An Exploration of Physics by Dimensional Analysis

The diversity of dimensions of physical quantities and their consequences on how the behaviors of physical systems are shaped, plays an essential role throughout all physics theories. This is somehow well-known in principle since long ago. However the explanatory power of such considerations is usually pitifully neglected in the practice of many physics courses. Dimensions are there, they qualify each quantity. We are told that we must respect them, and we have some equations of physical laws using them. Quantities appear in formulas. Formulas give some results which come somehow as a surprise for each particular problem.

Let's not keep this as a black box giving surprising results for each particular problem. Let's more systematically explore how the fundamental laws of physics relate diverse quantities in very general classes of situations. And let us present this as a stimulating introduction to these laws of physics and to how things go in the universe.

(To make things all clean, a first step should be to present mathematical foundations of what is dimensional analysis: what the concept of "different dimensions" for quantities mean mathematically, and how operations between quantities can be defined in principle. Such foundations will be developed in this site later. In the below, this will be assumed as known. The focus here will be on physics.)

List of independent physical quantities

Here we shall qualify different physical quantities as "independent" if we can effectively find a class of different physical systems with a similar behavior, and between which these quantities vary independently of each other.
The resulting classification of quantities is a little different from the traditional conventions inherited from history.
And the very interpretation of this physical definition of independance between quantities, is not always clear and simple.

Typically, physical quantities are least independent in more fundamental theories (indeed the fundamental constants of c, G of General relativity, and h of quantum physics, together provide absolute units for all physical quantities, reduced to real numbers); we get a wider independence between quantities in more phenomenological theory (classical macroscopic physics), where the constants of the fundamental physics which link the different quantities are considered "very small" or "very big" and therefore their being bigger or smaller than they are would not yield practical consequences.

Let us first give the list in bulk (or rather, one possible presentation of this list ; two lists are equivalent if they form different basis of the same free abelian group):

The amount of substance counts the very large number of atoms or molecules that appear in macroscopic situations. Thus its deep meaning is that of natural numbers, but too big for the elementary unit (an individual atom or molecule) to be of any significance. This quantity was introduced from the observation that chemical reactions happen in definite proportions of the ingredients, which was the first established fact from the beginning of the 19th century, while its explanation in terms of atoms was only clearly established later that century.

The standard unit for the amounts of substance is the mol, whose "true value" is given by the Avogadro number N= 6.0221023 mol−1. Generally the definition is that 1 mol of some pure substance contains about 6.0221023 molecules of this substance, so that 1 mol of carbon-12 weights 12 grams, thus roughly, 1 mol of hydrogen atoms weights 1 gram (= 0.001 kg).

Usual lists also include the temperature as another independent quantity. However the gas constant is always near in any experiment with temperature, even when considering solids instead of gases. Thus through this constant, we can consider temperature as a composite of other quantities. Namely, a temperature is an energy per amount of substance. In fact, the thermic energy contained in an object usually has the same order of magnitude as the product of its temperature by the amount of substance it contains, but is not identical to it. This will be detailed later.

Different quantities are naturally produced in different contexts. They may be universal constants associated with these specific processes, or they may be quantities describing the specific case of a given experiment, such as the acceleration of gravity on the ground that depends on the mass and size of the Earth.

Let us explore different constants and their effects.

(Some sections of this exploration have been moved to separate pages)

The speed of light and astronomical distances


The Planck constant

This constant governs physical phenomena at the atomic level, together with the specific constants describing the particles involved (masses and charges). It is the natural unit of action, since the deep nature of quantities of action is that of oscillations number : the principle of least action that governs classical mechanics comes from the fact that scenarii with similar action (those near an extremum of action) benefit a constructive interference from a quantum viewpoint. Depending on context, we may use ℏ or h= 2πℏ = 6.62610−34 Js = 4.135710−15 eVs, as ℏ is used for writing the partial differential equations of the wavefunctions, while h is for counting the numbers of oscillations.
This constant


Quantum oscillators

On the atomic scale, things can oscillate. On the one hand, electrons can somehow "oscillate" between several positions (visit several orbitals), or even leave an atom. On the other hand, a molecule can oscillate as its atoms have an elastic move with respect to each other.
However, for the same thing that oscillates in one same direction, 2 kinds of "oscillations" should be distinguished depending on their cause:
We shall need to compute and compare both, and tell which one dominates in each circumstance.

Many cases of oscillations can be approximately described as an harmonic oscillation : that is when the potential function is approximated as a 2nd degree polynomial, so that the oscillation period does not depend on amplitude of oscillation, but only on the mass and the coefficient of the 2nd degree term of this polynomial.

