Thermodynamic potentials


From Callan, we have two equivalent principles of thermodynamics:.

  1. Entropy maximum principle. The equilibrium value of any unconstrained internal parameter is such as to maximize the entropy for the given value of the total internal energy.
  2. Energy minimum principle. The equilibrium value of any unconstrained internal parameter is such as to minimize the energy for the given value of the total entropy.1

The basic formula for the internal energy as a function of S and V (and the chemical potentials, Ni) is

U = TS - pV - \mu N

where N stands for the sum over different chemical potentials

\sum_{i} \mu_{i}N_{i}

representing the quasi-static chemical work and

dU = TdS - pdV - \mu dN

Depending on the problem at hand, It may be simpler to solve for the equilibrium state if the formula had different independent variables. That is what Legendre transformations do.

Legendre transformations

Given a formula

Y = Y(X)


P=\frac{\partial Y}{\partial X}

is the slope of the curve at a given point. We would like an equivalent representation of the system but one where the independent variable is P rather than X. We will use the intercept of the Y axis of a tangent of constant slope at X.

Then the slope is given by

P = \frac{Y-\psi}{X-0}

which can be arranged to give the Legendre transformation

\Psi = Y - PX


d\Psi = dY - PdX -XdP = -XdP


-X = \frac{\partial \Psi}{\partial P}

so the inverse transformation is just

Y = XP + \Psi

In general, for

Y = Y(X_0,X_2...X_i)

the partial slope of this hypersurface is given by

P_k = \frac{\partial Y}{\partial X_k}

and the Legendre transformation is

\Psi = Y -\sum_{k} P_kX_k

Example from Callan, problem 5.2-1:

y = \frac{x^2}{10}    gives    P = \frac{\partial Y}{\partial X} = \frac{x}{5}


\Psi (P) = y - Px = -\frac{5}{2}P^2


-X = \frac{\partial \Psi}{\partial P} = -5P


Y(X) = \Psi + XP = \frac{X^2}{10}

Principle thermodynamic potentials

The most used thermodynamic potentials are the following and are all Legendre transforms of the internal energy U as shown in the table.

Name Symbol Formula Natural variables Constant Interpretation
Internal energy U TS – pV + μNi S, V, Ni Total internal energy
Helmholtz free energy F (or A) U – TS T, V, Ni T Max energy at constant T, V after system pays “entropy tax”
Enthalpy H U + pV S, p, Ni p  Total heat (caloric) energy availabe in system at const. P after V change
Gibbs free energy G U + pV -TS T, p, Ni T and p  Max energy available at const. T, P after V change and “S tax”

We can identify each term with a type of energy:

  • TS with heat (or caloric) energy,
  • pV with volume-change energy (as when a piston is pushed) and
  • the sum \sum_{i} \mu_{i}N_{i} as the energy of chemical bonding.

We will ignore such things as electrical energy.

The above-cited energy-minimum principle applies to all these potentials under specific circumstances.

Helmholtz Potential Minimum Principle. The equilibrium value of any unconstrained internal parameter in a system in diathermal contact with a heat reservoir minimizes the Helmholtz potential over the manifold of states for which T = Tr [at constant T].1)Callan, 155

Enthalpy Minimum Principle. The equilibrium value of any unconstrained internal parameter in a system in contact with a pressure reservoir minimizes the enthalpy over the manifold of states of constant pressure (equal to that of the pressure reservoir).2)Callan, 156

Gibbs Potential Minimum Principle. The equilibrium value of any unconstrained internal parameter in a system in contact with a thermal and a pressure reservoir minimizes the Gibbs potential at constant temperature and pressure (equal to those of the respective reservoirs).3)Callan, 157

We can attribute physical meanings to these potentials.

In the case of enthalpy, a system at constant P can expand or contract, therefore losing energy to move around molecules. This volume-change energy is exactly what is subtracted from U in order to derive H. So enthalpy is the energy released as heat by the system at constant P. Or, in the other direction,

…heat added to a system at constant pressure and at constant values of all the remaining extensive parameters (other than S and V) appears as an increase in the enthalpy.4)Callan, 161

As for the Helmholtz free energy,

… the work delivered in a reversible process, by a system in contact with a thermal reservoir [so at constant T], is equal to the decrease in the Helmholtz potential of the system.5)Callan, 159

The Helmholtz free energy is thus the available work at constant temperature and volume. The term TS represents the “entropy tax” which the system must pay in order for the total entropy of the universe to remain zero.

The Gibbs free energy is useful when T and P are constant, which is the usual case of chemical reactions open to the atmosphere, which acts as a reservoir of T and P. Gibbs free energy is thus much beloved of chemists. Callan gives a more interesting example, pointing out that it may also be true “… in a small subsystem of a larger system that acts as both a thermal and a pressure reservoir (as in the fermentation of a grape in a large wine vat).”6)Callan, 167

Spontaneous reactions

Suppose there is only one thermodynamic system in the universe and that its T and V are constant. Then the sum of the entropy changes of the system and of its surroundings is the entropy change of the universe.

\Delta S_{univ} =\Delta S_{surr} +\Delta S_{sys}

If the energy of the system changes by an amount U, then the surroundings will change by -U, and

\Delta S_{surr} = - \frac{\Delta U}{T}


\Delta U_{univ} = \Delta S_{sys} - \frac{-\Delta U}{T}

which means

\Delta U_{univ} = T\Delta SU_{univ} = -(\Delta U -T\Delta S) = -\Delta F

So the entropy maximum principle says that for constant T and V, the change in the Helmholtz free energy must be negative, so that \Delta S_{univ} = \frac{\Delta U_{univ}}{T} > 0.

Similarly, if only a spontaneous chemical reaction occurs at constant T and P, then a change in heat energy (enthalpy) of the system means the surroundings change by the negative of that., i.e., -\Delta H_{sys}. So equation (1) gives

\Delta S_{univ} = -\frac{\Delta H_{sys}}{T} + \Delta S_{sys}

The only change in the universe is due to the dispersal of energy in a quantity we may call -ΔG. Therefore

\Delta S_{univ} = -\frac{\Delta G}{T} = -\frac{-\Delta H_{sys}}{T} + \Delta S_{sys}

which may be rewritten as

\Delta G = \Delta H_{sys} - T\Delta S_{sys}

which is the Gibbs free energy.


Atkins, Peter, The laws of thermodynamics: A very short introduction. Oxford: Oxford University Press, 2010. Print.

Callan, Herbert B. Thermodynamics and an introduction to thermostatistics. New York: John Wiley and Sons. 1985, 2005. Print.

Shankar, R. Fundamentals of Physics. Mechanics, Relativity, and Thermodynamics. New Haven: Yale UP, 2014. Print.

Notes   [ + ]

1. Callan, 155
2. Callan, 156
3. Callan, 157
4. Callan, 161
5. Callan, 159
6. Callan, 167

Galaxies, clusters, super-galaxies

Today, there are hundreds of billions of galaxies, each containing hundreds of billions of stars. And that is just for the observable universe.

