Archive for the ‘physical chemistry’ 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.

osm16

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

A couple of blocks down from the Metro station Jussieu in Paris’s 5th arrondisement lies Rue Cuvier, which runs along the north-western edge of the botanic gardens which houses the Natural History Museum. The other side of the road is bordered by various institutes of the Sorbonne, notably UPMC (formerly Pierre and Marie Curie University).

The Curies have historical associations with a number of streets in the Latin Quarter, and Rue Cuvier in particular. Pierre Curie was born at No.16 and it was in a science faculty building in this street that the Curies conducted their fundamental research on radium between 1903 and 1914. The building still exists, shielded from public curiosity by a set of prison-style metal gates, and it was in this laboratory that the first pioneering research into what would later be recognized as nuclear energy was conducted in 1903.

Yet it was not the renowned husband-and-wife team which carried out this experiment. It was in fact Pierre Curie and his young graduate assistent Albert Laborde who did the work and reported it in Comptes Rendus in a note entitled Sur la chaleur dégagée spontanément par les sels de radium (On the spontaneous production of heat by radium salts). The note, which barely covers two pages, was published in March 1903.

The laboratory in Rue Cuvier where the Curies and Laborde worked was at No.12. Just across the street is No.57, which once housed the Appled Physics laboratory of the Natural History Museum. It was here in 1896 that Henri Becquerel serendipitously discovered the strange phenomenon of radioactivity.

Between that moment of discovery on one side of Rue Cuvier and Curie and Laborde’s remarkable experiment on the other, lay the years of backbreaking work in a shed in nearby Rue Vauquelin where the Curies, together with chemist Gustave Bémont, processed tons of waste from an Austrian uranium mine in order to extract a fraction of a gram* of the mysterious new element radium.

*the maximum amount of radium coexisting with uranium is in the ratio of their half-lives. This means that uranium ores can contain no more than 1 atom of radium for every 2.8 million atoms of uranium.

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The Curie – Laborde experiment

Albert Laborde (left) and Pierre Curie, in 1901 and 1903 respectively

Pierre Curie and Albert Laborde were the first to make an experimental determination of the heat produced by radium because they were the first to have enough radium-enriched material to make the experiment practicable. It was a close-run thing though. Ernest Rutherford and Frederick Soddy had been busy working on radioactivity at McGill University in Canada since 1900, but they were hampered by lack of access to radium and were using much weaker thorium preparations. This situation would quickly change however when concentrated radium samples became available from Friedrich Giesel in Germany. By the summer of 1903, Soddy (now at University College London) and Rutherford would have their hands on Giesel’s supply. But Curie and Laborde had a head start, and they turned their narrow time advantage to good account.

Methodology

To determine the heat produced by their radium preparation, they used two different approaches – a thermoelectric method, and an ice calorimeter method.

This diagram of their thermoelectric device, taken from Mme Curie’s Traité de Radioactivité (1910), Tome II, p272, unfortunately lacks an explanation of the key, but the set-up essentially comprises a test ampoule containing the chloride salt of radium-enriched barium and a control ampoule of pure barium chloride. These are marked A and A’. The ampoules are placed in the cavities of brass blocks enclosed in inverted Dewar flasks D, D’ with some unstated packing material to keep the ampoules from falling down. The flasks are enclosed in containers immersed in a further medium-filled container E supported in a space enclosed by a medium F, all of which was presumably designed to ensure a constant temperature environment. The key feature is C and C’ which are iron-constantan thermocouples, embedded in the brass cavities, with their associated circuitry.

The current produced by the Seebeck effect resulting from the temperature difference between C and C’ was measured by a galvanometer. The radium ampoule was then replaced by an ampoule containing a platinum filament through which was passed a current whose heating effect was sufficient to obtain the same temperature difference. The equivalent rate of heat production by the radium ampoule could then be calulated using Joule’s law.