Take a particle with mass m in a potential energy field E=k.x2/2.
A mass times a speed squared, equals k times a distance squared.
Thus k is a mass divided by a time squared.
So, the classical oscillation period is t=2π m/k.

In the phase space (x,p) where p=mv is the momentum, the particle with energy E follows the ellipse with equation k.x2 + p2/m = 2E.
The area of this ellipse is 2E π m/k.

Quantum physics identifies any oscillation period t to a quantity of energy (a quantum of energy, that is the difference between 2 energy levels)
E0 = h/t = ℏ k/m.

Quantum physics does not allow the energy of an harmonic oscillator to take any value as classical mechanics would allow, but only multiples of this quantum of energy. More precisely, the possible values of the energy are E= (n+ 1/2)E0 where n is a natural number that we shall call the number of phonons of this oscillation (though the word "phonon" is traditionally reserved for essentially the same concept but applied to the oscillations of a crystal instead of a single oscillating mass), so that the energy difference between any states is a multiple of E0:
(m+ 1/2)E0 − (n+ 1/2)E0= (mn)E0

Between energies 0 and nE0 there are the n possible states of oscillation, with energies (m+ 1/2)E0 for any number of phonons m<n, so with values from E0/2 to (n −1/2)E0. In the phase space, they "occupy" the inside of the above ellipse with energy nE0, with area
2nE0 π m/k = nh.

The uncertainty d on position for the ground state (zero phonon, energy E= E0/2) corresponds to the amplitude of a classical oscillation with energy E0 :
d2= E/k = ℏ / km = ℏ2/mE0.
More precisely, the wavefunction as a function of the position x, is proportional to
exp(-x2/(2d2))
so that its expression as a function of the momentum p (the Fourier transform of the latter) is proportional to
exp(-p2d2/(2ℏ2))
making it fit in the same way with the kinetic energy function p2/2m, as the wavefunction of position fits with the potential energy function : (ℏ/d)2/m = ℏ k/m = E0
The probability density function, square of the wavefunction, is exp(-x2/d2)/dπ
where (dπ)2 = h/2 k.m.

The energy of nuclear reactions + The radius of nuclei


Electromagnetism


The gravitational constant


Effects of General Relativity


The parameters of the atomic structure

Atoms, in their chemical properties, are dominated by the electrostatic interaction of electrons with other electrons and with nucleus in the framework of quantum physics.
So let us take the constant ε 0 = 8.85410−12 C2.J−1.m−1 of electrostatics, the elementary charge e=1.6021810−19 C, and the Planck constant h= 2πℏ = 6.6260710−34 J.s.
From them let us write the quantity (does it have any standard name and notation ?)
ve = e2/(4πε 0ℏ) = e2/(2h ε 0) = 1.60218e-19^2/(2*6.62607e-34 *8.8542e-12) = 2187.7 km/s.
This is a high speed, not quite far from the speed of light, that is the typical speed of electrons in atoms.
Its ratio with the speed of light takes the name of the fine-structure constant (which has some small effects on atoms...):
α = ve/c
The Rydberg unit of energy is formally defined as the kinetic energy of an electron, with mass me = 9.1093810−31 kg, going at this speed:
1 Ry = meve2/2 = me(e2/ hε 0)2 / 8 = 13.6057 eV = 2.179910−18 J
as 1 eV= 1.6021810−19 J.
We can first describe the hydrogen-like atoms made of one electron linked to one nucleus with charge (atomic number) Z.
Their energy levels are the quantum mechanical versions of the elliptical Kepler orbits : an electron with mass me in the orbital with quantum numbers  < n may be figuratively understood as being in a fuzzy and undetermined elliptical Kepler orbit, with
Indeed we can verify in the classical case of a circular orbit that its speed is v = veZ/n (as the energy is proportional to v2 and it has the right value for n=Z=1) thus its angular momentum is L = a mev = n ℏ.

We may also compute in the interpretation by Kepler orbits, the orbital period T, independent of excentricity thus computable from the circular case :
T = 2π a/v = (2π a/ve) n/Z = (h/meve2)(n3/Z2) = (h/2Ry)(n3/Z2)

to be compared with the difference between 2 nearby energy levels for large values of n : En+1 - En =  Ry Z2(1/n2 − 1/(n+1)2) ≈ 2 Ry Z2/n3 = h/T.
Indeed we can understand that an electron in circular orbit has a sinusoidal movement and thus can only emit a photon with a definite frequency, the one that will reduce his energy level by only one unit. This photon will also carry the one unit of angular momentum that the electron must lose, since the lower energy level must have lower angular momentum too.
On the other hand, the movement in an elliptical orbit is not sinusoidal but has a range of other harmonics ; these correspond to the different possible energies of the first photon that might be emitted. These higher harmonics are specifically due to the faster movement of the electron near the kernel. Photons emitted from this origin specifically take their energy from the speed of the electron in this section of the orbit. The electron being slowed down the there is driven into a less excentric orbit. This is normal because each photon can only carry one unit of angular momentum.