Not only are stars grouped into galaxies, galaxies are grouped into clusters and clusters into superclusters. Space is really, really big. Stars in a cluster generally have about the same age and were probably formed from the same nebula. Our galaxy, the Milky Way, is a member of a group — too small to be called a cluster — called the Local Group.

Local galactic group, by Andrew Z. Colvin via Wikimedia

Local galactic group, by Andrew Z. Colvin via Wikimedia

Studies of the large-scale structure of the universe — on scales where galaxy clusters look tiny — indicate that galaxies are spread out in a filament-like structure. It is sometimes compared to soap bubbles, where the galaxies are distributed mainly on the surfaces of the bubbles.

Panoramic view of the entire near-infrared sky reveals the distribution of galaxies beyond the Milky Way. by IPAC/Caltech via Wikimedia.

Panoramic view of the entire near-infrared sky reveals the distribution of galaxies beyond the Milky Way. by IPAC/Caltech via Wikimedia.

Although the interior of the “bubbles” is less bright, so that they are also called voids, there are galaxies inside them, including our Local Group.

Reconstruction of inner parts of d2f survey by William Schaap via Wikimedia Commons

Reconstruction of inner parts of d2f survey by William Schaap via Wikimedia Commons

Large-scale structure and geometry of the universe

Assuming the validity of the Cosmological Principle, that the distribution of objects in the universe is homogeneous and isotropic (the same everywhere and in every direction), the equations of GR have different solutions for the geometry of the universe. These solutions correspond to three possible sorts of curvatures of the universe (in the GR sense).

Geometry of the universe, from WMAP via NASA

Geometry of the universe, from WMAP via NASA

The geometry depends on a parameter called the critical density of the universe, ρcrit. One often refers to Ω0, the ratio of the observed density to the critical density, .

Ω0 = ρobscrit.

  • Ω0 = 1, means the universe is flat and the sum of the angles of a triangle is 180°.
  • Ω0 < 1 means the universe has negative curvature and the sum of the angles of a triangle is less than 180°.
  • Ω0 > 1 means the universe has positive curvature and the sum of the angles of a triangle is greater than 180°.

For Ω0 >= (greater than or equal to) 1, space is infinite. If space is infinite now, it always has been, including at the Big Bang. That was indeed the universe in a grain of sand!

Different forms of matter and energy

It is now thought that the universe is composed of three sorts of matter and energy.

Ordinary matter is the classic one and once was thought to be the only kind. This is the matter we are familiar with and which is composed mainly of protons, neutrons and electrons. Contrary to previous ideas, we now know that it only makes up 4.6% of the energy density of the universe1)All percentages in this section come from WMAP.. The rest is made up of dark matter and dark energy.

It has been observed that the outer parts of galaxies rotate too fast; the amount of matter in the galaxy is insufficient to generate a gravitational force capable of keeping them from spinning off into space. But dark matter could counteract that centrifugal force. The mass of dark matter provides the gravitational pull needed to hold the galaxy together. Similar phenomena are seen for galaxies in clusters and for the strength of gravitational lensing.2)Bending of light waves from a distant star by another star or a planet, like by a lens.

Dark matter must be composed of something which interacts only weakly with ordinary matter and is not visible to us, but its composition is still unknown. It may be non-baryonic matter left over from the Big Bang. Hypothetical dark-matter particles are referred to as WIMPs, Weakly Interacting Massive Particles. Recent studies suggest several types of such particles.3)Astronomers Measure the Density of Dark Matter in Galaxy Clusters. SciTechDaily. Dark matter constitutes 24% of the energy density of the universe.

Dark energy is stranger yet. It has been observed from SNIa red shifts that the expansion rate of the universe has been increasing over about the last 7 billion years. The equations of GR contain what at first seemed an arbitrary constant,4)Deemed the Cosmological Constant. which has the interesting – in this case, very handy – property that it causes space to be suffused with a dark energy of negative pressure which exerts a push, not a pull, on matter. After inflation ended, ordinary gravity attracted matter together and slowed down the expansion. But expansion continued nonetheless. As matter became more diffuse in space, gravity was no longer a match for the dark-energy pressure, which is thought to be a property of space and so is constant and not diluted. Since the universe celebrated about its 7 billionth birthday, the dark-energy push has been stronger than the attraction of gravity and the expansion of the universe has accelerated. Dark energy has been calculated to furnish 71.4% of the energy density of the universe. Almost amazingly, this is just the total energy density required in order for Ω0  to be approximately equal to 15)WMAP finds a value of 1.02+-0.02. so that the universe is “flat”.

Universe makeup pie chart, from WMAP via NASA

Universe makeup pie chart, from WMAP via NASA

The bad news is that theoretical calculations of the constant come up with a value far, far too large to be true. Work continues.

In summary, dark matters attracts and can be diluted by the expansion of space. Dark energy repulses and never dilutes.

It is quite astonishing that only of 4.6% of the

Formation of our solar system

About 10 Gya, a new protostar was formed in the outer fringe of the the galaxy we call the Milky Way. As usual, it was turning. Since the centrifugal force was greater in the plane perpendicular to the axis of rotation, a flat disk of dust formed around the big lump in the middle. The lump became the proto-Sun. The matter in the disk also clumped together into rotating chunks called planetesimals. The largest of these chunks formed the planets, and other, smaller pieces, became asteroids and comets. When the centrifugal force of their motion around the Sun balanced out gravity, they were maintained in elliptical orbits at nearly constant distances from the Sun – another example of an equilibrium state. Slowly, the planets swept their orbits clean, merging the encountered rocks with each protoplanet’s surface. The subsequent history of the Earth is treated in the following chapter (What geology tells us).

Future of the solar system and the universe

According to currently prevailing cosmological theory, the ultimate state of the universe is quite literally not bright. There are minority opinions to the contrary.

As we have seen, in 5 billion or so years, the Sun will flare up into a red giant and engulf the Earth, including us, unless we can escape to another solar system, a voyage of at least several generations. But even that would be a temporary respite. Meanwhile, our solar system finally

On a broader scale, galaxies will disperse as stars explode into dust floating throughout expanding space-time. As the universe goes on expanding under the impulsion of Dark Energy, the dust eventually will be too diffuse for another star to form from it under the pull of gravity. The universe will then become a thin “soup” of particles diffused randomly throughout an enormous and ever-expanding void. The expansion of space will prevent any communication between celestial objects.6)This does not violate Special Relativity. It is not a signal  moving faster than light, it is space expanding. Besides, how does one define velocity in expanding space…? There will be only cold and darkness, with no starlight to warm and delight us and no “us” there to be warmed or delighted. It will be totally silent, not a bird will sing, nor a cat meow. “The Big Freeze” (or “The Big Sleep”.)7)This will be the fate of the universe if it is indeed flat, as is now thought. If the energy density is greater than it is thought to be, then the universe may eventually collapse back onto itself in a “Big Crunch”. Feel better?

Nevertheless, the understanding of the universe around us underscores and nourishes our astonished appreciation of its beauty. The knowledge that time for life and beauty, although vast, is limited can remind us to appreciate that we are here – now, once – so it is our only chance to make the best possible use of our limited lifetimes – individual or collective. Perhaps we even will be pushed to develop ways of living which will leave the billions of years of life still left on Earth a place pleasing to our descendants – whatever they may be.

