Archive for the ‘thermodynamics’ Category

Credit: Microbiology Online Notes

Although the Carbon Cycle is a well-accepted concept illustrated by countless graphics on the internet such as the one shown above, I wonder if it deserves to be called a cycle in the general sense of uninterrupted circular motion. Because the fossil fuels (coal, oil and gas) formed over millions of years from atmospheric CO2 are actually end products. Left to themselves they would remain as coal, oil and gas and the cycle would stop turning. To say that the cycle is completed by human interference sounds somewhat contrived.

But despite this criticism, the Carbon Cycle has an obvious value: it helps us to see a bigger picture. And this broader understanding can be further enhanced by looking at another aspect of carbon which changes during its journey around the cycle – namely its oxidation state.

I have not come across a graphic that includes this, so here is one I drew to illustrate the idea.

Rather than describing carbon in terms of its sequence of physical transformations, this cycle shows how carbon’s oxidation state reflects changes in the nature of its chemical bonds. Oxidation states are conveniently if somewhat abstractly represented by dimensionless numbers, which in the case of carbon range from +4 (most oxidized state) to –4 (most reduced state). The lower the number, the more energy is present in carbon’s chemical bonds. So in the process of carboniferous fuel creation the oxidation state number decreases. Conversely, the process of energy release from carboniferous fuel results in an increase in oxidation state number. The natural abundance of carbon with its wide range of oxidation states centered about zero is what gives carbon its usefulness as both a source and a store of energy.

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Energy in, Energy out

Below is a quantified illustration of how carboniferous fuels store and release more energy as the oxidation state number decreases.

Energy release in kJ per mole of CO2 formed. Numbers are indicative

The green arrow shows the oxidation state decrease from +4 to 0 associated with photosynthesis in terrestrial plants and marine plankton. The carbon in the repeating molecular unit is reduced by hydrogenation, and combusting this fuel e.g. in the form of cellulose releases 447 kJ per mole of CO2 formed. Further reduction to oxidation state –1 is associated with the creation of coal (not illustrated) which releases about 510 kJ per mole of CO2 formed. Continued reduction to oxidation state –2 is associated with petroleum which releases around 610 kJ per mole of CO2 formed. Finally the formation of natural gas represents the lowest possible oxidation state of carbon, –4. On combustion natural gas releases the maximum energy of 810 kJ per mole of CO2 formed.

These numbers illustrate a general (inverse) relationship between the magnitude of the carbon oxidation state and the amount of energy generated by combustion (see Appendix 1 for more data). For each mole of CO2 released, natural gas (–4) produces nearly twice as much energy as cellulosic biomass (0). That is an appreciable difference, which perhaps deserves more attention in public discourse about greenhouse gas emissions than it receives.

It is also worth noting that although it takes millions of years for natural gas to be formed from atmospheric carbon dioxide in Nature’s Carbon Cycle, the same carbon transformation can be achieved by human beings on a vastly accelerated timescale using a process known as the Sabatier reaction. This has been recently demonstrated in a remarkable Power-to-Gas (P2G) project conducted in Austria.

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Underground storage and conversion

In Pilsbach, Upper Austria, energy company RAG Austria AG conducted a P2G project called Underground Sun Storage in which excess electricity production from wind and solar was converted by electrolysis of water to hydrogen gas which was then pumped down into a depleted natural gas reservoir at a depth of 1 km. Following the successful conclusion of this project, a second P2G project called Underground Sun Conversion was then initiated in which carbon dioxide sourced from biomass combustion or DAC was co-injected with hydrogen into the gas reservoir.

According to RAG, the pores in the matrix of the underground reservoir contain micro-organisms which within a relatively short time convert the hydrogen and carbon dioxide into natural gas, recreating the process by which natural gas originates but shortening the timescale by millions of years. In its project description RAG Austria claims that “this enables the creation of a sustainable carbon cycle”.

How the micro-organisms effect the reaction between H2 and CO2 is not described in the material I have seen. Perhaps microbial enzymes serve as catalysts – the Sabatier reaction is spontaneous and indeed thermodynamically favored under the temperature and pressure conditions of the reservoir (313K, 107 bar).