The Rydberg unit of energy would characterize the interaction between 2 charges both equal to e. However this is not exactly the dominant case of interaction between charges. Indeed the Pauli exclusion principle that does not allow 2 electrons to be very close to each other, usually behaves in atoms like a repelling force that is stronger than the electrostatic force : it keeps electrons with the same spin orientation far enough from each other to make the electrostatic force between them small and irrelevant.

As for electrons with opposite spin, they can somehow still meet through quantum tunelling : the probability density of their relative position does not even cancel near the zero vector; still of course it is lower there, so that this interaction lowers its own effects. Anyway electrons naturally stay together in pairs of opposite spins because the rest of the atomic structure which determines the state of lowest energy for any electron, usually gives them both the same offer.

The dominant phenomenon that governs the behavior of atoms is the attractive electrostatic interaction between the electrons on the higher (external) energy levels, and the nucleus. So, between a charge = -e and a charge that is many times e.

Still not as many times as Z, because of the shielding effect by the negative charge of the electrons on lower energy levels (and a partial shielding from those on the same level), on the behavior of the electrons more far from the kernel, that determine chemical interactions. And in the formula of a, proportional to n2/Z, the higher value of Z which would shrink the size of the atom, is balanced by the higher value of n (which roughly measures how much the quantum mechanical properties can be approximated into classical ones, by the least action principle with an action equal to n.h).

Now let us give the value of the coefficient in the formula of a :
a0= ℏ /meve = 5.292 10−11 m = 0.5292 Å (Angstrom)
This is the fundamental unit of distance from which all sizes of atoms are derived. For example in a water molecule H2O, the distance between the O and H atoms, is 0,958 Å = 1.810 a0.

We can also look at the volume taken by each molecule of H2O in liquid water : its molar mass is 18.015 g/mol, and its density is 1 g/cm3,
thus 1 cm3 contains N/18.015 = 3.343 1022 molecules. The volume per molecule is that of a cube with size 3.1 Å.

In the case of diamond, the molar mass is 12.01 and the density is 3.52 g/cm3. Each atom thus takes the volume of a cube of 1cm(12.01/3.52 N)1/3 = 1.783 Å. The distance of each atom with each of its 4 closest neighbors is 1.544 Å.
 

Let us explain why the nucleus is much smaller than the atom : both sizes are determined by the wavelength ℏ /mv of a particle with a mass m at a speed v. The difference is mainly that the mass involved for the size of atoms is the mass of an electron, that is much smaller than the mass of protons and neutrons, mp/me = 1,836.15. (There is currently no explanation for this value)
A smaller contribution to the ratio is that the typical speed of electrons in atoms, 2,187.7 km/s, is slower than the typical speed of protons and neutrons inside the nucleus, which we calculated above as 85,000 km/s. (The ratio between these speed also determines the amplitude of how neutrons become more numerous than protons in heavy atoms, and finally the fact that too heavy atoms are unstable).

The compressibility of condensed matter

Let us now compute how strongly can condensed matter (liquids and solids) resist compression. We saw in the case of diamond that the size of semi-major axis roughly expressed by a =  (ℏ /meve) (n2/Z), kept values close to ℏ /meve (a distance of 3 ℏ /meveb between atoms in a covalent bound can be understood as 2 excentric orbits meeting in the middle) while n=2 so that the shielding effects may have decreased the effective value of Z to something like Z=4 despite the basic value Z=12. But for the value of the energy − Ry Z2/n2 we have a further factor of Z, so that we have several times Ry per electron. Moreover, usually each binding electron binds 2 atoms but each atom is bound by more than 2 electrons. Thus the total binding energy per atom may be several Ry. But in the absence of covalent bounds holding all atoms together on the large scale, the effective binding energy can be lower. Still the resistance to compression can remain as it is just a matter of volume per molecule and does not depend on the precise configuration between molecules.

The compressibility of condensed matter is the proportionality coefficient between the compression rate (an infinitesimal dimensionless quantity) and the pressure. Its inverse, called the bulk modulus, is a pressure. A pressure is homogeneous to an energy density (an energy per volume). We can get such a coefficient by multiplying the binding energy per atom, by the number of atoms per volume.