So much for cosmology. On to read about what geology tells us.

Notes   [ + ]

1. All percentages in this section come from WMAP.
2. Bending of light waves from a distant star by another star or a planet, like by a lens.
3. Astronomers Measure the Density of Dark Matter in Galaxy Clusters. SciTechDaily.
4. Deemed the Cosmological Constant.
5. WMAP finds a value of 1.02+-0.02.
6. This does not violate Special Relativity. It is not a signal  moving faster than light, it is space expanding. Besides, how does one define velocity in expanding space…?
7. This will be the fate of the universe if it is indeed flat, as is now thought. If the energy density is greater than it is thought to be, then the universe may eventually collapse back onto itself in a “Big Crunch”. Feel better?

The life of stars

Stars did not start forming before ~200 My ABB.1)After Big Bang Before this time, only simple atoms existed, mostly single protons (hydrogen) and a small number of proton-neutron pairs (helium).They formed clouds of particles which are referred to by the Latin word, nebulae. There are nebulae out there today, but their composition has changed since the first stars started to form. We will see how.

We can consider the life cycle of stars in a series of steps.

Formation of protostars

After about 200 My, the  universe had cooled enough that the gravitational force could finally come into play, so particles of matter were attracted to each other. Dust particles stuck together and formed tiny clumps which in turn adhered to from larger ones. Thus, a larger clump of gaseous matter was built up. As its density increased, it became warmer. If the mass of such a clump exceeds the so-called Jeans mass2)The Jeans mass depends on temperature and density, but is typically many thousands of solar masses., the matter composing it becomes so dense that any generated radiation (photons) cannot escape, so the clump becomes opaque. The trapped photons cause the clump to heat even faster. The clump has now become a protostar.

So first, the strong force first formed nuclei, then the electromagnetic force formed atoms. And now the gravitational force forms clusters of matter which will become stars, galaxies, planets and black holes.

The opposing forces of pressure and gravity have opposite thermodynamic effects. Gravity pulls matter into a more organized state, so one of lower entropy. However, it can be shown that the radiation from the outward pressure ejects more than enough energy into space to maintain a global increase in entropy.

A protostar may or may not go on to become a star. If its mass is too low (<8% of the mass of our Sun), its temperature never becomes high enough for fusion to begin and the protostar becomes what astronomers call a brown dwarf.

Main-sequence stars

In larger stars, when the temperature rises to 107 Kelvin, a series of nuclear fusion reactions referred to as the proton-proton chain uses protons to produce helium. First, the protons fuse together to form deuterium (2H, with one proton and one neutron in its nucleus), then 3He (2 protons and one neutron) and finally 4He (2 protons and two neutrons).3)A word on notation. A superscript before an element’s symbol denotes its atomic mass, essentially the total number of neutrons plus protons in the nucleus. When fusion starts to take place, the protostar has becomes a star. This first occurred only when the universe was some 200 million years or more old.

The fusion process is often referred to as “burning” of hydrogen, but it has nothing to do with ordinary terrestrial burning, or oxidation. It is really 4 protons forming 2 protons and 2 neutrons held together to form a helium nucleus. Since the nucleus weighs somewhat less than the initial 4 protons, the extra mass is converted into a great deal of energy4)If only we could harness this fusion energy here on Earth! due to the large size of the speed of light, c, in the famous equation from GR,

E = mc2.

Some of this energy is emitted in the form of light emitted by the the stars, so from the moment fusion began, the universe was not only transparent, there was light (i.e., photons) in it.

Astronomers visualize the formation and subsequent life of stars by a two-dimensional plot called the Hertzsprung-Russell  diagram (hereafter abbreviated HR). The HR is somewhat analogous to the Periodic Table of the elements in that it arranges stars by their properties in such a way as to show similarities between them. It is a scatter plot (points in a 2-dimensional space) of luminosity (or brightness, on the Y-axis) versus temperature (or spectral class, on the X-axis). Stars close to one another on the diagram are found to share many properties. As stars evolve, they move from one point to another on the diagram, unlike elements, which maintain their place in the periodic table. About 90% of stars fall on the diagonal – called the main sequence – between hot, luminous stars in the upper-left-hand corner and cool, dim ones in the lower-right-hand corner. Our Sun lies near the center of the main sequence – for the moment (where the “moment” will last about another 5 Gy). A star’s position in the main sequence is mainly a function of its mass. The current state of the Sun is taken as the standard luminosity.

Hertzsprung-Russell Diagram, Chandra X-ray Observatory via NASA

Hertzsprung-Russell Diagram, Chandra X-ray Observatory via NASA

Red giant phase

A star is maintained by the so-called hydrostatic equilibrium between the outwardly-directed pressure due to the fusion in the core and the inwardly-directed force of gravity due to the star’s mass. Larger stars burn faster and so have briefer lives than smaller stars. But in all stars, the hydrogen fuel in the core is eventually depleted. The pressure then is reduced to where it can no longer counteract the force of gravity and the core contracts. As it does so, it heats up and generates increasing outward pressure. If the star is very small, it just radiates the energy away. But for most stars, there is still some hydrogen in a thin layer on the outside of the core and the core-generated heat causes it to begin to fuse into helium in hydrogen shell fusion, as opposed to the hydrogen core fusion which has taken place until now. Observation of the envelope, which is all we can see, will first show a slight cooling with only a small increase in brightness, but then the increased pressure from the core will bring about an increase in size of a hundredfold or more at approximately constant temperature. This is  shown on the HR diagram below. The star’s color will change to orange and it will become a red giant. When the Sun reaches this stage, it will probably engulf the inner four planets – including ours.

Evolutionary track for a Sun-like star to the red giant phase, Creative Commons image from Pennsylvania State University

Evolutionary track for a Sun-like star to the red giant phase, Creative Commons image from Pennsylvania State University

From the red-giant phase on, what happens depends on the mass of the star. There are many variants, of which we will mention only a few. For the moment, let us go on considering a star the size of the Sun.

Triple-alpha or red-supergiant phase

The star’s core still is not in equilibrium and continues to contract until it hits the limit imposed by the QM exclusion principle, which keeps its electrons from occupying the same QM state. Within the core, the helium resulting from the preceding hydrogen core fusion begins to fuse into heavier elements,8Be, 12C (and 16O, in still heavier stars). Since 12C formation requires three alpha particles (4He nuclei), the process is called the triple-alpha process and the star is in the core helium fusion phase. The star now has two layers, the inner one fusing to heavier elements than the outer one.

For the Sun, this stage will last less than about 1 billion years, during which it will “burn” 4He into 12C and 16O. A star of about this size will live for a total of around 10 Gy5)Most authors would say 10 billion years, but in order to be consistent with later chapters, we will say 10 Gy. Since the Sun has already lived for almost 5 Gy (as we shall see in the geology chapter), it has (and we have) about 5 Gy left.

The core helium is used up faster than its hydrogen was and when this occurs, the core contracts again. The increased temperature causes a thin shell of remaining helium just inside the hyrdogen-fusion shell to go on fusing. The core now has three layers: an inner layer of carbon and oxygen, a middle layer of helium fusion and an outer shell of hydrogen fusion. The star expands again and becomes a red supergiant.