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Energy in, Energy out (again)

Ok, so let’s take a look at the thermodynamics of the processes by which RAG Austria turns carbon dioxide into natural gas. Applying Hess’s Law, it can be seen that the sum of the electrolytic process and the Sabatier reaction is equivalent to a reversal of methane combustion and corresponds to the energy stored underground in the C-H bonds of the methane molecule – note that the oxygen is formed above ground during electrolysis and is vented to the atmosphere*

*the sum of reactions bears a curious similarity to the process of photosynthesis: 6CO2 + 6H2O → C6H12O6 + 6O2

Energy loss is intrinsic to both parts of this methane synthesis program. The electrical efficiency of water electrolysis using current best practises is 70–80%, while the Sabatier reaction between H2 and CO2 taking place in the underground reservoir is exothermic and loses around 15% of the energy used to form hydrogen in the initial stage.

On the face of it, underground storage in a natural gas reservoir of hydrogen alone would seem to offer better process economics. But carbon capture and underground conversion can be titrated to achieve a variable quotient between stored hydrogen and converted methane. Both have their economic attractions; what methane lacks in terms of process inefficiencies can be compensated for in several ways in relation to hydrogen. Superior energy density, more efficient transportation as LNG and compatibility with existing energy supply infrastructures are some of them. And then there is the larger issue of the value that society places upon the desire for carbon neutrality in existing energy systems on the one hand, and the promise of carbon-free energy systems on the other.

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Appendix 1

Carbon oxidation state and heat of combustion per mole of CO2 produced


Negative numbers in the last column indicate exothermic reaction i.e. heat release. Units are kJmol-1

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Suggested further reading

EURAKTIV article on energy storage projects June 2019
https://www.euractiv.com/section/energy/news/four-energy-storage-projects-that-could-transform-europe/

RAG Austria AG website – Underground Sun Conversion
https://www.underground-sun-conversion.at/en/project/project-description.html

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P Mander June 2019

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Credit: Extinction Rebellion

On Tuesday 2nd April 2019, environmental activists from the Extinction Rebellion group staged a parliamentary protest in London by glueing their bottoms to the windows of the viewing gallery of the House of Commons during a Brexit debate.

They used this cheeky tactic to call on British politicians to act on the ‘Climate and Ecological Crisis’. On its website, Extinction Rebellion wrote that “Government must act now to halt biodiversity loss and reduce greenhouse gas emissions to net zero by 2025”.

There is an irony here in that the call to immediate action was delivered at the precise moment when the UK government was in a state of complete paralysis over Brexit. Then again, the demonstrators’ demands weren’t meant to be taken entirely at face value. Even the most optimistic environmental biologists and chemical engineers would shake their heads at what was proposed to be accomplished in the space of just six years.

What the demonstration did achieve was to focus attention on the task in hand and the current state of progress. And it prompted CarnotCycle to pen this post on one promising technology for carbon capture and conversion (CCC) in which UK researchers are playing a key role.

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It is a well-known fact, not to mention a subject of serious environmental concern, that cows burp methane as a result of anaerobic fermentation of the grass they eat. And being aerobic respirers they also exhale carbon dioxide. Both these substances are potent greenhouse gases associated with human activity whose atmospheric levels, according to the best available science, must at least be stabilized in order to stand a chance of keeping global warming within manageable limits.

The cow in our picture is asking us an interesting question. What if it were possible to react carbon dioxide and methane together to form products that are not greenhouse gases? Even better, what if the reaction products could be put to useful purposes? Just think how cute that would be!

Putting our physical chemistry hats on for a moment and looking at the above equation, we notice that the oxidation states of carbon in the two molecules are at opposite ends of the scale. Methane has the most reduced form of carbon (-4) while carbon dioxide has the most oxidized form of carbon (+4). A redox reaction between the two looks possible and indeed is possible, albeit at elevated temperatures:

This process – called dry reforming of methane or DRM – was first introduced by Germany’s dynamic duo, Franz Fischer and Hans Tropsch, in 1928 but extensive investigation only started in the 1990s when increasing concerns about the greenhouse effect were raised by the international scientific community.

Notice how DRM simultaneously converts two greenhouse gases into two non-greenhouse gases which together make valuable syngas (1:1), a key industrial intermediate in the production of chemicals and clean fuels.

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Well, sort of. The big difficulty in making DRM viable relates to side reactions. The DRM reaction proceeds above 918K but then so does the thermal decomposition of both methane and carbon monoxide which results in carbon being deposited on the catalyst, clogging up the pores and thereby deactivating it.

