The
science of chemistry – chemistry, the whole thing –
is analysed in terms of systems thinking. First, the idea of the system
is introduced, along with the concepts of complexity, fractal, automata,
emergence & chaos. After this, some familiar chemistry topics with
simple, linear behaviours are described, before moving on to regions of
chemistry space where things get more complicated...
Note that Chemistry & Complexity extends over three web pages: 1,
2, 3.
Chemistry
& Complexity is dedicated to the memory of my old friend. Alan
was fascinated by emergent behaviour, and was read widely on the
subject. I would have greatly valued Alan's input into this page,
but unfortunately this was not to be...
Systems Thinking
During the twentieth
century it became apparent that while science is excellent at explaining
and modelling linear systems, the real world is non-linear and complicated.
We can calculate
with exquisite precision what will happen when a sample of gas is compressed,
but we cannot predict next week's weather or oil price.
The crucial word
is system, and since the 1940's
many researchers have looked at systems, types of system and patterns
of behaviour exhibited by systems.
A system
can be:
• a sample
of gas in a piston
• the gas laws
• a single cell micro-organism
• a cat
• a pride of lions
• a hive of bees
• an ecosystem
• a planet
• a solar system
• a small business
• a computer operating system
• a chemical reaction
• the set of chemical reactions
Systems are objects that exist
in system space. Systems are dynamic and they evolve with time.
The evolution
may be trivial and null: the pressure of a sample of gas in a sealed
container held at constant temperature does not change.
The evolution may be predicable and linear, like the motion of
the moon round the earth and planets orbiting the sun (Newtonian celestial
mechanics)'
Or the evolution may be very complicated, like the development
and history of a pride of lions over a 10 year period.
Systems
theory is concerned with types of system and patterns of behaviour
exhibited by different types of system.
One important aim
of systems research is to develop mathematical tools and computer software
able to model various types of system. Largely by a process Darwinian
selection in the marketplace – appropriate models and software
are used and promoted, inappropriate models and software are not – it has been found that different types/classes of software are suited
to different types of system:
Computer
file and software applications are handled by the local operating
system: Windows, Mac, LINUX, etc.
Business systems are generally modelled using relational database
(RDMS) technology.
Engineered items are designed and manufactured using CAD/CAM
software.
Systems thinkers
are particularly interested in complex systems, and an important branch
of systems theory is complexity
theory. It has been found that large and complicated systems exhibit
emergent behaviours which, while difficult to predict beforehand
may be blindingly obvious with hindsight!
A
trivial example: Bend a rod and eventually it fails (breaks)...
More
subtly: When multi-lane highways were first designed and built it was
not envisaged that at high traffic densities, vehicles would bunch together
and that waves of bunching would move independently of the traffic speed.
As Peter Cochrane writes:
"Anyone
who drives on motorways will have experienced traffic waves created
by some unseen event ahead. Probably the best place to experience
this phenomena in the UK is on the M25 when, for no apparent reason,
the traffic speed can oscillate between 10 and 70 mph for long periods.
Sometimes the traffic comes to a complete halt and then lurches forward
to 40 mph and back down to zero.
"This is
the classic behaviour of a system of independent entities in a serial
queue having a delay between observation and action. In this case
the observation might be an accident, a breakdown, or someone driving
foolishly. The delay is between our eye, brain and foot. As soon as
we see something and we reach for the break pedal then very shortly
afterwards so does everyone else, and so the wave starts. "
As
a recent Nature editorial asked rhetorically, In
Pursuit of Systems , Nature, Vol 453, 5th May 2005, pp1:
"What is
the difference between a dead cat and a live cat?
"A dead cat
is a collection of its component parts.
"A live cat
is the emergent behaviour of the system incorporating those parts."