Diamond
According to this site, the inverse of diamond's compressibility has almost the highest value of all materials. It is 443 GigaPascals, thus 4.431011 Pa = 4.431011 J/m3.
This value corresponds to an energy per atom equal to 2.51110-18 J = 1.15 Ry. A more precise calculation taking account of the exact meaning of things, is presented below.

Water
[From Wikipedia] : The compressibility of water is a function of pressure and temperature... At the zero-pressure limit, the compressibility reaches a minimum of 4.410−10 Pa−1 around 45 C before increasing again with increasing temperature.
As the pressure is increased, the compressibility decreases, being 3.910−10 Pa−1 at 0 C and 100 MPa (=1000 atmospheres).
The low compressibility of water means that even in the deep oceans at 4 km depth, where pressures are 40 MPa, there is only a 1.8% decrease in volume.
The bulk modulus is thus 2.2109 Pa , corresponding to an energy per molecule = 6.810−20 J = 0.031 Ry.
Seen in this way, water appears as quite a compressible substance after all.


Other
We find intermediate values between the above for glass (35 to 55 GPa) and steel (160 or 170 GPa).

The speed of the sound in condensed matter

Dividing the bulk modulus by the density of mass (in kg/m3), gives an energy per mass, thus a square of a speed. The square root of this quantity gives the speed of the sound in that substance. The more compressible is a substance, the slower is the speed of sound there.
Examples: in water, the speed of sound is 1,484 m/s = √2.2109 /1000 - 1497 m/s at 25 C in fresh water, and 1560 m/s in sea water (without bubbles or other things), decreasing to a minimum of 1480 at about 800m depth, then increasing again.

In steel, with 7700 kg/m3 , we have a longitudinal velocity of 6000 m/s.
In granite it is 5000 m/s.

However there is a lower speed for transversal waves (S-waves), as solids are usually more "compressible" by distortions (change of shape) than by uniform compression (change of volume)).
The velocity of seismic waves tends to increase with depth, and ranges from approximately 2 to 8 km/s in the Earth's crust (starting in the Tibetan Plateau with a 16 km thick upper crust with P-wave velocity 5.55 km/s and S-wave velocity 3.25 km/s, to 8km/s of P-wave in the upper mantle), and values go up to 13 km/s down in the deep mantle, thus even larger than the value of the speed of sound in diamond from the value of its bulk modulus mentioned above (11.2 km/s). This is because at very high pressures, the precise configuration of atoms becomes irrelevant, as the atoms just resist being smashed in volumes : all electrons in the external layers contribute to resist pressure as they need their space (because of Pauli's exclusion principle); while at low pressures, molecules who had all the space could take their "confort" away from each other and be only weakly bounded to their neighbors in subtle ways.

This speed is much slower than the typical speed of electrons in atoms ve = 2187.7 km/s. This is because the kinetic energy Ry =  meve2/2 of electrons at ve is replaced by some energy E per atom with the same order of magnitude but now converted back into a speed v by the formula E = mv2 where m is the much heavier mass of the whole atom instead of the mass of the electron.

Now we can explain why it is at the size of the Earth that rocky planets undergo significant compression by their own weight:
Take the velocity of the seismic waves in the deep mantle 13 km/s, that gives the order of magnitude for the compression of matter at high pressures, and multiply it by the gravitational time of condensed matter, that was for the Earth 805 s. The result is 10.500 km, comparable with the Earth's radius of 6,356 km. So the Earth undergoes significant compression because its radius is not small compared to 10.500 km. (There is the superficial difference of pressure and thus compression from the surface to the mantle, that drives the velocity of seismic wave from small to large values; then there is the compression we speak about that happens more far deep, governed by the velocity of 13 km/s).

Now what about the speed of the sound in the air ? In dry air at 20 C (68 F), the speed of sound is 343.2 metres per second, thus 4 times smaller than that in water. But this comes from a very different law, that we shall present now.

Temperature

At the macroscopic level, a temperature is an energy per entropy.
At the microscopic level (from the viewpoint of a few particles), a temperature is an energy per quantity of information.
Because the microscopic nature of entropy is that of a quantity of information.
To understand this, let us present Boltzmann's law, general description of physical systems in thermodynamical equilibrium with an environment at a temperature T (this law can itself be deduced from the first principles of quantum physics):

Boltzmann's law: For any physical system whose range of possible states (configuration space) can be described in the form of (an abstract split of its configuration space, into) a list of its possible distinct elementary states with definite values of their energies, it is said to have temperature T if it is fully described as obeying the only probability law on such a list of states, so that the probability to be in each state is proportional to exp(-E/kT) where E is the energy of this state, and k is the Boltzmann constant (that does the conversion between units of temperature and energy).