The second method used was a Bunsen calorimeter, which was known to be capable of very exact measurements using only a small quantity of the test substance. For details of the operational principleof this calorimeter, the reader is referred to this link:

http://thewaythetruthandthelife.net/index/2_background/2-1_cosmological/physics/j9.htm

The above diagram of the Bunsen calorimeter is taken from Mme Curie’s Traité de Radioactivité (1910), Tome II, p273.

Results

For most of their experiments, Curie and Laborde used 1 gram of a radium-enriched barium chloride preparation, which liberated approximately 14 calories (59 joules) of heat per hour. It was estimated from radioactivity measurements – no doubt using the quartz electrometer instrumentation invented by Curie – that the gram of test substance contained about one sixth of a gram of radium.

Measurements were also made on a 0.08 gram sample of pure radium chloride. These yielded results of the same order of magnitude without being absolutely in agreement. Curie and Laborde made it clear in their note that these were pathfinding experiments and that their aim was solely to demonstrate the fact of continuous, spontaneous emission of heat by radium and to give an approximate magnitude for the phenomenon. They stated:

» 1 g of radium emits a quantity of heat of approximately 100 calories (420 joules) per hour.

In other words, a gram of radium emitted enough heat in an hour to raise the temperature of an equal weight of water from freezing point to boiling point. And it was continuous emission, hour after hour for year after year, without any detectable change in the source material.

Curie and Laborde had quantified the capacity of radium to generate heat on a scale which was far beyond that known for any chemical reaction. And this heat was continuously produced at a constant rate, unaffected by temperature, pressure, light, magnetism, electricity or any other agency under human control.

The scientific world was astonished. This phenomenon seemed to defy the laws of thermodynamics and the question was immediately raised: Where was all this energy coming from?

Speculation and insight

In 1903, little was known about the radiation emitted by radioactive substances and even less about the atoms emitting them. The air-ionizing emissions had been grouped into three categories according to their penetrating abililities and deflection by a magnetic field, but the nature of the atom – with its nucleus and orbiting electrons – was a mystery yet to be unveiled.

Illustration from Marie Curie’s 1903 doctoral thesis of the deflection of rays by a magnetic field. Note the variable velocities shown for the β particle, whose charge-mass ratio Becquerel had demonstrated to be identical to that of the electron.

Radioactivity had been discovered by Henri Becquerel as an accidental by-product of his main area of interest, optical luminescence – which is the emission of light of certain wavelengths following the absorption of light of other wavelengths. By association luminescence was seen as a possible explanation of radioactivity, that radioactive substances might be absorbing invisible cosmic energy and re-emitting it as ionizing radiation. But no progress was made on identifying a cosmic source.

Meanwhile, from her detailed analytical work that she began in 1898, Marie Curie had discovered that uranium’s radioactivity was independent of its physical state or its chemical combinations. She reasoned that radioactivity must be an atomic property. This was a crucial insight, which directed thinking towards the idea of conversion of mass into energy as an explanation of the continuous and prodigious production of heat by radium that Pierre Curie and Albert Laborde had observed.

One of the major theories in physics at this time was electromagnetic theory. Maxwell’s equations predicted that mass and energy should be mathematically related to each other, and it was by following this line of thought that Frederick Soddy, previously Ernest Rutherford’s collaborator in Canada, came to the conclusion that radium’s energy was obtained at the expense of its mass.

Writing in the very first Annual Report on the Progress of Chemistry, published by the Royal Society of Chemistry in 1904, Soddy said this:

” … the products of the disintegration of radium must possess a total mass less than that originally possessed by the radium, and a part of the energy evolved must be considered as being derived from the change of a part of the mass into energy.”

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A different starting point

While Pierre Curie and Albert Laborde were conducting their radium experiment in Rue Cuvier, Paris, Albert Einstein – a naturalized Swiss citizen who had recently completed his technical high school studies in Zurich – was working as a clerk at the Patent Office in Bern. Much of his work related to questions about signal transmission and time synchronization, and this may have influenced his own thoughts, since both of these issues feature prominently in the conceptual thinking that led Einstein to his theory of special relativity submitted in a paper entitled Zur Elektrodynamik bewegter Körper (On the electrodynamics of moving bodies) to Annalen der Physik on Friday 30th June 1905.