"The interaction
of many parts, giving rise to difficulties in linear or reductionist
analysis due to the nonlinearity of circular causation and feedback
effects." (Calresco)
"The study
of complex phenomena in natural systems. Basic themes include the dynamics,
interactions, emergence, adaptation, learning, and evolution of a system" (dddmag.com)
"A dynamical
system that is extremely sensitive to its initial conditions." (Word
Net)
"A system whose
long term behaviour is unpredictable, tiny changes in the accuracy of
the starting value rapidly diverge to anywhere in its possible state
space. There can however be a finite number of available states, so
statistical prediction can still be useful. " (Calresco)
"Complicated
patterns that are not truly random. Chaos is a cryptic form of order,
what a random-number generator produces. There is, as the phrase goes,
'a sensitive dependence on initial conditions.' Because chaos was defined
in a paradoxical way ('It may look random, but it's merely chaotic'),
it is a term often misused or misunderstood." (William
Calvin)
"Apparent disorder;
more accurately, a state of unpredictability of events. The term also
applies to a specific branch of mathematics relating to non-linear systems
and other infinitely complex relationships." (Tom
Graves)
"Emergence
is the process of deriving some new and coherent structures, patterns
and properties in a complex system. Emergent phenomena occur due to
the pattern of interactions between the elements of a system over time.
Emergent phenomena are often unexpected, nontrivial results of relatively
simple interactions of relatively simple components. What distinguishes
a complex system from a merely complicated one is that in a complex
system, some behaviours and patterns emerge as a result of the patterns
of relationship between the elements." (Wikipedia)
"Emergence
is the process of complex pattern formation from simpler rules. This
can be a dynamic process (occurring over time), such as the evolution
of the human brain over thousands of successive generations; or emergence
can happen over disparate size scales, such as the interactions between
a macroscopic number of neurons producing a human brain capable of thought
(even though the constituent neurons are not themselves conscious).
For a phenomenon to be termed emergent it should generally be unpredictable
from a lower level description. Usually the phenomenon does not exist
at all or only in trace amounts at the very lowest level. Thus, a straightforward
phenomenon such as the probability of finding a raisin in a slice of
cake growing with the portion-size does not generally require a theory
of emergence to explain. It may however be profitable to consider the
emergence of the texture of the cake as a relatively complex result
of the baking process and the mixture of ingredients."
(Wikipedia)
Catastrophe,
Complexity, Emergence, Fractals, Automata, Butterflies & The Mandelbrot Set
In the 1960s Rene
Thom's catastrophe
theory described a variety of systems that had domains that were prone
to suddenly flipping between states, instead of gradually moving between
states in a linear manner.
In systems prone to catastrophe, the equilibrium
surface is said to have a cusp:
Examples of systems
prone to catastrophe include:
A rod of
brittle material snapping
A ship and its tendency to capsize
The crystallisation of a super-cooled fluid
The collapse of a bridge
However it was soon realised that
while illuminating, the catastrophe approach was not general and that
real systems, like earthquakes, show even more complex behaviours. For example, while a single earthquake can be modelled by catastrophe theory, in reality sequences of larger
earthquakes are always followed by clusters of smaller aftershocks.
Click here
for the near real-time IRIS seismic monitor, one of the most impressive
scientific sites on the web.
In the 1970s, Benoit
Mandelbrot, at the time an IBM research mathematician, explored the looping of
very simple equations. In this new mathematical space – for practical
purposes only available with the advent of programmable computers – Mandelbrot discovered the extraordinary pattern that bears his name. The Mandelbrot set has been described as the most complex object in
mathematics.
The two books
by Mandelbrot: Les objets fractals. Forme, hasard, dimension (1989
ed3) and The fractal geometry of nature
(1982)
introduced the idea of the "fractal"
and "fractal geometry" to more general audiences.
The perfect example
of fractal behaviour is the Mandelbrot set itself. It is possible
to infinitely zoom into the fractal edge of the Mandelbrot set and there find an infinite number
of mini-Mandelbrot sets hiding in the infinite detail:
A sequence of 15 zooms into the edge of the Mandelbrot set, see
here.