This problem can however be mitigated in a very neat way by combining DRM with another methane-reforming process, namely steam reforming (SRM). Coupling reactions in this way not only reutilizes the deposited carbon but also adds a product stream with an H2/CO ratio of 3:1 which enables the syngas ratio to be adjusted for the synthesis of methanol, ethanoic acid or dimethyl ether (DME), which has promise as a sulfur-free diesel fuel, or towards the (2n+1):n H2/CO ratio required for Fischer-Tropsch synthesis of alkane fuels

The other problem is that all these reactions are endothermic (heat requiring). This energy has to be obtained from somewhere, and now here comes the next neat idea. Adding oxygen to the reactant stream allows partial oxidation of methane (POM) and catalytic combustion of methane (CCM) to take place, which are exothermic reactions that can supply the necessary heat

Putting three reforming agents – carbon dioxide, water and oxygen – together in the reactant stream with a methane feedstock creates a sufficiently energy-efficient overall process known as ‘tri-reforming’.

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To complete the conceptual scheme, the tri-reforming process is integrated into a cycle where the syngas output is utilized in the tri-generation of fuels, industrial chemicals and electricity, with the flue gases from these processes being fed back after nitrogen purging to the tri-reforming reactor. Carbon dioxide can also be fed into the cycle from external sources such as power plants and cement works.

Note that in principle, carbon dioxide can be fed into the cycle from established carbon capture and storage (CCS) processes. In this way underground reservoirs of anthropogenic carbon dioxide can be utilized as a feedstock for additional tri-generation.

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Almost. I mentioned at the start that the UK was involved in CCC research so I should say a few words about that. Scientists at Oxford and Cambridge are working with the King Abdulaziz City for Science and Technology in Saudi Arabia and the National Natural Science Foundation of China on the tri-reforming/tri-generation technology detailed above.

Since China is the world’s largest CO2 emitter and Saudi Arabia is the world’s largest oil producer, the Anglo-Sino-Saudi initiative seems a sensible geoscientific cluster. The news was announced on 28 January 2018, more than a year before the Extinction Rebellion demonstrations in London.

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Suggested further reading

A mini-review on CO2 reforming of methane

Published in June 2018 this is a useful and easily readable grounder covering the thermodynamic, kinetic, catalysis and commercial aspects of the subject.

Turning carbon dioxide into fuel

A paper co-written by the Oxbridge scientists involved in the Anglo-Sino-Saudi initiative. It was published in 2010, which shows that these guys have been on the case a while. Climate activists take note, and read their stuff.

Turning carbon dioxide into fuel – a new UK-China-Saudi Arabia initiative

The January 2018 press release referred to above. I reckon my poly-alliterative Anglo-Sino-Saudi sounds better.

Tri-reforming: a new process for reducing CO2 emissions

A bedrock paper (January 2001) from Chunshan Song at Penn State. The process diagram featured above is taken from this paper. If you don’t read anything else, read this one.

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P Mander May 2019

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I dare say most of you will remember this classroom demonstration, in which water passes through a semi-permeable membrane and causes the liquid level to rise in the stem of the thistle funnel. The phenomenon is called osmosis, and at equilibrium the osmotic pressure is equal to the hydrostatic pressure.

Historical background

This experiment has its origins way back in the mid-18th century, when a French clergyman named Jean-Antoine Nollet tied a piece of pig’s bladder over the mouth of a jar containing alcohol and immersed the whole thing in a vat of water. What prompted him to do this is not known, but we do know the result of his experiment. The bladder swelled up and ultimately burst from the internal pressure.

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Jean-Antoine Nollet 1700-1770

Nollet published his findings in Recherches sur les causes de Bouillonement des Liquides (1748) in which he gave a correct interpretation of the phenomenon, which arises from the much more marked permeability of the bladder to water as compared with alcohol.

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Moritz Traube 1826-1894

The actual measurement of osmotic pressure had to wait for over a century, until the German chemist Moritz Traube showed in 1867 that artificial semipermeable membranes could be made using gelatin tannate or copper ferrocyanide. Traube’s compatriot Wilhelm Pfeffer, a botanist, succeded in depositing the latter in the walls of a porous jar, which when filled with a sugar solution, connected to a mercury manometer and then plunged into pure water, provided a means of measuring osmotic pressures.