(This definition is actually independent of the choice of such a list; this choice is no more unique when there exists more than one elementary state with the same definite value of the energy − the energy of a system is no more definite if it has nonzero probabilities to be in elementary states with different definite values of the energy).

Thus, between 2 states A and B with energies E and E' such that E'-E=kT*ln(2), the state A is twice more probable than A.
But if we have another state B' with the same energy as B, then being in state (B or B') will be as probable than being in state A. So, the state (B or B'), where B and B' have each probability 1/2, has more energy than A (the difference is kT*ln(2)) but it has entropy = 1 bit of information = ln(2), while A, as a pure state, has zero entropy.
Finally, A is equiprobable with (B or B') because both have the same free energy.

However in many cases, the quantity of information is just given by the quantity of matter, and can thus be just replaced by it.
For example, if you compress a gas by dividing its volume by 2 and then cool it to keep its initial temperature, then you have just substracted 1 bit of indetermination on the position of each molecule of gas : the cooling must have taken away a quantity of entropy equal to n*ln(2) where n is the quantity of molecules (their number divided by the Avogadro number). It went away as a heat, thus with a quantity of energy that is the product of this quantity of entropy by the temperature.
For the temperature to stay constant, a decrease of volume V of n moles of gas by a small fraction dV/V must be accompanied by a release of entropy, that, counted in units of information (1 bit = ln(2)), equals to nN dV/V where N is the Avogadro number. Thus as a quantity of heat with temperature T, it goes with a quantity of energy dE= kTnN dV/V = Tn R dV/V where R=kN is the gas constant (8.314 JK−1mol-1). This energy that goes out, came in as a work of pressure PdV, thus the equation of ideal gases PV=nRT.

Macroscopically, quantities of entropy S are counted so that for a quantity of heat (flowing from a place to another) at temperature T, its energy E and its entropy S are related by E=ST. This unit of entropy is converted into the unit of information (1 bit = ln(2)) by the Boltzmann constant k : 1 conventional unit of entropy (1 joule per kelvin) = 1/k units of information or 1/(k ln(2)) bits of information, so that the number of units of information, to enter Bolzmann's law, is E/kT = ((1/k) times the number of units of entropy)


Energy of a quantum harmonic oscillator with a given temperature


If the typical energy kT of a given temperature is large compared to the quantum of energy of oscillation, then the mean "amplitude" of the movement is approximated by classical mechanics : the density of probability in the phase space is proportional to exp(-E/kT). For a one-dimensional oscillator, regions in the phase space limited by values of the energy of oscillation, have their area proportional to the interval of energy (difference between chosen max and min of energies). Thus the number of states is just proportional to this interval. The mean value of the energy is equal to kT.

Let us exactly compute the mean value of the energy of a quantum oscillator (with a probability distribution of the phonons number given by Bolzmann's law) above its ground energy .

It is the product of the value of the energy of phonons E0 with the average value of the number of phonons.
This mean number can be computed as the sum for all natural numbers j>0, of the probabilities pj to have n no smaller than i. Indeed, in this way, the probability of having each possible number n of phonons, is counted n times in this sum, once for every 0<jn.
Then we find pj = exp(-jE0/kT)).
Indeed p0=1 and this formula satisfies the Bolzmann's law that the probability (pn - pn+1) of having exactly n quanta of energy, is proportional to exp(-nE0/kT)).

Then the sum of the pj for all nonzero values of j, gives 1/(exp(E0/kT)−1)
leading to a mean energy E0/(exp(E0/kT)−1)

If E0 is very small compared to kT then this is approximately E0/(E0/kT) = kT, or more precisely, kT-1/2, corresponding to the classical oscillator with thermic agitation (neglecting the quantum effects).
If E0 is very large compared to kT then this is approximately E0 exp(-E0/kT) : this is as if we just had a possibility of the first phonon with probability exp(-E0/kT).
Let us compare these formulas, for E0=1 and variable kT:

kT 0.2
0.5
0.75
1
1.442695 2
4
100
kT-1/2 -0.3
0
0.25
0.5
0.942695
1.5
3.5
99.5
1/(exp(1/kT)−1) 0.00678
0.1565
0.358
0.582
1
1.5415
3.521
99.5
exp(-1/kT) 0.00673
0.1353
0.2636
0.3678
0.5
0.60653
0.7788
0.99

In particular for E0=5kT, the oscillating energy 0.00678 is only 3.39% of the classical oscillating energy of kT=0.2, and can thus be considered very small (shut down by the quantization of movement).

