Posts Tagged ‘reaction rate’

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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

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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

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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:

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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

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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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The Arrhenius equation explains why chemical reactions generally go much faster when you heat them up. The equation was actually first given by the Dutch physical chemist JH van ‘t Hoff in 1884, but it was the Swedish physical chemist Svante Arrhenius (pictured above) who in 1889 interpreted the equation in terms of activation energy, thereby opening up an important new dimension to the study of reaction rates.

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Temperature and reaction rate

The systematic study of chemical kinetics can be said to have begun in 1850 with Ludwig Wilhelmy’s pioneering work on the kinetics of sucrose inversion. Right from the start, it was realized that reaction rates showed an appreciable dependence on temperature, but it took four decades before real progress was made towards quantitative understanding of the phenomenon.

In 1889, Arrhenius penned a classic paper in which he considered eight sets of published data on the effect of temperature on reaction rates. In each case he showed that the rate constant could be represented as an explicit function of the absolute temperature:

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where both A and C are constants for the particular reaction taking place at temperature T. In his paper, Arrhenius listed the eight sets of published data together with the equations put forward by their respective authors to express the temperature dependence of the rate constant. In one case, the equation – stated in logarithmic form – was identical to that proposed by Arrhenius

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where T is the absolute temperature and a and b are constants. This equation was published five years before Arrhenius’ paper in a book entitled Études de Dynamique Chimique. The author was J. H. van ‘t Hoff.

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Dynamic equilibrium

In the Études of 1884, van ‘t Hoff compiled a contemporary encyclopædia of chemical kinetics. It is an extraordinary work, containing all that was previously known as well as a great deal that was entirely new. At the start of the section on chemical equilibrium he states (without proof) the thermodynamic equation, sometimes called the van ‘t Hoff isochore, which quantifies the displacement of equilibrium with temperature. In modern notation it reads:

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where Kc is the equilibrium constant expressed in terms of concentrations, ΔH is the heat of reaction and T is the absolute temperature. In a footnote to this famous and thermodynamically exact equation, van ‘t Hoff builds a bridge from thermodynamics to kinetics by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions

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Substituting this quotient in the original equation leads immediately to

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van ‘t Hoff then argues that the rate constants will be influenced by two different energy terms E1 and E-1, and splits the above into two equations

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where the two energies are such that E1 – E-1 = ΔH

In the Études, van ‘t Hoff recognized that ΔH might or might not be temperature independent, and considered both possibilities. In the former case, he could integrate the equation to give the solution

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From a starting point in thermodynamics, van ‘t Hoff engineered this kinetic equation through a characteristically self-assured thought process. And it was this equation that the equally self-assured Svante Arrhenius seized upon for his own purposes, expanding its application to explain the results of other researchers, and enriching it with his own idea for how the equation should be interpreted.

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Activation energy

It is a well-known result of the kinetic theory of gases that the average kinetic energy per mole of gas (EK) is given by

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Since the only variable on the RHS is the absolute temperature T, we can conclude that doubling the temperature will double the average kinetic energy of the molecules. This set Arrhenius thinking, because the eight sets of published data in his 1889 paper showed that the effect of temperature on the rates of chemical processes was generally much too large to be explained on the basis of how temperature affects the average kinetic energy of the molecules.

The clue to solving this mystery was provided by James Clerk Maxwell, who in 1860 had worked out the distribution of molecular velocities from the laws of probability. Maxwell’s distribution law enables the fraction of molecules possessing a kinetic energy exceeding some arbitrary value E to be calculated.

It is convenient to consider the distribution of molecular velocities in two dimensions instead of three, since the distribution law so obtained gives very similar results and is much simpler to apply. At absolute temperature T, the proportion of molecules for which the kinetic energy exceeds E is given by

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where n is the number of molecules with kinetic energy greater than E, and N is the total number of molecules. This is exactly the exponential expression which occurs in the velocity constant equation derived by van ‘t Hoff from thermodynamic principles, which Arrhenius showed could be fitted to temperature dependence data from several published sources.

Compared with the average kinetic energy calculation, this exponential expression yields very different results. At 1000K, the fraction of molecules having a greater energy than, say, 80 KJ is 0.0000662, while at 2000K the fraction is 0.00814. So the temperature change which doubles the number of molecules with the average energy will increase the number of molecules with E > 80 KJ by a factor of more than a hundred.

Here was the clue Arrhenius was seeking to explain why increased temperature had such a marked effect on reaction rate. He reasoned it was because molecules needed sufficiently more energy than the average – the activation energy E – to undergo reaction, and that the fraction of these molecules in the reaction mix was an exponential function of temperature.

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The meaning of A

But back to the Arrhenius equation

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A clue to the proper meaning of A is to note that e^(–E/RT) is dimensionless. The units of A are therefore the same as the units of k. But what are the units of k?

The answer depends on whether one’s interest area is kinetics or thermodynamics. In kinetics, the concentration of chemical species present at equilibrium is generally expressed as molar concentration, giving rise to a range of possibilities for the units of the velocity constant k.

In thermodynamics however, the dimensions of k are uniform. This is because the chemical potential of reactants and products in any arbitrarily chosen state is expressed in terms of activity a, which is defined as a ratio in relation to a standard state and is therefore dimensionless.

When the arbitrarily chosen conditions represent those for equilibrium, the equilibrium constant K is expressed in terms of reactant (aA + bB + …) and product (mM + nN + …) activities

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where the subscript e indicates that the activities are those for the system at equilibrium.

As students we often substitute molar concentrations for activities, since in many situations the activity of a chemical species is approximately proportional to its concentration. But if an equation is arrived at from consideration of the thermodynamic equilibrium constant K – as the Arrhenius equation was – it is important to remember that the associated concentration terms are strictly dimensionless and so the reaction rate, and therefore the velocity constant k, and therefore A, has the units of frequency (t^-1).

OK, so back again to the Arrhenius equation

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We have determined the dimensions of A; now let us turn our attention to the role of the dimensionless exponential factor. The values this term may take range between 0 and 1, and specifically when E = 0, e^(–E/RT) = 1. This allows us to assign a physical meaning to A since when E = 0, A = k. We can think of A as the velocity constant when the activation energy is zero – in other words when each collision between reactant molecules results in a reaction taking place.

Since there are zillions of molecular collisions taking place every second just at room temperature, any reaction in these circumstances would be uber-explosive. So the exponential term can be seen as a modifier of A whose value reflects the range of reaction velocity from extremely slow at one end of the scale (high E/low T) to extremely fast at the other (low E/high T).

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P Mander September 2016