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Wilhelm Pfeffer 1845-1920

Following Pfeffer’s osmotic pressure measurements using sucrose solutions, on which JH van ‘t Hoff based his famously flawed gaseous theory of solutions, there were two notable teams of experimentalists – one on each side of the Atlantic – which provided high quality osmotic pressure data to test the ideas of theoreticians. In the USA, Harmon Northrop Morse and Joseph Christie Whitney Frazer led a team at Johns Hopkins University, Baltimore, Maryland from 1901 to 1923. In Britain meanwhile, the aristocrat-turned-scientist Lord Berkeley and co-worker Ernald Hartley set up a private research laboratory near Oxford which operated (with gaps due to war service) from 1904 to 1928.

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Ernald Hartley (1875-1947) Besides being a research chemist, Hartley was an amateur clarinetist who played in the Oxford orchestra for many years. The photo dates from 1925.

While Morse and Frazer used the same principle as Pfeffer, albeit with a more advanced electrochemical method of depositing the membrane in the pores, Berkeley and Hartley reversed the arrangement of solvent and solution, applying measured pressure to the latter to attain equilibrium.

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Theoretical development in Europe

In Europe, the rigorous application of thermodynamics to the phenomenon of osmosis started in 1887 with Lord Rayleigh, who combined the use of the ideal gas law PV = nRT with the idea of a reversible isothermal cycle of operations in which the sum of work in the complete cycle is zero.

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Lord Rayleigh 1842-1919

Being essentially an attempt to provide hypothesis-free support to van ‘t Hoff’s troubled gaseous theory of solutions, the solute in Rayleigh’s cycle was a mole of ideal gas, which was first dissolved in the solution by applied pressure and then recovered from the solution by osmotic pressure to return the system to its original state.

Rayleigh’s approach, using a zero-sum cycle of operations, was thermodynamically sound and continued to form the basis of theoretical development in its next phase, which in Europe focused on vapor pressure following the influential papers of Alfred Porter in 1907 and Hugh Callendar in 1908.

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Alfred Porter 1863-1939

By 1928, the theoretical model in JAV Butler’s popular textbook The Fundamentals of Chemical Thermodynamics was close to the familiar classroom demonstration of osmosis shown at the head of this post, in which the hydrostatic pressure acting on the solution counteracts the tendency of the solvent to pass through the semi-permeable membrane. At equilibrium, the hydrostatic pressure P is equal to the osmotic pressure.

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JAV Butler 1899-1977

To obtain a thermodynamic relation for osmotic pressure in terms of vapor pressures, Butler uses Rayleigh’s idea of a reversible isothermal cycle of operations together with a semipermeable membrane in the form of a movable piston between the solution and the solvent:

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The diagram shows a solution under hydrostatic pressure P which is equal to the osmotic pressure. Below the semi-permeable piston is pure solvent. Butler then applies the following argumentation:

1] Vaporize 1 mole of the pure solvent at its vapor pressure p0, and expand it reversibly so that the vapor pressure falls to p equal to the partial pressure of the solvent in the solution (Butler assumes that p is not affected by P applied to the solution). Condense the vapor into the solution. Since the work of vaporization and condensation cancel out, the only work done is the work of expansion. Assuming the vapor obeys the ideal gas law, the work (w) done is given by the textbook isothermal expansion formula

2] Now move the semi-permeable piston up against the pressure P until a quantity of solvent equivalent to 1 mole of vapor has passed through it. If the decrease in the volume of the solution is ΔV, the work done is PΔV.

The cycle is now complete and the system has returned to its original state. The total work done is zero and we may equate the two terms

(1)

where P is the osmotic pressure, ΔV is the partial molal volume of the solvent in the solution, p0 is the vapor pressure of the pure solvent and p is the vapor pressure of the solvent in the solution. This thermodynamically exact relation, which involves measured vapor pressures, is in good agreement with experimental determinations of osmotic pressure at all concentrations.

There is a great irony here, in that this equation is exactly the one that JH van ‘t Hoff found his way to in Studies in Chemical Dynamics (1884), before he abandoned his good work and went completely off-track with his gaseous theory of solutions.