The speed of sound in the air

The most obvious effect of temperature, is its determination of the speed of sound in the air.
Namely, it has the same magnitude as the average speed of molecules. Or more precisely, as the average value of the component of the speed in the direction of propagation.
The distribution of probabilities of the component vx of the speed of a molecule with mass m in a given direction x, is given by Boltzmann's law: exp(-mvx2/2kT). This gives an order of magnitude for the speed of sound, of vx=sqrt(kT/m).

But let us look for a more exact formula.
The exact formula is based on the compressibility of the air : if a volume of air decreases by some proportion, how much does its pressure increase ?
We have the formula of ideal gases : PV=nRT.

The variation of pressure itself depends on 2 things : the variation of volume (directly given) and the variation of temperature. The problem is to compute the variation of temperature.
The compression is adiabatic : it preserves the quantity of entropy. The decrease of entropy of positions is given by the given relative decrease of volume. Now this entropy is converted into other possible forms corresponding to the increase of temperature.
Any given relative increase of temperature dT/T goes together with entropy increases as follows :
In the case of a diatomic molecule such as O2 or N2, that are the two main components of air, this count gives:
With the free movements alone, a temperature increase dT/T of n moles of gas "contains" an entropy increase
dS = (5/2)nR dT/T.
If we count the elastic movement, then it is dS = (7/2)nR dT/T.

Thus the entropy decrease of positions dS = -nRdV/V (as dV/V is negative, this quantity is positive) goes with a temperature increase (without elastic movement)
dT/T=(2/5)(dS/nR)= -(2/5)dV/V
Finally the relative pressure increase is dP/P= dT/T - dV/V = (1+2/5)(-dV/V)
The bulk modulus is B=dP (V/-dV)= γP where γ=1.4 is the heat capacity ratio.

The speed of sound is sqrt(γPV/m) where m=mass of the volume V of air.
So, v=sqrt(γnRT/m)

With vibrations, γ= 1.2857

The proportion of water vapor is quite variable. Let us take an air with about 1.2% of water vapor (that is an ordinary value), for its further dimensions of movement to cancel the effects of lesser ones of the 9,3% part of monoatomic Argon (3 dimensions of free movement).
Let us compute the molecular mass of air. Taking for example 77.1% of nitrogen with molecular mass 28;  then 20.75% oxygen with molecular mass 32 ; then 0.93% Argon with atomic mass 40; finally, 1.22% water vapor with molecular mass 18. We get m/n= 28.82 g/mol (the value is 28.96 for dry air).

The above becomes  v= γRT/0.02882 where R=8.314 JK−1mol-1, so v= 288.48γT

The vibration does not take place at usual temperatures, but only at higher temperatures. Let us investigate this now.

The strength of bonds between neighbor atoms or molecules

The rigidity of the covalent bounds as oscillators (oscillations of the value of the distance between 2 atoms due to the form of the potential energy as a function of this distance), was already involved in the former study of the speed of sound in solids.
Take a crystal, or any sort of condensed matter. The rigidity of bonds between neighbors can be deduced from a measure of the speed of sound there, by the following reasoning.

Imagine a crystal with atoms or molecules configured in horizontal slices.
First slice is steady
Second slice oscillates up-down
Third slice is steady
Forth slice oscillates down-up
Fifth slice is steady
and so on.

This situation can be equivalently interpreted in 2 ways.
One way is that each moving slice of atoms oscillates as blocked by both neighbor slices
The other way is to notice that this forms a stationary wave of sound (=a superposition of 2 waves in opposite direction).
Thus, the classical oscillation period for a typical bound between neighbor atoms or molecules there, is the time for the wave to go through 4 slices (it is π√2 =4.44 for a more exact analysis). But there are several bounds there, on each side (up and down). To represent the effect of only 1 bound, the period is longer (2π).

The rigidity of bonds in diamond


But let us make a separate computation the case of diamond. It has a bulk modulus of K=443 GigaPascals, defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. Take a cubic piece of diamond with size x, and compress it to a size x(1-ε) with a small ε. Its volume V is compressed as V(1-3ε). The pressure after compression is 3Kε. The potential energy of compression is 9VKε2/2.
We said, each atom takes the volume of a cube of z=1.783 Å. Thus the potential energy per atom is 9z32/2.
Each atom has 4 bonds, while each bond connects 2 atoms. We calculated z3K= 1.15 Ry.
So the potential energy per bond is 9z32/4 = ε2(2.59 Ry).

The distance of each atom with each of its 4 closest neighbors is 1.544 Å.
So in the formula E=k.x2/2 for a covalent bond in diamond, we have k = 2.17 Ry / Å2.