Nebula phase

After this, the loss of mass to solar wind and the reduced supply of nuclear fuel will lower the star’s pressure to where gravity shrinks it to approximately the size of the Earth. Its outer layers will be cast outwards into an expanding shell of gas and for a while it will be a hot core of carbon surrounded by a planetary nebula (cloud) of ionized gas.6)Note that a planetary nebula has nothing to do with planets. UV radiation from the cooling core lights up the gaseous nebula.

The Ring nebula, M57, NGC 6720, from NASA

The Ring nebula, M57, NGC 6720, from NASA

After the dust drifts away, all that will be left of the Sun’s core will be a relatively cool and invisible white dwarf. The star is supported against collapse only by the QM exclusion principle. Eventually, the white dwarf will cool into a hunk of cold iron – an unglamorous end to quite a stellar career.  This will be the fate of any star less than about 1.4 times the mass of the Sun, a figure known as the Chandrasekhar limit.

One more time…

Within the matter ejected into space as nebula, the whole process can start over again with agglomeration of dust to form protostars, but with two important differences.

  1. The nebula dust now contains heavier elements like carbon and oxygen.
  2. Exploding stars, in particular, the supernovae to be discussed shortly, may send out shock waves and they may provoke the formation of new protostars.

Variable stars – cepheids

The main-sequence expansion to a red giant does not necessarily go smoothly. Some stars will be in unstable equilibrium, like a wobbly tight-wire walker. Pressure may cause them to overshoot and become too big, then gravity will take over and they may become too small. And so on, back and forth. Such stars are called either Cephiad variables or RR Lyrai variables.

Massive stars

For stars distinctly more massive than the Sun, the sequence may go further. In a later stage, 12C and 16O may fuse to 20Ne, 23Na, 24Mg and 28Si. In turn, 28Si may fuse to 56Fe, but this can only take place in the most massive stars. In any case, that is where the process stops; Fe does not fuse to anything else. This is the second step of nucleosynthesis.

The fate of a star depends on its mass, from Chandra X-ray Observatory, via NASA/CXC/SAO.

The fate of a star depends on its mass, from Chandra X-ray Observatory, via NASA/CXC/SAO.

As massive stars pass from step to step of the fusion process, burning successively carbon and oxygen, then other elements, each step takes place in the core’s center, pushing the previous step’s reactants outwards. So the star comes to have a number of layers, like an onion, with different fusion reactions taking place in different layers. Each successive step takes less time, the fusion of Si into Fe taking only on the order of four days. Once the Si is converted to Fe, though, the pressure drops and nothing can keep the star from collapsing. The outer layers “drop” towards the center and bounce off the core, causing the whole structure to explode and become a Type II supernova (SNII). Such an enormous amount of energy is released that the light is as bright as that as of several billion Suns. For some days or weeks, it may be the brightest object in the sky, depending on how far away it is. Light from the supernova gets brighter over a period of about 20 days and then decreases slowly over a year or so.

Supernova 1994D, NGC 4526 from Hubble Gallery

Supernova 1994D, NGC 4526 from Hubble Gallery

Many neutrons have been created by this time and some of them react with the Fe to create heavier, radioactive elements which subsequently are expelled into the supernova. These radioactive elements then decay with emission of gamma rays. (It has been proposed that such gamma rays from a huge supernova may have been the cause of the Ordovician-Silurian extinction around 447-443 Mya; see the geology chapter.) Be that as it may, many of these atoms combine by simple chemical reactions and form a new, but richer, dust cloud in space, richer because it contains all those heavier elements produced by the first-generation stars. These elements include C and O which are necessary for life as we know it. This cloud may be the source for new stars. This is the third and final step in nucleosynthesis.

To resume, nucleosynthesis takes place in three steps:

  1. In Big-Bang nucleosynthesis, the lightest atoms, H, He and Li are formed.
  2. In stellar fusion, heavier elements are forged, through C, O and up to Fe.
  3. In nebulae, still heavier elements are formed, including radioactive ones.

The remaining core of the exploded star is reduced to a small, extraordinarily dense clump of neutrons. Such neutron stars rotate rapidly and have strong magnetic fields. If they are oriented in such a way that we can detect their rapidly changing magnetic fields here on Earth, they are called pulsars.

However, if the collapsed core outweighs three solar masses, gravity is so strong that nothing can escape it, not even light. The core becomes a black hole.

Back to the Sun – Type I supernovae

Many – maybe most – stars in the universe are binary, meaning that two relatively nearby stars revolve around the center of gravity of the pair. If one of them is below the Chandrasekhar limit and so becomes a white dwarf, it may pick up matter radiated by its twin. If enough such matter is accreted to its surface, it may become massive enough, compared to the Chandrasekhar limit, that the electron degeneracy7)Another name for the Exclusion Principal can no longer support it and it explodes into a Type Ia supernova (SNIa). Such objects are extraordinarily bright.8)The most recently viewed one in our galaxy was observed in 1572 by the young Danish astronomer Tycho Brahe. They are important to cosmologists as they all burn with the same luminosity, making them standard “candles” which can be used to estimate their distance from the Earth (by their apparent brightness) and their speed (from the Doppler shift of their spectra, known as the red shift). In this way, astronomers have measured the speed of galaxies and discovered that the expansion of space has accelerated for the last 7 billion years.

Stars are important, but there are larger and smaller structures in the universe. We live on one of them. So on to galaxies, clusters and large-scale structures.

Notes   [ + ]

1. After Big Bang
2. The Jeans mass depends on temperature and density, but is typically many thousands of solar masses.
3. A word on notation. A superscript before an element’s symbol denotes its atomic mass, essentially the total number of neutrons plus protons in the nucleus.
4. If only we could harness this fusion energy here on Earth!
5. Most authors would say 10 billion years, but in order to be consistent with later chapters, we will say 10 Gy
6. Note that a planetary nebula has nothing to do with planets.
7. Another name for the Exclusion Principal
8. The most recently viewed one in our galaxy was observed in 1572 by the young Danish astronomer Tycho Brahe.

The Big Bang

Cosmology is the study of the origin and evolution of the cosmos, where by cosmos we mean everything – no less. This is a subject which has always fascinated mankind and has given rise to many fanciful and often farcical stories. It is one of the principle reasons for the invention of religion. We looked at one such story in the introduction, the cyclic creation, maintenance and destruction of the cosmos by Brahma, Vishnu and Shiva. Now let’s get serious.

What we know about the Big Bang

We know from astronomical observation that the universe – space – is expanding. The stars, galaxies and other distant celestial bodies are moving away from us (and we, from them). Observation of type 1a supernovae (explained later) has shown that for the last 7 Gy the expansion has been accelerating slowly.

If we do a backwards extrapolation of the measured expansion, we find that about 14 Gya all spatial objects were in the same place. This figure has recently been refined to 13.8 Gya.1)According to the WMAP project, the age of the universe is 13.77 ± 0.059 billion years. The ESA Planck probe finds 13.799 ± 0,021 billion years. We will settle for 13.8 Gya. The entire visible universe once occupied a very small region, smaller than the size of an atom.