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JH van ‘t Hoff 1852-1911

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Theoretical development in America

In the US, the theory of the semipermeable membrane and the ‘equilibrium of osmotic forces’ was the work of one supremely gifted man, Josiah Willard Gibbs, who more or less single-handedly laid the theoretical foundations of chemical thermodynamics in his milestone monograph On the Equilibrium of Heterogeneous Substances.

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J Willard Gibbs 1839-1903

But before delving into the powerful idea he introduced, let us return to the subject of equilibrium in a system subject to osmotic pressure with a set-up that is slightly different to that used by Butler. In the diagram below, the piston supplies pressure Psoln to the solution which is just enough to stop solvent passing through the membrane and bring about equilibrium at constant temperature; the osmotic pressure is defined as the excess pressure Psoln – p01.

osm13

The question can now be asked: Does the condition of osmotic equilibrium coincide with equality of a thermodynamic variable on either side of the membrane? Clearly it cannot be pressure or volume, nor can it be temperature since constant temperature does not prevent osmotic disequilibrium.

The P, V, T variables do not provide an affirmative answer, but in his monumental masterwork, Gibbs supplied one of his own invention which did – the chemical potential, symbolized μ. It is an intensive variable which acts as a ‘generalized force’, driving a system from one state to another. In the present context the force drives chemical components, capable of passing through a membrane, from a state of higher potential to a state of lower potential.

So given a membrane dividing solution from solvent and permeable only to the latter, we can understand the osmotic force driving the solvent (designated by subscript 1) through the membrane into the solution in terms of movement to a region of lower potential since

Now the difference in potential can be calculated according to the textbook formula

(2)

where x1 is the mole fraction (<1) of the solvent in the solution. To achieve equilibrium, the chemical potential of the solvent in the solution must be increased by the amount –RTlnx1 (a positive quantity since lnx1 is negative). This can be done by increasing the pressure on the solution since

is always positive (V1 is the partial molar volume of the solvent in the solution).

The osmotic pressure is defined as the excess pressure Psoln – p01. As can be seen from the diagram below, this is the pressure required to raise the chemical potential of the solvent in the solution so that it becomes equal to the chemical potential of the pure solvent.

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Since the slope is V1, it follows that

(3)

Combining (2) and (3) and designating the osmotic pressure by P gives the desired equilibrium relation

This is exactly equivalent to equation (1) derived by Butler, since by his terminology

The two methods of proof are thus shown to be equivalent – we can regard osmotic pressure as the excess pressure required to increase either the chemical potential or the vapor pressure of the solvent in the solution. But Gibbs saw an advantage in using potentials, which he voiced in an 1897 letter to Nature entitled Semi-Permeable Films and Osmotic Pressure:

“The advantage of using such potentials in the theory of semi-permeable diaphragms consists … in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm.”

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Tottenham Court Road, London WC2 in 1880

In the study of chemical reactions, thermodynamics enables us to calculate changes in state functions such as enthalpy, entropy and free energy, and determine the direction in which a reaction is spontaneous. But it tells us nothing about the speed of reaction; that is the province of chemical kinetics. Thermodynamics and chemical kinetics can be viewed as complementary disciplines, which together provide the means by which the course of a reaction can be elucidated.

A classic case which exemplifies the dual application of thermodynamics and chemical kinetics is the Tottenham Court Road gas explosion which occurred in July 1880.

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The incident

It was a time of great expansion of the network for gas pipeline transport in London. Gas lighting of streets and buildings was well-established, but now the gas stove was about to become a commercial success, and new gas mains were being laid to supply the anticipated demand.

The Gas Light and Coke Company, which supplied coal gas from a number of gasworks in London, had laid a new 1.2 kilometer (0.75 mile) section of main from Bedford Square to Fitzroy Square, the pipeline crossing Tottenham Court Road at the junction with Bayley Street and running along Percy Street before turning north along the entire length of Charlotte Street.

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On the evening of Monday 5th July 1880, workmen were preparing to connect the new main to the existing network at Bayley Street. Unknown to them however, a faulty valve at the other end of the new main was leaking coal gas, which had mingled with the air in the pipe to form an explosive mixture. In a presumed act of carelessness by one of the workmen at Bayley Street, a flame or other ignition source came in close proximity to the pipe.