On the basis of electromagnetic theory, supplemented by the principle of relativity (in the restricted sense) and the principle of the constancy of the velocity of light contained in Maxwell’s equations, Einstein proves Doppler’s principle by demonstrating the following:

Ist ein Beobachter relativ zu einer unendlich fernen Lichtquelle von der Frequenz ν mit der Geschwindigkeit v derart bewegt, daß die Verbindungslinie “Lichtquelle-Beobachter” mit der auf ein relativ zur Lichtquelle ruhendes Koordinatensystem bezogenen Geschwindigkeit des Beobachters den Winkel φ bildet, so ist die von dem Beobachter wahrgenommene Frequenz ν’ des Lichtes durch die Gleichung gegeben:

If an observer is moving with velocity v relatively to an infinitely distant light source of frequency ν, in such a way that the connecting “source-observer” line makes the angle φ with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency ν’ of the light perceived by the observer is given by:

where Einstein uses V (not c) to represent the velocity of light. He then finds that both the frequency and energy (E) of a light packet (cf. E=hν) vary with the velocity of the observer in accordance with the same law:

It was to this equation Einstein returned in a paper entitled Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Does the inertia of a Body depend on its Energy Content?) submitted to Annalen der Physik on Wednesday 27th September 1905.

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Mass-energy equivalence

Marie Curie and Albert Einstein, Geneva, Switzerland, 1925

Einstein’s paper of September 1905 – the last of the famous set published in Annalen der Physik in that memorable year – is less than three pages long and constitutes little more than a footnote to the preceding 30-page relativity paper. Yet despite its brevity, it is a difficult and troublesome work over which Einstein brooded for some years.

The paper describes a thought experiment in which a body sends out a light packet in one direction, and simultaneously another light packet of equal energy in the opposite direction. The energy of the body before and after the light emission is determined in relation to two systems of co-ordinates, one at rest relative to the body (where the before-and-after energies are E0 and E1) and one in uniform parallel translation at velocity v (where the before-and-after energies are H0 and H1).

Einstein applies the law of conservation of energy, the principle of relativity and the above-mentioned energy equation to arrive at the following result for the rest frame and the frame in motion relative to the body, the light energy being represented by a capital L:

At this point, things start getting a little tricky. Einstein subtracts the rest frame energies from the moving frame energies for both the before-emission and after-emission cases, and then subtracts these differences:

These differences represent the before-emission kinetic energy (K0) and after-emission kinetic energy (K1) with respect to the moving frame

Since the right hand side is a positive quantity, the kinetic energy of the body diminishes as a result of the emission of light, even though its velocity v remains constant. To elucidate, Einstein performs a binomial expansion on the first term in the braces, although he makes no mention of the procedure; nor does he show the math. So this next bit is my own contribution:

Let (v/V)2 = x

The appropriate form of the binomial expansion is

Setting x = v2/V2 and n = ½

The contents of the braces in the kinetic energy expression thus become

Now back to Einstein. At this point he introduces a new condition into the scheme of things, namely that the velocity v of the system of co-ordinates moving with respect to the body is much less than the velocity of light V. We are in the classical world of v<<V, and so Einstein allows himself to neglect magnitudes of fourth and higher orders in the above expansion. Hence he arrives at

This equation gives the amount of kinetic energy lost by the body after emitting a quantity L of light energy. In the classical world of v<<V the kinetic energy of the body is also given by ½mv2, and since the velocity v is the same before and after the light emission, Einstein is led to identify the loss of kinetic energy in his thought experiment with a loss of mass:

Gibt ein Körper die Energie L in Form von Strahlung ab, so verkleinert sich seine Masse um L/V2. Hierbei ist es offenbar unwesentlich, daß die dem Körper entzogene Energie gerade in Energie der Strahlung übergeht, so daß wir zu der allgemeineren Folgerung geführt werden: Die Masse eines Körpers ist ein Maß für dessen Energie-inhalt.