It is important to understand that we can see only part of the cosmos, the visible universe, or what we call “our Universe”. It is limited in size because of the time light takes to get from a distant point to us and because of the age of the universe. Light travels with a finite speed, so the more distant an object or event, the more time light takes to reach us and the farther back in the past the event we are seeing. The most distant objects we can see are therefore the oldest, those which emitted their light at the time of the Big Bang, about 14 Gya. Because space has been expanding in the meantime, such objects are now about 41 Giga light-years away, which is thus the current radius of the visible universe. It is, of course, getting bigger all the time.

Current best hypothesis – inflation

Astronomers, cosmologists and physicists – the folks who worry about this sort of thing — generally accept the Inflationary Big Bang as the explanation of the origin of our universe, simply because it is capable of accounting for many facts left unexplained by the original, non-inflationary Big Bang hypothesis.

The inflationary Big Bang framework says that the universe has evolved in a two-step process:

  • a brief and extremely rapid expansion, called inflation, followed by
  • subsequent, slower expansion powered by the negative pressure of gravity (the cosmological constant).

In a nutshell, our universe started out smaller than the current size of an atom. It was roughly uniform but with tiny quantum-mechanical fluctuations (often referred to as “jitters”). Almost instantly, on a human time scale, it inflated (expanded) enormously to about the size of a grapefruit.2)Tegmark (2014), 117. The rapid expansion isolated the quantum jitters before they could resolve. When inflation stopped, a mainly uniform space had been filled with matter formed from the spread-out quantum fluctuations. Since that time, it has expanded well beyond what we can see of it.

It must be stressed that when we speak of the expansion of space, we mean the “fabric” of space itself. The expansion is a property of space only: As space expands, the push to expand remains the same at each (new) point and does not dilute away. Only space expands and only on very large scales. On smaller scales, the other physical forces, such as gravity, keep particles and planets together. Although the space around our galaxy, the Milky Way, is expanding, the galaxy is not. Nor are the stars, nor the planets, nor you nor I.3)If you are expanding, you cannot blame it on universal inflation.

One of the strong points of the inflationary Big Bang is that it explains how the expansion started. It was due to the properties of the inflaton field. Read on.

The origin of inflation is thought to be due to a field called the inflaton field (or false vacuum). This field was originally in a metastable state of high energy in which it exerted a huge outwardly-directed force which caused space to expand enormously, in fact exponentially, doubling its size every 10-37 seconds.

Such an increase in size rapidly becomes huge. At that scale, doubling its size every 10-37 seconds, one hundred doublings would have taken about 10‑35 seconds, but would have resulted in a total increase by a factor of 2100 = 1.3×1030.4)Greene (2004), 308, Guth (1997), 173.  Max Tegmark says the doubling may have occurred every 10‑38  seconds.

Here’s more detail. The inflaton field was in a state of unstable equilibrium. Anyone who has ever been delicately balanced, say on a taut rope, so that if she leans one way or the other she starts to fall, understands what it means to be in unstable equilibrium. At about 10-35 seconds of age5)Greene (2004), 285., the field “lost its balance” and started to “fall” (metaphorically) off its high value. After about 10-32 seconds, space had “fallen” as far as it could in the inflaton field, effectively hitting “bottom”. More precisely, it reached its point of minimum energy in the inflaton field. The inflation stopped but the universe went on expanding, but more slowly, under the outwardly-directed pressure of gravity (the cosmological constant). The energy released by the “falling” inflaton field provided the energy and matter that constitute the universe today. As each bit of matter exerted a gravitational attraction on every other bit, the rate of expansion slowed down. When the universe attained the age of about 7 billion years, the expansion rate began accelerating. More on that later.

At its pre-inflation size, the universe was so small that it must be explained using quantum mechanics, according to which states of energy normally forbidden by the Law of Conservation of Energy can exist for very short periods of time, as explained in the theory chapter. So there were a great number of relatively very tiny places where the energy was higher or lower than the average. The rapid inflationary expansion caught these energy fluctuations unaware, so to speak, and scattered them out over macroscopic scales before they could merge and even out the energy. They gave rise to particles and antiparticles. This tiny non-uniformity has been observed in the cosmic microwave background radiation (explained shortly).

The inflationary Big Bang solves the so-called horizon problem: How the universe managed to reach large-scale uniformity without there having been enough time for light to traverse it. Before the inflaton field started “falling”, the universe was so tiny that light could travel from any part to any other, so it managed to reach a state of internal equilibrium in which it was all at practically the same temperature everywhere. As the universe inflated, this region of almost constant temperature did too, so the universe we see today is approximately homogeneous, at least on a large scale, a fact which is confirmed by satellite observation.6)NASA WMAP project, These results have been refined by the more recent Planck probe.

Inflation also resolves the so-called flatness problem, since it was so fast that the resulting universe looks flat. (Think of the surface of a balloon which is rapidly inflated to a very large size.7)Better yet, glue some 2-centime pieces to the balloon, before you inflate it more. You will see how the coins move apart without getting any larger.) In cosmology, flatness is very important, as it means the expansion will continue; otherwise, the universe would begin to collapse back together, terminating in a “Big Crunch”.8)The flatness problem is more complex than this says, but it gives the idea. (More on flatness later.)

There are several reasons to think inflationary exponential expansion took place. It explains why expansion took place (the inflaton field), as well as why space has a uniform temperature everywhere on the average. And it shows how perfectly “normal” quantum-mechanical phenomena (briefly-existing spots of non-conserved energy) gave rise to packets of energy and eventually particles.

This much is agreed on by almost all versions of the inflationary Big Bang theory, but there are several versions of it.

Time line of the inflationary Big Bang, from WMAP via NASA

Time line of the inflationary Big Bang, from WMAP via NASA

Infinite expansion and the inflationary multiverse

No one knows if the universe infinite, but it seems hard to imagine that it is not and most cosmologists today accept that it is.

Indeed, in most versions of inflation, expansion goes on forever in time, too. If the inflaton field decays with a well-defined lifetime, like radioactive elements, then regions of the universe regularly fall off their metastable energy peak to give up energy to form … universes! Each one looks like ours to the extent that it has been born out of an inflationary burst. This process has been compared to cell division9)Guth (1997), 251., but does this suggest that the inflationary multiverse should be regarded as an evolutionarily revolving population of universes? Perhaps not.

Such universes within the inflationary multiverse have been called “pocket” or “bubbleuniverses. Each one is infinite. Although the pocket may seem finite from the outside, time is transformed into space inside due to the effects of relativity, making the interior infinite!

Since inflation continues forever in this scenario, the Big Bang at the origin of our universe was really the moment when inflation in our part of the multiverse stopped.

This multiverse hypothesis is a suggestion on the extreme edge of cosmology, but it is taken seriously by many cosmologists. In fact, there are other multiverses proposed. For more info, see the multiverse page.