The excellent PC
program Fractint
allows very deep zooming into the Mandelbrot set by bypassing
a computer's 16/32/64 bit maths routines, both OS & CPU. The image
of the mini-Mandelbrot set shown below [and captured from
here], is obtained at a magnification of 4 x 10^58:
4 x 10^{58}
is a simply HUGE magnification. It is a zoom of:
x 4,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
This YouTube clip gives a deep zoom and is called: "A Mandelbrot the size of the known universe".
The very highly recommendedPowers of 10web site zooms from the edge of the known universe to deep inside of a proton, and that is only a magnification factor of 10^{39}. A Homer Simpson spoof version can be found here.
Mandelbrot showed that fractals are patterns have the property of being self-similar with respect to scale, and so when looking at a fractal pattern it is not possible to deduce the scale of the image because self-similarity can occur over many orders of magnitude. Furthermore, Mandelbrot pointed out that
many features of the natural world are fractal and self-similar. The famously
rugged coastline
of Great Britain is fractal over the scale from metres to hundreds of
kilometres. Move your mouse over the image below:
Three outline
maps show parts of the Great Britain coastline, specifically parts
of Wales. The map on the left shows about 1 mile, the middle map
about 10 miles and the right map about 100 miles.
One implication of fractal
and self-similar geometry is that is is not possible to come up with
an accurate measure of the length of the coastline of Great Britain
because the answer will be dependent upon the scale at which the measurement
is taken. Thus, fractal geometries can lead to real problems for classical
sciences which like to be able to provide precise numerical answers
to clear questions like: "How long is the coastline of Britain?".
The coastline diagrams bring
out an important feature of fractal geometry. Even though we may not know
the scale, coastlines are still clearly coastlines and we do not
confuse coastlines with other natural fractal system such as branching
trees, mountain ranges or clouds. Our brains are excellent at distinguishing between different types of natural fractal pattern, which is useful because the natural world is so intrinsically fractal:
Psychologists and neuroscientists do not yet understand how the brain is so good at distinguishing between different types of natural fractal.
Mandelbrot showed that simple, looping non-linear
equations can exhibit regular behaviour, or they can act in complex,
unpredictable, chaotic ways.
Consider the looping equation
(A * B * B) – 1 = B
Meaning that
the numerical result of (A * B * B) – 1 is used as the value of B on the next iteration of the
loop.
With certain values of A
and B (0.400 and 0.630) the series decays to zero, with some
it regularly oscillates between values, and with others it behaves chaotically
producing "random" numbers:
This spreadsheet model shows
the famous Butterfly
effect in operation: A butterfly flaps its wings in China and a hurricane forms in the Caribbean.
When the system is behaving chaotically, tiny
[butterfly wing flap] changes to the constant B, for example
1.553, 1.554 and 1.555, cause wide variation so that it is not possible
to predict the values of B after 20 or more iterations:
Download the Excel
spreadsheet and play with the sliders, here.
Cellular
automata are discrete dynamical systems whose behaviour is generative
and completely specified in terms of local rules. Complex emergent behaviours can
develop in even very simple systems. One of the best known and studied
examples is The Game of Life invented by the
British mathematician John Conway in the nineteen sixties.
The Game of Life consists
of a two dimensional array of squares on a computer screen and the "game"
is played over a series of steps [or generations] with two simple rules:
If a square is off,
it turns on if exactly three of its neighbours are on.
If a square is on, it stays on if exactly two
or three neighbours are on, otherwise it turns off.
Game of Life "runs"
develop in involved ways, but the individual objects always have
a few common types of ending:
On screen object
may grow for a few generations until it becomes static and unchanging.
An object may
flip or cycle between two or more forms and become a "blinker" [a Game of Life term].
An object may
grow for a few generations and then shrink, self-destruct and vanish.
An on screen
object may eject a "glider" [a Game of Life term], an object
which moves away from the place where it formed until it collides
with the edge of the playing area.
An interactive (java) Game
of Life website can be found here.
Click and have fun.
Go to this mathworld
link for an excellent discussion and history of Conway's work.
The science of cellular automata
has been rigorously explored by Stephen
Wolfram in his recent book, A New Kind of Science, and website,
NKS and here,
Read a fair –
but critical – review of this new analysis, here.
Lorenz strange attractors are useful when modelling the weather, here:
Fractal images
have a dimension of between 1 and 2
Fractal objects
have dimensions between 2 and 3
Mitchell
Feigenbaum, in the 1970s and like Mandelbrot also using computers
to study looping mathematical functions, found a universal constant for
functions approaching chaos via period
doubling.
It turns out that
the Feigenbaum and Mandelbrot approaches are closely related.
Systems that Crackle
Crackling noise
is associated with: earthquakes and the general fracture of disordered
materials, crumpling paper, magnetising of iron, shearing bubbles
in a foam, biological extinctions, fluids invading a porous medium,
fluctuations in stock markets, solar flares, sound emitted during
martensitic phase transitions, and even group decision making
Complex systems
are discussed in a Nature Insight collection of review articles: Nature,
410, 241-284 (2001), specially the first paper: Crackling Noise by
J.P. Sethna, K.A. Dahmen & C.R. Myers pp242-250.
Understanding
the nature of turbulence has long been recognised as a scientific
and engineering problem. It is now recognised that turbulence is fractal:
In 1988 James Gleick, a science
journalist, reviewed the new and emerging field in his well timed and
readable book Chaos: Making
a New Science. In a non-technical way, Gleick's book reviews and links
together the people who made the discoveries with the various parts of
[what is now known as] complexity theory.
Ian
Stewart, a mathematician and communicator has written extensively on the these matters with more depth
and with greater mathematical rigour.
View some extraordinary
images of computer generated mathematically complex objects on the Mathematica
gallery (below are
some captured thumbnails):
And, there is a heap
of good stuff to be found on the web. Explore some of the links from this
page, or explore this excellent page of chaos & fractal links, here,
or have a look at images generated by the commercial package, Untrafractal.
"Systems chemistry deals with the emergent properties of interacting chemical systems or networks. In other words, properties that result from the interaction between the components in a network, rather than any one species acting individually." RSC
One powerful systems way to
look at chemistry – as in "chemistry, the whole thing"
– is to consider chemistry to be a generational science
because all entities of chemical interest, with the notable exceptions
of the electron and the photon, are compound objects constructed from
simpler unit entities:
Up & down quarks
combine to give protons & neutrons
Protons and neutrons
combine to give atomic nuclei
Atomic nuclei and
electrons combine to give atoms and ions
Atoms with the
same proton number can be collected as bulk elements
Elements combine
to give diatomic molecules and binary compounds
Binary compounds
may be gases, liquids or solids; metallic, ionic, molecular or network
materials
Simple chemical
species react with each other to give larger and ever more complex molecules
C, H, N, O, P & S combine to give, amongst other entities, the nucleic and amino acids
The nucleic and
amino acids combine to give DNA, RNA, thousands of peptides, proteins...
and life
As the entities
become more complicated, emergent properties appear, as will be discussed
below.
The "chemistry
is a generational science" position is formally a digital
mechanics (Fredkin) view of the universe.
Chemistry is not unique
in being generational, consider language & literature:
Letters arrange
into words
Words arrange into
phrases
Phrases arrange
into sentences
Sentences arrange
into paragraphs, sections, chapters & books
Books are arranged
in a library
Literature mirrors
and influences society
Emergent properties
appear: We would not predict from the arrangement of letters in words
(spelling), the rules for constructing sentences or for writing successful
novels.
And motorcars:
Individual components
are arranged into the engine, chassis, bodywork subassemblies
Subassemblies are
combined into the working car
Cars are driven
on roads with other cars obeying the rules of the road
Emergent properties
appear: We would not predict from the arrangement of the nuts and bolts
that, that at high traffic densities on fast moving freeways traffic
bunching would occur.
As a generational science,
chemistry can – in large part – be regarded as a cone of increasing
complexity emanating from the periodic table of the elements:
The periodic table is a schema
that rationally displays the building blocks from which our physical world
is constructed, the chemical elements. The cone of increasing complexity
gives rise to:
Molecules & materials
Gases, liquids, solids
Pure substances & mixtures
of substances
Chemical interactions &
reactions between substances
Inorganic and organic materials
Rock, geology, biology
As a generational science consisting
of many interacting particles, chemistry behaves like a cellular automata.
As Stephen Wolfram and John
Conway show with with their very simple math cellular automata (discussed
above), cellular automata systems can develop in simple ways or they
can exhibit extremely complex behaviours. And so it is with "chemical
automata"that involve real interacting chemical species: atoms,
ions, molecules...
There are regions
in chemistry space where behaviour is linear and simple, including the
gas laws, ideal dilute solutions, periodicity, etc., as will be discussed
below.
And there are regions
where the behaviour is formally complex, like the Belousov-Zhabotinsky
(BZ) reaction, turbulence, mechanistic pathways, synthesis, etc.
Theory and
experience tell us that as systems "enlarge", behaviour
becomes more complex and new, emergent, complex properties appear, and
chemistry is no different.
Note
that here the term "enlarge" refers to variety of amongst
the types of chemical species and variety in the rules of chemical interaction,
rather than the total number of species.
The
behaviours of chemical species interactions are more involved that those
of the cells in Conway-Wolfram cellular automata.
Systems exhibiting
linear behaviour can be found in chemistry space... amongst the complexity!
Chemical
Systems
Chemistry
consists of multiple systems which together explore and define chemistry
space.
Consider the term "explore":
We learn about our house,
garden and the roads around it.
Millions of ants explore
every nook and cranny of their local environments.
Googlebots
and other crawlers and spiders explore web-space.
Chemical species explore
their local spaces by occupying the available quantum states:
Chemical species
are quantum objects that vibrate, rotate, translate (move), and at higher
energies undergo electronic transitions. The science is well understood
and an array of mathematical tools – based on quantum mechanics
and kinetic theory (statistical thermodynamics) – is available.
The techniques involved are covered by university level physical chemistry
courses and by textbooks such as Atkins'
Physical Chemistry.
Chemical species
explore their local chemical interaction & reaction environments.
This science is the primary concern of the
chemogenesis web book which takes the simplest chemical
species, the main group elemental hydrides, and employs a combinatorial
approach to explore the emerging reaction chemistry.
Chemical
species distribute themselves between – and thereby completely explore
– the local states according to temperature. This
argument is sometimes expressed as the totalitarian rule (after TH White,
Once and Future King):
"Everything
not forbidden is compulsory."
Or in a chemical
context: "Any reaction that can occur, will occur."
The equilibrium distribution
of species in a system with two states, i & j,and where i is higher in energy than j, is modelled
by the Boltzmann
relationship:
where:
n is the number
of particles (mole fraction) in the states ni
and nj E is the energy of the states Ei
and Ej k is Boltzmann's constant, 1.38066 x 10^{23} T the thermodynamic (Kelvin) temperature
The Boltzmann relationship
tells us that at equilibrium and at absolute zero (0.00 Kelvin or –273.15°C),
all species will be in the lowest energy state, Ej,
and that at higher temperatures [and at thermal equilibrium,
see below] there will always be more species in the lower energy
state than the higher energy state.
Consider the statement: "chemistry
consists of multiple systems":
Each of the following is
"a system":
An atom
The periodic table
A closed thermodynamic
system
Kinetic theory
The gas laws
A closed, half-filled
jar of water
Butane and its
conformers
A Bunsen burner
flame
A gas chromatograph
The manufacture
of methyl tertiary butyl ether, MTBE
Inorganic chemistry
Mechanistic theory
The Chemical Thesaurus
reaction chemistry database, here
The chemical literature
Etc., etc., etc...
Chemical systems explore their
local environments and in total, define the expanding boundary
of chemistry space.
At school and university,
chemistry is always taught on a topic-by-topic (system-by-system) basis.
Matter is the stuff
from which our world is constructed, however, chemical scientists make
the great simplification of – wherever possible – working
with simple matter: pure substances, homogeneous solutions and
reagent grade chemicals. Matter and its classification are discussed
elsewhere in this web book, here.
The change that occurs when
a sample of matter is processed: heated, compressed or mixed with another
type of matter can be be described in terms of a chemical equation, a
powerful metaphor because the entities may be of many types.
For example, a reaction equation
employ imaginary, generic species that have arbitrary symbols like X,
Y & Z.
The generic reaction equation:
X + Y
Z
can be used to model many
reaction processes, see here.
Or, the entities that make
up the equation may represent real chemical substances like hydrogen ,
oxygen and water:
2H2
+ O2
2H2O
When all the reacting entities
in a chemical equation are real reagent grade chemicals of known purity,
the reaction equation can be balanced in terms of mass, charge, free energy,
enthalpy and energy, below.
The reaction equation arrow:
Is the central & pivotal part of a chemical reaction equation.
Is equivalent to – but is not the same as – the equality or "equals sign" in maths =
Reactions are always
written and balanced with respect to the reaction arrow.
Thermochemical
data is always given with respect to the reaction as written,
ie. with respect to the arrow. This convention removes any apparent
ambiguity where the enthalpy of a reaction may be expressed "per
mol", but it is not be clear what the mole is. The decomposition
of ozone would be an example: 2O3
3O2. See here.
There are several
types of reaction arrow:
As written, a balanced chemical
reaction equation is a closed system that evolves with time, and
when we discuss a gas phase reaction involving real chemical reagents,
such as:
2H2(g)
+ O2(g)
2H2O(g)
the chemical reaction
is the system universe:
No edges are described
or defined in the reaction equation. These are constraints for us to
add.
We are reduced
to inert observers inside a reaction vessel of infinite size containing
only hydrogen and oxygen and/or water.
If the reaction
involves a physical phase change, for example the conversion of water
to vapour reaction system:
H2O(l)
H2O(g)
the phase surface
is an infinite surface, like the boundary between an infinite ocean
and an infinite atmosphere.
The reaction equation
is scale invariant: it can represent individual atoms or it can represent
billions of tons of bulk material.
It was stated above that chemistry
is a generational science. Each and all of these generational steps: quarks
to protons & neutrons; protons + neutrons to atomic nuclei; atomic
nuclei + electrons to atoms and ions; etc., can be described in terms
of balanced chemical equations.
Chemical reactions and the
associated symbolic reaction equations are the "unit systems"
from which chemistry is constructed. The set of chemical reactions represents
reaction space.
Summing
Up
There are a
number of theories:
Systems theory
Catastrophe theory
Chaos theory
Complexity theory
Briefly, when
academics first started looking systems they realised that systems could
be linear (simple) or non-linear and complicated.
Catastrophe
theory was one of the first approaches to dealing with difficult non-linear
systems, but the theory proved to be lacking in generality.
In the 1970s
it was discovered that looping computer code could generate patterns of
infinite complexity, and the term chaos was adopted to describe
the extraordinary fractal patterns produced.
Mandelbrot
realised that the new fractal patterns seemed to mimic the natural world,
and could exhibit rare beauty. By the mid 1980s the term chaos
theory entered popular culture.
It is now recognised
that the terms "complex" and "complexity" are actually
more useful – and carry less cultural baggage – than the term
"chaos", even though the terms are sometimes [confusing] used
interchangeably.
Due to complexity
theory, the word "random" has rather lost its meaning because
it is just a description, whereas the term complexity implies
that a non-linear process is operating. Physical and natural systems that
exhibit random behaviour are always complex.
Before
Mandelbrot's work, the coastline of Great Britain may have been described
as "a random collection of headlands and bays". Post–Mandelbrot,
coastlines are recognised as being complex, fractal and self-similar.
Chemistry can be studied using systems analysis, as discussed on these pages and by a Royal Society of Chemistry Instant insight: Systems chemistry by R. Frederick Ludlow & Sijbren Otto.