }+ [m_{k}^{2}v^{2}]c^{
2 }

Where [m_{k}] now reflects the relativistic or kinetic mass
associated with a velocity [v], rather than [m_{0}], which reflects
the rest mass. As such, the [m_{k}^{2}v^{2}]
term is normally replaced by momentum [ρ^{2}] based on the
argument that the only real form of mass is rest mass. However, we will
continue with the expanded form, highlighted by [m_{k}], 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=mc^{2}=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
[m_{0}=0] and [v=c]:

[16] E^{ }= m_{k}c^{2} = hf = hc/λ

m_{k} = E/c^{2}
= hf/c^{2} = h/λc

[17] λ = h/m_{k}c

*Note: This compares directly with Compton’s wavelength
given in [10]. In this case, we can justify the equality of [mc ^{2}=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 [m_{k}] and velocity [v=c], even though it has no rest
mass [m_{0}]. However, it should be highlighted that the
structure
of a photon is still an issue of debate. *

### Electron with rest mass [m_{0}] but zero velocity:

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

[18] E^{ }= m_{0}c^{2}

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

[19] λ = h/m_{0}c
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
[m _{0}c^{2}=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 [m_{0}=m_{e}]. 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*10^{20}, even though there is no obvious
associated velocity [v=c] other than via association to the value of
[c] in E=mc^{2}. 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 [m_{0}] and kinetic mass [m_{k}]
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 → m_{k}vc only as
[v] approaches [c]

However, [m_{k}] corresponds to kinetic mass, where [m_{k}=γm_{0}],
plus we can also substitute for E=hc/λ :

hc/λ = γm_{0}vc

[20] λ = hγ/m_{0 }v = h/m_{k}v

The form of equation [20] does differ from [17] and [19] because
unlike a photon, there is a rest mass [m_{0}] 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 [m_{k}>>m_{0}], 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>>m_{0}]. We might
also wish to rearrange [21] to provide an expression for the velocity
[v]:

[22] v = h/ m_{k}*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

We 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 = ke^{2}/mv^{2}

rv^{2} = ke^{2}/m

r(nh / 2π*mr)^{2} = ke^{2}/m

rn^{2}h^{2 }/( 2π)^{2}m^{2*}r^{2}
= ke^{2}/m

n^{2}h^{2 }/ (2π)^{2}m^{2*}r
= ke^{2}/m

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

[27] r = n^{2 } (^{ }h^{2
}/ 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 m^{2} C-^{2} |
1/(4πε) |

The rationalised form of equation [27] is:

[28] r = n^{2} * 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}/n^{2
}h^{2 })

[29] v = (1/n)*(2π ke^{2}/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*10^{6} m/s
Note: this is ~1% of [c]

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