L’après Big Bang — nucleosynthesis and background radiation

Once the inflaton field’s energy was spent, inflationary expansion ended. The universe went on growing under the pressure of the cosmological constant, but the gravitational attraction of all that mass limited the expansion. As the universe expanded, it cooled.10)Think of how a bicycle tire gets hot as you pump air into it; this is just the opposite phenomenon. All temperatures from Guth (1997), 86-94.

Time after Big Bang Temperature of universe
 10‑39 secs  1020K
 1 sec  1010K
 7 days  17×106K
 1 year  2×106K
 100 Kyrs  5.8×103K


Big-Bang nucleosynthesis, the creation of the light chemical elements, hydrogen, helium and lithium, took place in the period from about 1 sec to 3-4 minutes after the Big Bang.

At .1 sec ABB, it was too hot for atomic nuclei to form,. The universe was filled with rapidly moving protons and neutrons, electrons and positrons, neutrinos and antineutrinos and lots of radiation (photons). Protons and neutrons underwent reactions in which one changed to the other.11)The ratio of protons to neutrons was 1.61, due to the equilibrium rate of the reactions.

Remember the four basic forces of physics? At such high temperatures, their impact was negligible, even zero. But as temperatures fell and particles moved more slowly, the strong force was able to exert its influence – by forming nuclei (but not yet atoms).

At around one second of age12)Guth (1997), 94., the temperature was “cool” enough that the frenetic activity of all those particles calmed down. Protons stopped changing into neutrons. Neutrons still decayed into protons and electrons13)Since neutrons are slightly heavier than protons., but those which did not decay combined to make an isotope of hydrogen called deuterium (2H, one proton + one neutron). Electrons and positrons annihilated to form huge amounts of photons, visible today as the Cosmic Microwave Background radiation. By 30 seconds of age, about half the electrons and positrons had annihilated.

At about 3 minutes of age, the universe was a nuclear furnace. Deuterium nuclei combined to form helium. In this way, the lighter elements, including small amounts of lithium, were formed. BB nucleosynthesis was the first of three steps in the formation of the elements of the universe.

The Cosmic Microwave Background Radiation (CMB)

As the universe continued expanding and cooling, more heat was given off in the form of radiation, i.e., photons. But it was still too hot for electrons to be bound to nuclei as atoms, so the matter remaining was in the form of a plasma, mostly rapidly-moving free protons, neutrons and electrons. The electrons scattered the photons, which were thus like light in a cloud or fog. In a word, the universe was opaque.

After about 380,000 years, the temperature descended to around 3000 Kelvin, and the electromagnetic force became effective, binding electrons with nuclei to form atoms, a process called recombination by cosmologists. Photons now were no longer deflected by electrons and the universe became transparent, although the number of photons was still far too low for space to be called luminous. That radiation, which was emitted in all directions, is still “visible” today.14)Since it is in the microwave frequency, we cannot really see it. This Cosmic Microwave Background Radiation (CMB) is light from 13.3 Gya.

When we look up into the sky, we are looking into the past, because of the time light from distant celestial objects takes to reach us. But because the universe was opaque until 380 Ky ABB, we can see no farther back than that. The most distant light we can detect is composed of the remnants of radiation emitted then, now reduced now reduced by the expansion of space to microwave frequencies.

Just as we cannot see past the surface of a cloud, we cannot see past the time when the universe became translucid. From NASA

Just as we cannot see past the surface of a cloud, we cannot see past the time when the universe became translucid. From NASA

The great uniformity in the distribution of this radiation supports well the cosmological principle, the idea that the universe is uniform or homogeneous on very large scales, on the order of 106 light-years.

After about 200My, something happened to dispel the darkness. To find out what, read about the life of stars.

Notes   [ + ]

1. According to the WMAP project, the age of the universe is 13.77 ± 0.059 billion years. The ESA Planck probe finds 13.799 ± 0,021 billion years. We will settle for 13.8 Gya.
2. Tegmark (2014), 117.
3. If you are expanding, you cannot blame it on universal inflation.
4. Greene (2004), 308, Guth (1997), 173.  Max Tegmark says the doubling may have occurred every 10‑38  seconds.
5. Greene (2004), 285.
6. NASA WMAP project, These results have been refined by the more recent Planck probe.
7. Better yet, glue some 2-centime pieces to the balloon, before you inflate it more. You will see how the coins move apart without getting any larger.
8. The flatness problem is more complex than this says, but it gives the idea.
9. Guth (1997), 251.
10. Think of how a bicycle tire gets hot as you pump air into it; this is just the opposite phenomenon. All temperatures from Guth (1997), 86-94.
11. The ratio of protons to neutrons was 1.61, due to the equilibrium rate of the reactions.
12. Guth (1997), 94.
13. Since neutrons are slightly heavier than protons.
14. Since it is in the microwave frequency, we cannot really see it.

Simpler overviews

This thread gives presentations that are simpler and necessarily less complete overviews of the subjects of main articles. As such, they may not contain enough information to be quite convincing. Sorry about that. These are sometimes counter-intuitive subjects.

So we suggest that you start with the full version and only come here in case of need or, maybe, to review. Appropriate links exist in the main articles.

We will start with the Big Three theories of physics.


Thermodynamics is the branch of physics which started out talking about heat energy, in the early days of steam engines. Then other forms of energy, like electromagnetic or nuclear energy, came up and were included, so today it is the general theory of energy. We will be talking a lot about energy, so thermodynamics is important to us.

Thermodynamics posits three laws:

  1. The energy of the universe does not change; it is always conserved.
  2. In a physical process, the entropy of the universe always increases.
  3. The entropy of a system at a temperature of absolute zero is zero.

Law number one says that energy is always conserved. In fact, like speed limits, this one can be violated provided the perp does not get caught. This happens in Quantum Mechanics and we will look at that shortly. For anything bigger than an atom, though, energy is always conserved.

Law number two says in any process involving transformation of energy, you are probably going to lose some. This is equivalent to saying that the universe goes from a relatively ordered state (such as large plates stacked together, small plates stacked together, no large ones mixed with small ones) to a relatively unordered state (all the plates stacked together in any old order). Take a look at the Entropy entry on the main menu bar for more (easy) details. Most simply put, entropy is a measure of disorder. Nature seeks disorder, and therefore increased entropy. Entropy thus serves to predict which direction a process will take, forwards or backwards in time.

A standard example is the breaking of an egg, in which the egg becomes more disordered and therefore in a state of higher entropy. To do the opposite, for the broken egg to come back together, would require a decrease in entropy. Have you ever seen a broken egg re-form itself?

We are victims of entropy too. We eat food to provide energy to maintain our bodies in their highly ordered state. Whenever metabolism stops, rot sets in and .. we return to unordered dust.

We will use entropy in the following pages to explain why lots of things happen the way they do, rather than the opposite way.

Law number three just gives a base value for entropy measurement. Zero Kelvin is really cold, so cold we have never been able to get there in a laboratory. Precisely, it is -273.15° Celsius or zero Kelvin. We don’t do Fahrenheit.

Maybe you would like to check the full version now. If not, just go on.

Quantum Mechanics

Quantum mechanics is the theory we use to talk about atoms and elementary particles. i.e,  of what happens at very small dimensions, on the order of 10-30 meters or less! On that scale, things are very unusual.