If a body gives off the energy L in the form of radiation, its mass diminishes by L/V2. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that: The mass of a body is a measure of its energy content.

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Testing the theory

The pavilion where Curie and Laborde did their famous work

When Einstein wrote Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? in 1905, he was certainly aware of the phenomenon of continuous heat emission by radium salts as measured by Curie and Laborde, and confirmed by several others in 1903 and 1904. In fact he saw in this a possible means of putting relativity theory to the test:

Es ist nicht ausgeschlossen, daß bie Körpern, deren Energieinhalt in hohem Maße veränderlich ist (z. B. bei den Radiumsaltzen) eine Prüfung der Theorie gelingen wird.

It is not impossible that with bodies whose energy content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.

In hindsight, it was unlikely that Einstein could have made this test work and he soon abandoned the idea. Not only would the mass difference have been extremely small, but also the process of nuclear decay was conceptually different to Einstein’s thought experiment. In Curie and Laborde’s calorimeter, the energy emitted by the body (radium nucleus) was not initially in the form of radiant energy; it was in the form of kinetic energy carried by an ejected alpha particle (helium nucleus) and a recoiling radon nucleus.

But Einstein had a knack of getting ahead of himself and ending up in the right place. The mass-energy equivalence relation he obtained from his imagined light-emitting body turned out to be valid also in relation to the kinetic energy of radioactive decay particles.

To see this in relation to Curie and Laborde’s experiment, consider the nuclear reaction equation

Here Q is the mass difference in atomic mass units (u) required to balance the equation:
Mass of Ra = 226.02536 u
Mass of Rn (222.01753) + He (4.00260) = 226.02013 u
Mass difference = Q = 0.00523 u
The kinetic energy equivalent of 1 u is 931.5 MeV
So Q = 4.87 MeV

The kinetic energy is shared by the ejected alpha particle and recoiling radon nucleus. Since the velocities are non-relativistic, this can be calculated on the basis of the momentum conservation law and the classical expression for kinetic energy. Given the masses of the Rn and He nuclei, their respective velocities must be in the ratio 4.00260 to 222.01753. Writing the kinetic energy expression as ½mv.v and recognizing that ½mv has the same magnitude for both nuclei, the kinetic energies of the Rn and He nuclei must also be in the ratio 4.00260 to 222.01753. The kinetic energy carried by the alpha particle is therefore

4.87 x 222.01753/226.02013 = 4.78 MeV

This result has been confirmed by experiment.

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Links to original papers mentioned in this post

Sur la chaleur dégagée spontanément par les sels de radium ; par MM. P. Curie et A. Laborde
Comptes Rendus, Tome 136, janvier – juin 1903

http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3091&I=673&M=tdm

Zur Elektrodynamik bewegter Körper; von A. Einstein
Annalen der Physik 17 (1905) 891-921

https://archive.org/stream/annalenderphysi108unkngoog#page/n1020/mode/2up

Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? von A. Einstein
Annalen der Physik 18 (1905) 639-641

https://archive.org/stream/annalenderphysi143unkngoog#page/n707/mode/2up

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Postscript

In Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Einstein arrived at a general statement on the dependence of inertia on energy (Δm = ΔE/V2, in today’s language E = mc2) from the consideration of a special case. He was deeply uncertain about this result, and returned to it in two further papers in 1906 and 1907, concluding that a general solution was not possible at that time. He had to wait a few years to discover he was right. I include links to these papers for the sake of completeness.

Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie; von A. Einstein
Annalen der Physik 20 (1906) 627-633
http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1906_20_627-633.pdf

Über die vom Relativitätsprinzip geforderte Trägheit der Energie; von A. Einstein
Annalen der Physik 23 (1907) 371-384
http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1907_23_371-384.pdf

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P Mander June 2017