The gas mixture detonated and the explosion ripped through the entire length of the new 1.2 kilometer main. A number of people were killed and injured in the blast, and 400 houses were damaged by flying debris. The entire incident lasted about 12 seconds.

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The investigation

A singularly worrying feature of the Tottenham Court Road gas explosion was that it had ripped through over a kilometer of pipeline in a matter of seconds. How could this happen? And how easily could this happen again? For the safety of millions of Londoners, answers had to be found.

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Augustus Vernon Harcourt (1834-1919)

The authorities turned to one the country’s leading chemists, Augustus Vernon Harcourt, who was conducting a program of research in chemical kinetics at Oxford University. Together with his student Harold Baily Dixon (1852-1930), Harcourt began to investigate the rates of propagation of gaseous explosions.

In what sounds like a rather risky experiment, they set up long metal pipes under the Dining Hall of Balliol College Oxford to measure the speed with which explosion waves travel when a mixture of air and coal gas detonates.

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The Dining Hall of Balliol College, Oxford

Twenty three years earlier, the German chemist Robert Bunson (of Bunsen burner fame) had investigated the rate of propagation for the ignition of coal gas and oxygen and concluded that the flame front velocity was less than 1 meter per second. From the experiments at Balliol however, Harcourt and Dixon arrived at a very different answer. In a report to the Board of Trade on the Tottenham Court Road blast, Harcourt concluded that the velocity of a coal gas/air explosion wave exceeded 100 yards per second (91 meters per second).

From the safety point of view, Harcourt and Dixon had shown how absolutely essential it was to prevent air becoming mixed with coal gas in the gas pipeline network. But it would take decades before sufficient theoretical progress was made to allow a detailed understanding of what exactly happened in the great gas explosion of 1880.

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Branching chains

The development of chemical kinetics involved many different contributors in the decades after Harcourt and Dixon’s pioneering work at Oxford. Theories were advanced on several different aspects of the subject, but one piece of theoretical work had particular relevance to the study of explosions.

In 1921, a Danish physical chemist by the name of Jens Anton Christiansen (1888-1969) completed his PhD studies in reaction kinetics at Copenhagen University. In his thesis he incorporated an idea first suggested by Bodenstein in 1913 and introduced the term “kædereaktion”. This term, and the conceptual idea behind it, attracted considerable attention and the equivalent English expression “chain reaction” came into use. Two years later, Christiansen and the Dutch physicist Hendrick Anthony Kramers (1894-1952) published a paper in which they suggested the possibility of branching chains. Their idea was that a chain reaction could involve steps in which one chain carrier (an atom or radical) might not only regenerate itself but also produce an additional chain carrier. If such chain branching occurred, the number of chain carriers could increase extremely rapidly and result in an explosion.

The idea proved to be well-founded, and was further developed by Nikolai Semyonov (1896-1986) and Cyril Norman Hinshelwood (1897-1967). Their work also showed that chain carriers were removed at the walls of the reaction vessel. If the rate of removal of the chain carriers was fast enough to counteract the effect of chain branching, a steady reaction ensued. But if the removal rate could not keep pace with the chain branching rate, an explosion would result.

On the basis of their thinking, the reaction rate expression assumed the form

where F is a function of the concentrations characteristic of the chain branching step, fa is a function determining the removal of chain carriers, and fb is a function expressing the branching nature of the chain reaction.

In steady reaction conditions, fa is sufficiently greater than fb. But if conditions change so that fa and fb converge, a point will be reached where the difference between them becomes vanishingly small. The reaction rate will soar towards infinity however small F may be, and the evolution of heat in the system will be so great as to cause an explosion.

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Semyonov and Hinshelwood were awarded the Nobel Prize in 1956 for their work on reaction rates

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Piecing the facts together

From the information contained in newspaper reports, and the application of kinetic theory and thermodynamics, it is possible to arrive at a likely explanation of why the great gas explosion of 1880 happened in the way it did.

It is known that coal gas leaked into the newly laid main at its northern end, and that detonation occurred at the other end in Bayley Street. From this it can be inferred that the entire pipeline between these two points contained coal gas admixed with the air that the pipe originally contained. On the assumption that the leaking valve was introducing coal gas at a modest and steady rate, it is likely that the partial pressures of the gases in the pipe were being brought into equilibrium as the coal gas seeped along the pipe.

Newspaper reports stated that the new main between Bayley Street and Fitzroy Square was a metal pipe of fixed (3 ft/0.91 m) diameter. The ratio of the surface area to the enclosed volume or, which is the same thing, the ratio of the circumference to the cross-sectional area

was therefore constant along its length*.

*assuming the geometry of the bend had no effect on fa. This point is examined later.

At the moment of detonation at Bayley Street, it is a reasonable hypothesis that the function F in the Semyonov-Hinshelwood rate expression was not subject to large variations along the length of the new main. The same can be said of fb, and since the ratio of the circumference to the cross-sectional area of the pipeline was constant, the function fa determining the removal of chain carriers at the walls of the pipe was also constant. In short, the reaction rate expression applying at the end of the pipe – where detonation is known to have occurred – applied at every other point along its length.

At this juncture, it is convenient to recall the combustion reactions of the principal components of coal gas, namely hydrogen, methane and carbon monoxide:

We observe that from a stoichiometric perspective, none of the reactions involves an increase in volume; in fact two of them result in a decrease. The overall entropy of reaction is negative, and this tells us that the conversion of reactants into products, however rapidly it took place, could not in itself have resulted in any pressure increase under the constant volume conditions of the pipe.

From an enthalpy of reaction perspective however, the situation is very different. The above reactions are all significantly exothermic processes – the calorific value of coal gas is typically around 20 megajoules per cubic meter. In the circumstances of detonation, the virtually instantaneous release of a large amount of heat would result in a similarly rapid rise in temperature, causing sudden compression of the adjacent volume element in the pipe and heating it to the point of detonation. This sequence would be repeated from one volume element to the next, with a wave of adiabatic compression intensifying the pressure as it traversed the pipe. A continuously propagating explosion would then follow the pressure wave along the course of the main as the pipe ruptured.

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The bend in the pipe

The junction of Percy Street with Charlotte Street was the only point along the entire length of the new main which deviated from a straight line. Here the pipeline executed a 90 degree turn, and it raises the question of how a detonation wave can go round corners. The exact construction of the bend is not recorded, but it is likely that an elbow joint was used.

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Geometrically, the bend itself is a quadrant of a torus, whose geometry is such that regardless of whether the elbow has a long or short major radius R, the ratio of the surface area to the enclosed volume is constant

This is the same ratio as that of the straight pipe. The bend at the junction of Percy Street with Charlotte Street introduced no changes to the fa term in the Semyonov-Hinshelwood rate expression, and thus the conditions for detonation were met at every point of the bend.

So the 90 degree elbow made no difference to the detonation wave. It simply turned sharp right and carried on up to Fitzroy Square, at a velocity of almost 100 meters per second.

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Estimating the power of the explosion

It is known from the analysis of coal gas that one volume of coal gas requires approximately 10 volumes of air for its complete combustion. This means that an explosive mixture with air cannot be formed at coal gas concentrations much above 9%, since there would be insufficient oxygen to support the necessary rate of reaction. Below 7% coal gas concentration, the mixture is also non-explosive, for other reasons.

An average coal gas concentration of 8% throughout the pipeline is therefore a fair estimate, and seems plausible given that the new main contained air when laid and that coal gas was introduced at a modest rate from a leaking valve. We know that the new 1.2 kilometer main had a radius of 0,455 meters, giving a total volume of 780 cubic meters. At the moment of detonation, coal gas is estimated to have filled 8% of this volume i.e. 62 cubic meters. The calorific value of coal gas is typically 20 megajoules per cubic meter, so we can conclude that the Tottenham Court Road gas explosion released around 1,240 MJ in the 12 seconds it took to traverse the pipeline. The power of the explosion was therefore 1240/12 = 103 MW.

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The 3×2 flagstones used on London sidewalks weigh around 70 kg each. The energy released by the Great Gas Explosion of 1880 was sufficient to blast 59,000 flagstones to a height of 30 meters.

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Contemporary accounts

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Charlotte Street after the blast

Newspaper accounts remarked on the rapid progression of the explosion, with one commenting:

“[The main pipe at Bayley Street] burst with a terrific report, and sheets of flame issued suddenly from the earth. Instantly the report seemed to run along Percy Street, which was torn up for sixty or seventy yards (ca. 60 meters), the paving stones flying on each side against the houses.”

“At the corner of Charlotte Street the basements of two houses were shattered. The paving stones were here also sent into the air, falling on and through the roofs of the houses opposite. Further on, the pipe burst again, near the corner of Bennett Street, where there is a large gap in the roadway. Another burst-up occurred near the corner of Howland Street, and at the corner of London Street (now Maple Street) still further on…”

One eye-witness was in Percy Street when the explosion occurred. He experienced the effect of not only the pressure wave from the bursting pipe, but also the decompression wave which followed in its wake:

“I was walking down Percy Street, when I felt the ground shaking under my feet. I immediately saw the centre of the street rising in the air. A tremendous report followed, and then there was a shower of bricks and stones. I felt myself lifted from the ground, and the next moment I was lying among the debris at the bottom of a deep hole in the roadway.”

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P Mander December 2015

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I can’t think of a better introduction to this post than Ludwig Boltzmann gave in his Vorlesungen über Gastheorie (Lectures on Gas Theory, 1896):

“General thermodynamics proceeds from the fact that, as far as we can tell from our experiences up to now, all natural processes are irreversible. Hence according to the principles of phenomenology, the general thermodynamics of the second law is formulated in such a way that the unconditional irreversibility of all natural processes is asserted as a so-called axiom … [However] general thermodynamics (without prejudice to its unshakable importance) also requires the cultivation of mechanical models representing it, in order to deepen our knowledge of nature—not in spite of, but rather precisely because these models do not always cover the same ground as general thermodynamics, but instead offer a glimpse of a new viewpoint.”

Today, the work of Ludwig Boltzmann (1844-1906) is considered among the finest in physics. But in his own lifetime he faced considerable hostility from those of his contemporaries who did not believe in the atomic hypothesis. As late as 1900, the kinetic-molecular theory of heat developed by Maxwell and Boltzmann was being vigorously attacked by a school of scientists including Wilhelm Ostwald, who argued that since mechanical processes are reversible and heat conduction is not, thermal phenomena cannot be explained in terms of hidden, internal mechanical variables.

Boltzmann refuted this argument. Mechanical processes, he pointed out, are irreversible if the number of particles is sufficiently large. The spontaneous mixing of two gases is a case in point; it is known from experience that the process cannot spontaneously reverse – mixed gases don’t unmix. Today we regard this as self-evident, but in Boltzmann’s time his opponents did not believe in atoms or molecules; they considered matter to be continuous. So the attacks on Boltzmann’s theories continued.

Fortunately, this did not deter Boltzmann from pursuing his ideas, at least not to begin with. He saw that spontaneous processes could be explained in terms of probability, and that a system of many particles undergoing spontaneous change would assume – other things being equal – the most probable state, namely the one with the maximum number of arrangements. And this gave him a new way of viewing the equilibrium state.

One can see Boltzmann’s mind at work, thinking about particle systems in terms of permutations, in this quote from his Lectures on Gas Theory:

“From an urn, in which many black and an equal number of white but otherwise identical spheres are placed, let 20 purely random drawings be made. The case that only black spheres are drawn is not a hair less probable than the case that on the first draw one gets a black sphere, on the second a white, on the third a black, etc. The fact that one is more likely to get 10 black spheres and 10 white spheres in 20 drawings than one is to get 20 black spheres is due to the fact that the former event can come about in many more ways than the latter. The relative probability of the former event as compared to the latter is the number 20!/10!10!, which indicates how many permutations one can make of the terms in the series of 10 white and 10 black spheres, treating the different white spheres as identical, and the different black spheres as identical. Each one of these permutations represents an event that has the same probability as the event of all black spheres.”

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By analyzing the ways in which systems of particles distribute themselves, and the various constraints to which particle assemblies are subject, important links came to be established between the statistical properties of assemblies and their bulk thermodynamic properties.

Boltzmann’s contribution in this regard is famously commemorated in the formula inscribed on his tombstone: S = k log W. There is powerful new thinking in this equation. While the classical thermodynamic definition of entropy by Rankine and Clausius was expressed in terms of temperature and heat exchange, Boltzmann gave entropy – and its tendency to increase in natural processes – a new explanation in terms of probability. If a particle system is not in its most probable state then it will change until it is, and an equilibrium state is reached.

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P Mander April 2016