There is no way to make QM intuitive. So here goes…

According to quantum mechanics, what is “out there” is a vast amount of space – not an empty backdrop, but actually something. This space is filled with particles so small that the distance between them is huge compared to their own sizes. Not only that, but these particles are actually waves, or something else which acts sometimes like waves and sometimes like particles. Light sometimes diffracts like waves (think prism) and sometimes leaves traces like particles.

As if that were not bad enough, it is impossible to measure simultaneously where they are and how fast they are moving (or how much energy they possess and when). This last effect is referred to as indeterminacy, or the Uncertainty Principle, one of the more uncomfortable and, simultaneously, fruitful results of the theory. We already have mentioned this exception to the First Law of Thermodynamics.

The three main difficulties most people have with QM are the following:

  1. the so-called wave-particle duality;
  2. the existence of discrete quanta for values of physical parameters;
  3. the Uncertainty Principle;
  4. the Exclusion Principle.

We have already mentioned numbers 1 and 3. Number 2 comes from the math. In fact, QM is a mathematical formalism with an equation one can (try to) solve for any given object of study. In general, the equation only has solutions for certain values of the parameters of the system. These might be the energy or the angular momentum or other things. For an atom, only certain energies are possible. Such allowed values are called quanta.

The Exclusion Principle says that certain particles known as fermions are constrained in such a way that no two of them can occupy the same QM state. Electrons are fermions. So this phenomenon, called the Exclusion Principle, is at the root of solid-state physics and therefore of the existence of transistors and all the technologies dependent thereupon – portable computers, mobile telephones, space exploration and the Internet, just as to mention a few examples. So QM has indeed revolutionized modern life, for the better and for the worse (think of wasteful and dangerous nuclear bomb proliferation).

The exclusion principle is also responsible for the fact that electrons in a collapsing super-dense star cannot all be in the same state, so there is a pressure effectively keeping them from being compressed any further. We will read more about that in the cosmology chapter.

Maybe you would like to check the full version now. If not, just go on.


Relativity is not the idea that ‘everything is relative.” It is about relative motion.  The first, or Special Theory of Relativity says that if you are in a moving train and I am standing still outside, we will both use the same equations to describe what is going on, although the values we get may be different. I think you are moving, but if you have just waked up from a deep nap, you may think you are standing still and I am moving. But the oddball thing is that if you and I both measure the speed of light, we will come up with the same value, independently of how fast either of us is moving with respect to the other.

Normally, if you walk along your train car from the back to the front of the train, you will figure you are walking at something like 4 km per hour. But if the train is moving at 100 km per hour, I will see you moving at 104 km per hour. But if it is a beam of light from a laser which you shine down the length of the train, we will both measure the same value, about 300,000 km per hour. Bizarre, non?

The other odd thing of Special Relativity is that time and space are not independent, but go together in what is called space-time. And only massless, objects can travel at the speed of light, like photons, the particles of light.

If a massive object like my sister traveled at high speeds, near the speed of light, it would seem to me that she was getting heavier but thinner (perpendicular to the direction of her motion) and her clock would run more slowly. This includes her bodily clock — her heart. This is the heart of the Twin Paradox. If she takes a very fast spaceship on a joy ride, when she gets back, she will be younger than any hypothetical twin she may have left behind on earth. This has been tested (with high-speed particles) and is definitely true.

The other Relativity theory is called General Relativity. Whereas Special Relativity is the theory of space-time and light, General Relativity is the theory of gravity. GR says that space-time is curved and that it is more curved where gravitational forces are stronger. In fact, gravity is the curvature of space-time. Think of a plane surface with a depression in it. Put a ball on it and the ball will roll into the depression. Try to visualize that in four dimensions (Good luck!) and you’ve got GR.

Please take note: QM, SR and GR are all tested and confirmed theories. They are not hypotheses, they are true — at least for the moment. Any subsequent theory which amends them will have to include or explain their results.

Maybe you would like to check the full version now.

For the moment, I do not see how to simplify the next subject any more than I have already.  So go on to the Standard Model of Elementary Particles.

Cheat sheet

Some generally useful information you may want to look up occasionally.

Geological time scale, eons, eras, periods and epochs

Geological time scale and

Geological time scale — red lines are mass extinctions, past or to come…

Types of hominins

Timeline and grouping of principal fossil hominid species

Timeline and grouping of principal fossil hominid species

Biological species classification

Classification of modern humans and house cats, after Wikipedia

Classification of modern humans and house cats, after Wikipedia

The periodic table of the elements

Periodic table of the elements

Periodic table of the elements

Particles of the standard model

Standard Model particle zoo

Standard Model particle zoo

Hominoid clades

Hominoid families with dates.

Hominoid families with dates.

Phylogenetic tree

Phylogenetic tree By MPF [Public domain], via Wikimedia Commons

Phylogenetic tree By MPF [Public domain], via Wikimedia Commons



Now we are ready to understand how it is that carbon is such a versatile element. It is at the basis of all organic chemistry and, in particular, biochemistry. In living organisms, the four most abundant elements by number of atoms are hydrogen, oxygen, nitrogen and carbon. Together, they comprise over 99% of the mass of most cells.

We saw that the carbon atom’s electron-shell configuration was

12C: 1s22s22p2

so it has four electrons in its valence shell (n=2). That enables it to share four electrons with four others from other atoms. The bonds tend to be equally spaced around the carbon atom in the form of a tetrahedron, like those little creamer packets you get in cheap restaurants. A carbon atom can bond with four hydrogens, sharing each of its four valence electrons with one hydrogen, so each hydrogen has two and the carbon has a total of eight and everybody is happy. This is called methane and looks like this.

"Methane-2D-stereo" by SVG version by Patricia.fidi - Own work. Licensed under Public Domain via Wikimedia Commons.

Methane molecule, CH4 by Patricia.fidi via Wikimedia Commons.

You should see one of the lower-right-hand hydrogens as pointing up out of the page; the other, down into it. The angles between any two adjacent connecting lines (which of course are only imagined by us) are about 109.5°. Carbon’s versatility in binding is illustrated by the examples in this diagram.

Versatiliy of carbon bonding, after Lehninger.

Versatility of carbon bonding, after Lehninger.

The dots represent valence electrons and the right-hand column is another way of looking at the product in terms of bonds rather than electrons. Each line between atoms is a shared pair of electrons. Note the double and triple inter-carbon bonds in the last two examples. This large number of ways of bonding is the key to carbon’s versatility. In fact, compared to the huge number of such molecules possible, only a relatively small number of the same biomolecules occur in living organisms. This is the first example we see of nature using the same set of techniques or tools all over the biosphere.

Single bonds between carbons also exist, of course, and have the particular advantage that the carbons and whatever is bonded to them can rotate around the axis linking the two carbons. This is more important than one might think. It turns out that some proteins function differently in their left-handed and right-handed versions. Since rotation can change the shape of the molecule, this enables biomolecules with hundreds of atoms to take on specific shapes with definite mechanical or fluid properties. (We will see some of this in the biochemistry chapter.)

The importance of water is not just because we drink it. Let’s go look at that.