The vibrations of the nitrogen molecule

Now let us come back to the vibrations of a diatomic molecule, with potential energy E=a.x2/2.
The quantum of vibration has energy ℏ k/m.
On the one hand, the potential energy of the bond is rather strong as we saw. There are several reasons for this: a bond has 2 electrons, and the effective charge of each nucleus is several times the elementary charge.
This (influence on k) is somehow balanced by the fact this potential is counted along an inter-atom distance of 1.544 Å that is about 3 times the fundamental distance a0= 0.5292 Å.

But on the other hand, the mass m involved here is the reduced mass, = half the mass of an atom, instead of the mass of an electron. An electron has a mass that is approximately 1/1836 that of the proton.
Thus 1/22030 times that of the carbon atom, or 1/29380 times that of the oxygen atom (1/14690 times the reduced mass for dioxygen).
Then we need to get the square root of this, and get a fraction of something like 1/100.

Now let us look at the exact strength of bond for the nitrogen molecule (N2).
It is observed in the form of its absorption spectrum: the main wavelength of absorption in the infrared, forms a peak around 4.3 μm (or 4.24 ?). (For carbon monoxide it is 4.8 μm).

So the classical oscillation period for nitrogen is t=2π m/k = (4.3 μm)/c.
The reduced mass m for nitrogen is about 14/2=7 times the mass of a proton, = 1.16293 10-26 kg.
Thus k = m(2πc/4.3 μm)2
 (from google calculator, with 1Ry= 2.179910−18 J)
k = (14.0067 u * (((2 * pi * c * (1e-10 m)) / (4.3 micrometers))^2)) / (2*2.1799e−18 J) = 10.24 Ry/Å2

That is, nearly 5 times the value of the rigidity of the bond between carbon atoms in diamond that we obtained above.

The distance between nucleus is 1.10 Å for N2, but 1.21 Å for O2 (reference)

For O2 the frequency is (from this article) 1580 c/cm, so the period is 6.33 μm/c
k = (16 u * (((2 * pi * c * (1e-10 m)) / (6.33 micrometers))^2)) / (2*2.1799e−18 J) = 5.40 Ry/Å2

This is hardly more than a half of the value for nitrogen. The difference of distances between nucleus and the fact that N2 has a triple bond while O2 only has a double bond, would not seem sufficient to explain this difference.
We can explain this by the fact that N2 is a very stable molecule, where electrons happen to remarkably fit at a low energy level for the precise value of the distance between atoms, while they don't fit so well with O2 (it does not have so low energy level, so that O2 is very reactive in combustions).

You may say : the value 5.40 Ry/Å2 is still quite strong compared to the one 2.17 Ry / Å2 we found for diamond. But, not only the bond for diamond is a simple bond, but its inter-atom distance (1.544 Å) is significanly larger (than 1.21 Å); and the typical energy of electrons is quite sensitive to such distances: for a free electron, doubling its wavelength means a division by 2 of its momentum, thus a division by 4 of its kinetic energy. And a potential energy function with that height but spread in twice the space, has its rigidity divided by 16.
Namely, the kinetic energy of an electron is 1 Ry when its half-wavelength is πa0= h /2meve = 1.66 Å.
Now let us compute the quantum uncertainty on the inter-atom distance (d2= ℏ / km = ℏ2/mE = ℏλ/(2πc*m) where E is the quantum of vibrational energy and λ is the wavelength):
For N2 it is d2= ℏ*4.3 μm/(2πc*7u)
that is d= (hbar*(4.3 micrometers/(2*pi*c*7*u)))^0.5= 0.0455 Å

For O2 it is d= (hbar*(6.33 micrometers/(2*pi*c*8*u)))^0.5= 0.0516 Å

Finally, let us see at which temperature do the vibrational states become significant.
For this, all we need is to compare the vibrational frequencies, with the energies corresponding to the temperatures, which also appear in the form of the spectrum of the black-body radiation of these temperatures - but there is a significant constant numerical factor between them.
Each oscillation period t=λ/c defines a quantum of energy E= h/t=hc/λ that corresponds to a characteristic temperature T= hc/kλ (where k is the Boltzmann constant).

For λ=4.3 μm we have T=hc/(k*4.3 micrometers) = 3 346 K.
For λ=6.33 μm we have T=2273 K.
The temperature of the surface of the Sun where the visible light comes from, T=5778 K corresponds to a wavelength
2pi*hbar*c/(k*5778 Kelvin)= 2.490μm
The one of melting ice, T= 273.15 K, corresponds to the wavelength 52.67 μm

The black-body radiation

You may wonder why, if the temperature of the surface of the Sun represents an energy corresponding to the wavelength 2.490μm, the main radiation from there, namely visible light, has quite shorter wavelengths : 390 nm (violet) to 700 nm (far red) and even shorter ones : ultraviolet (which is visible for birds, and also affects us otherwise). This seems surprising, especially as in temperature T, the probability of a state as a function of its energy E is proportional to exp(-E/kT), suggesting that its presence becomes quickly insignificant as E takes values larger than kT.

So let us explain the gap here.
It is due to the fact that, while higher energy states are individually less probable, they are more numerous, letting larger collections of possibilities with small individual probabilities take a share of importance over lower energy ones that are individually more probable but fewer.

Let us describe this as concerns the black-body radiation, that is the radiation from an ideally black body with a given temperature. (By a simple reasoning of thermodynamic equilibrium, we can see that a body that would have another color, absorbing a proportion a<1 of the light of a given wavelength, and diffusing the remaining 1-a, would also radiate a times the black-body intensity at this wavelength ; the black color means a=1).

For the concerns of radiation, the important measure is the volume in the configuration space of photons, that has an absolute unit (number of locations). We previously made such a count for the configuration space of protons and neutrons in an atomic nucleus. But here we consider photons that are bosons (while protons and neutrons are fermions), so that the Pauli exclusion principle does not apply here : each location in the configuration space has its state described by the number of photons it contains. Indeed, this number of photons is the number of quanta of oscillations for the electromagnetic field at this location.
Like with protons and neutrons, there are in fact 2 effective locations in the configuration space per space-time location, because there are 2 states of spin (polarizations).
We calculated the mean energy of a quantum oscillator with energy E0 at temperature T, to be = E0/(exp(E0/kT)−1).

A particle has 3 coordinates of position and 3 coordinates of momentum, forming 3 pairs of coordinates (x,px) (y,py) (z,pz) each made of 1 position and 1 momentum, and the unit of area inside each pair is given by the Planck constant h that, in terms of wave, represents a phase of 2π (one period of oscillation).
Now it is the same for the volume in the phase space of photons:
Two of the coordinates of momentum define the direction of propagation; the corresponding "position coordinates" represent the position of the photon on a screen orthogonal to this direction of propagation.
Take a photon whose direction of propagation is close to the z axis, and consider one of the pairs of coordinates in the phase space:
(x,θ) where θ measures angle in radians, of the deviation of the propagation axis in the x direction : xθ.z
The "unit of area" (number of locations) in the surface of the phase space limited by a given small interval dθ of values of θ, and an interval Dx of values of x, is then =Dx.dθ/λ where λ is the wavelength of the photon (because on a screen making an angle dθ with the direction of propagation, the picture of the field at a given time is that of a wave with length λ/dθ)

The last pair of coordinates can be expressed as (t,ν) (t= time, ν= frequency), a choice that makes its unit of area (its 2π phase) directly equal to 1. In other words, the number of locations = (interval of time) times (interval of frequencies).

Now let us deduce the function that compares the contributions of different wavelengths λ (or frequencies ν=c/λ) in the black-body radiation of a temperature T.
In each unit of surface S and each solid angle Ω (a small one, in steradians, roughly orthogonal to the surface S), the number of "locations" of the electromagnetic field with frequencies between ν and ν+dν (with dν much smaller than ν) radiating away in each time interval dt, equals to
(2 S.Ω.t/c2).ν2dν = (2 S.Ω.t/c2).ν3(dν/ν)

There are different ways to express the contributions of different wavelengths.
We may either be interested in visibility (number of photons), or in flow of energy.
The mean number of photons per unit of location is 1/(exp(hν/kT)−1).
The mean energy per unit of location is hν/(exp(hν/kT)−1).

We may either count them for intervals of frequency dν, or intervals of logarithm of frequency, (dν/ν).

Finally, up to constant multiplicative factors, 2 functions can be considered relevant for comparing the contributions of wavelengths:
Note that the total flow (summed over all wavelengths) of radiated photons (per surface per time) is proportional to T3, and the total flow of energy is proportional to T4.
 
Now let us fix the temperature and conventional units so that kT/h=1, and compare the values of the above function for different frequencies:
ν 1
2
3
4
5
6
7
8
9
ν3/(exp(ν)−1) 0.582
1.252
1.415
1.194
0.848
0.537
0.313
0.172

ν4/(exp(ν)−1) 0.582 2.504
4.244
4.776
4.240
3.220
2.191
1.375
0.810
In particular, contrary to the previous context, the contributions from the values around E0=hν=5kT cannot be considered "very small" but belong to the range of the main contributions to the black-body radiation.

The atmosphere


[More developments on the properties of matter will be written later]


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