Atomic energy levels and chemical bonding

Atomic structure is the basis of chemistry. It is explained by Quantum Mechanics, which is part of physics. We will see that physics explains chemistry, which explains physiology, which at least start to explain neurobiology. It’s one thing that leads to another.

In QM, the properties of a system, that is, a given object or set of objects, such as an atom, are given by the solution to the Schrödinger equation for the system. For atoms, there are a set of solutions, corresponding to different energy states of the atoms. What follows may smack of numerology.

Consider the hydrogen atom, composed of one negatively-charged electron in orbit around a nucleus containing one positively-charged proton. (This is an experimental result.) Look out, the orbit is not a well-defined path around the nucleus like those animations you see in TV ads, but rather a cloud of probability indicating the likelihood that the electron will be found at any given point in the cloud. This is due to the probabilistic character of QM. The different solutions to the Schrödinger equation express among other things the possible energy values of the atom. It is specified by a set of integer numbers called quantum numbers. In the case of atoms, they are the following:

  1. The principal quantum number, designated by the symbol n, takes on integer values from 1 on up, but in practice only to 7. It indicates the shell, or level of the cloud, in which the electron is found. The values 1-7 are often indicated by the letters K, L, M…Q.

  2. The orbital quantum number, l, indicates a level within the shell which is called the subshell, It can take on values from 0 up to n-1. The values 0-3 are often referred to as s, p, d and f. (It is really of no importance, but s, p, d and f are abbreviated forms of sharp, principal, diffuse and fundamental.)

  3. The orbital magnetic quantum number, m, refers to the magnetic orientation of the electron. It can range from -l up through +l.

  4. The electron spin, ms, can take on only two values, ½ or -½.

So the only allowed values for the quantum numbers are

n = 1, 2, 3, …

l = 0…n-1 (for a given value of n)

m = -l…+l (for a given value of l)

because those are the ones for which the Schrödinger equation has solutions. It is that simple.

The QM exclusion principle forbids two electrons to occupy the same state. So each set of values (n, l, m, ms) can only correspond to one electron. The result is illustrated in the following table.

n (shell)

l (subshell)

m (orbital)

Max no. electrons

1 0 0 2
2 0
3 0
-1, 0, 1
-2. -1, 0, 1, 2
4 0
-1, 0, 1
-2, -1, 0, 1, 2
-3, -2, -1, 0, 1, 2, 3

The fact that the quantum numbers do not vary continuously from, say, 0 to 0.001 and then 0.002 and on, but but jump from one integer value to another means that the energy of the electron in the electric field of the nucleus also takes on non-continuous values. These are called quantum states and are a feature, or if you prefer, a peculiarity, of QM.

The chemical properties of an atom depend only on the number of electrons. This is equal to the number of protons and is called the atomic number. All atoms except hydrogen have nuclei which also contain neutrons. The table summarizes the allowed values of quantum numbers for the first four shells.

In specifying which subshells are occupied by the electrons in an atom, one often uses the format


where l is specified as s, p, d or f and # is the number of electrons in the subshell. In its minimum energy state, called the ground state, the carbon atom (atomic number = 12, nucleus contains 6 protons and 6 neutrons) has the following electron configuration:

12C: 1s22s22p2

which indicates the maximum number of two electrons in shell 1, again in subshell s of shell 2 and the remaining two in subshell p of shell 2. Similarly, oxygen (atomic number = 16, 8 each of protons and neutrons) is

16O: 1s22s22p4

the meaning of which should now be clear.

What is interesting is that, for energetic reasons, each atom would like to have its outside subshell filled. If a few electrons are missing, it wants more; if most are missing, it might be willing to give up the rest in order to have an empty outside shell, referred to as the valence shell. (The number of electrons in this outer shell is called the valence.1)Officially, the maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted.) For instance, hydrogen

1H: 1s1

wants two electrons or none in its 1s shell, so it could give up its electron or gain one. What happens is, two H atoms share their electrons to make a molecule of H2, so each has two electrons half the time. Better than nothing.

Since oxygen already has shell 2 half-filled, it would probably prefer to gain electrons to fill it. And carbon… but carbon is special and will be considered in a moment.

Look at sodium (Na, atomic number 11) and chlorine (Cl, atomic number 17):

Na: 1s22s22p63s1

Cl: 1s22s22p63s23p5

Sodium could happily give up that 3s electron and chlorine could use it to fill up its 3p valence shell. And this is what happens in table salt, NaCl. But if you put salt in water, it separates (for reasons which will be discussed shortly) into charged ions, Na+ and Cl, because chlorine is greedy and keeps that negative 3s electron it took away from sodium. This attraction for electrons is called electronegativity.

Chemistry is the study of chemical systems (atoms, molecules) and chemical bonding between such objects. In the case of NaCl, the sodium and chlorine have opposite electrical charge and the attractive electric force is what holds the molecule together. This is called ionic bonding. Sometimes, when atoms cannot decide which has more right to an electron, the electron is shared between them, as in H2, making both atoms relatively happy. Bonding based on shared electrons is called covalent bonding; it is a sort of consensus situation, if we go on with the anthropomorphism.

Elements with the same number of electrons in their outer shells have similar chemical properties. So they are arranged in columns in that wonderful physical/chemical tool, the periodic table of the elements.

Periodic table of the elements

Periodic table from Wikimedia Commons

It is easy to see that the elements in the first column are like hydrogen in having one electron in their valence shell.

H: 1s1

Li: 1s22s1

Na: 1s22s22p63s1

K: 1s22s22p63s23p64s1

… and so on.

The extra elements in the middle are rule-breakers. Instead of filling one subshell before moving on to the next, they start one, add a small number (often only one) of electrons to the next, then go back to finish filling the next-to-last.

Columns in the table are called groups; rows, periods.

The subshell configurations we have been giving are for the lowest energy state of the atom, called the ground state, in which subshells are filled from the “bottom” up (with some exceptions, as just mentioned). But if that hydrogen atom is struck by a photon, enough energy may be transferred from the photon to the atom to push that 1s electron into a higher-energy subshell. The atom is then said to be in an excited state. The electron may then re-descend spontaneously to the lower subshell, emitting a photon of energy equivalent to the difference in energy levels of the subshells. In QM, photons behave like waves whose energy is a function of their frequency, so the frequency – equivalently, the color – of the light emitted is characteristic of the difference in energy of the two subshells. Any atom’s subshells will therefore correspond to a given set of photon frequencies emitted and these are seen as colors, although not all these colors will be visible to a human eye. The set of frequencies constitute the spectrum of the atom and may be used to analyze the identity of a light source. In this way, we can identify the chemical components of light-emitting objects like distant stars.

There are two other types of bonding. We will consider hydrogen bonds very shortly in the discussion of water. The fourth form is due to the shifting electron density distribution around an atom. At times, this may form a temporary dipole even in a neutral atom. This may in turn induce a dipole in nearby atom in such a way that the two dipoles attract each other very weakly. This is London, or van der Waal’s, bonding.

The functioning of all living things depends on water and on the versatility of the carbon atom. So let’s start with carbon.

Notes   [ + ]

1. Officially, the maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted.