Posts Tagged ‘Svante Arrhenius’

New kids on the science block: Wilhelm Ostwald and Svante Arrhenius in the late 1800s. Photo credit wikimedia.org

New kids on the science block: Wilhelm Ostwald and Svante Arrhenius in the 1800s. Photo credit wikimedia.org

For a binary solution in equilibrium with its own vapor, the Phase Rule tells us that the system possesses two degrees  of freedom. So if temperature is held constant, there will be a relation at equilibrium between pressure and composition, corresponding to the observed reduction of vapor pressure by solutes. And if the solution is in equilibrium with solid solvent, there will be a relation at constant pressure between temperature and composition, corresponding to the observed depression of freezing point by solutes.

Back in the 1870s, before the thermodynamics of colligative properties had been placed on a theoretical footing, these relations had been discovered in Grenoble, France, by physicist François-Marie Raoult in connexion with his work on solutions, which occupied the last two decades of his life.

François-Marie Raoult (1830-1901), whose work on the freezing-point depression of solutes had an  unexpected influence on the history of physical chemistry. Raoult's Law is named after him. Photo credit wikimedia.org

François-Marie Raoult (1830-1901), whose work on the freezing-point depression of solutes had an unexpected influence on the history of physical chemistry. Raoult’s Law is named after him. Photo credit wikimedia.org

Raoult’s first paper describing solute-mediated depression of freezing point was published in 1878. It was a fertile area of study which Raoult appears to have had to himself, and not surprisingly he was the first to discover empirical relationships between quantities. One such relation he found was between the depression of freezing point and the molality of the solute:

Freezing-point depression = ΔTfp = Kfp x msolute

where Kfp is the cryoscopic constant for the solvent in degrees per molal. Raoult conducted many measurements of solutes in various solvents, and over time built up a body of data in support of the above relation. But in 1884, Raoult discovered a curious exception to the rule when he used sodium chloride as a solute – the effect on the freezing point of water was nearly twice as large as it should be. There was something peculiar about the behavior of common salt in solution.

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Svante Arrhenius in 1884, the year he submitted his doctoral dissertation on electrolytic conductivity that would ultimately lead to a Nobel Prize. The adjudicating committee at Uppsala University gave him the lowest possible grade for his work. Photo credit climate4you.com

Svante Arrhenius in 1884, the year he submitted his doctoral dissertation on electrolytic conductivity that would ultimately lead to a Nobel Prize. The adjudicating committee at Uppsala University gave him the lowest possible grade for his work. Photo credit climate4you.com

In the same year that Raoult in France discovered the anomalous colligative actions of sodium chloride as a solute, a 24-year-old student in Sweden named Svante Arrhenius submitted to Uppsala University a 150-page doctoral dissertation on electrolytic conductivity in aqueous solution. In it, he advanced the thesis that the conductivity of solutions of salts in water was due to the dissociation of the salt into oppositely charged particles, to which Faraday had given the name “ions” fifty years earlier. But while Faraday believed that ions were only produced during electrolysis, Arrhenius asserted that even in the absence of an electric current, solutions of salts contained ions. And he went as far as proposing that chemical reactions in solution were reactions between ions.

The professors at Uppsala were incredulous, and duly gave Arrhenius and his far-fetched ideas the minimum mark. But they underestimated the self-belief and resolve of their young student. Arrhenius followed up on their rebuttal by sending his dissertation to cutting-edge figures in Europe such as Wilhelm Ostwald and Jacobus Henricus van ‘t Hoff, who were actively developing the new science of physical chemistry. They were far more impressed, and following the award of a travel grant from the Swedish Academy of Sciences, Arrhenius was able to study with Ostwald in Riga, Kohlrausch in Germany, Boltzmann in Austria, and van ‘t Hoff in Amsterdam.

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J H "Haircut" van't Hoff (1852-1911), a founding figure in physical chemistry, a pioneer in chemical thermodynamics, and the first winner of the Nobel Prize in Chemistry (1901). Photo credit nobelpreis.org

J H “Haircut” van ‘t Hoff (1852-1911), a founding figure in physical chemistry, a pioneer in chemical thermodynamics, and the first winner of the Nobel Prize in Chemistry (1901). Photo credit nobelpreis.org

Raoult’s discovery of the anomalous effect of common salt on the freezing point of water attracted the interest of JH van ‘t Hoff, who in 1887 subjected this peculiar result to detailed study. Investigating a number of ‘misbehaving’ salts, van ‘t Hoff found in each instance that the ratio of the measured freezing-point depression to the expected value approached a whole number as the solutions became increasingly dilute.

In the case of common salt, sodium chloride, the limiting ratio was 2. For sodium sulfate on the other hand, the ratio was 3, and for aluminum sulfate it was 5.

At the time when van ‘t Hoff found these whole number relations, Svante Arrhenius just happened to be visiting Amsterdam on his study tour. Arrhenius saw in these results the affirmation of his doctoral thesis, and could immediately supply the explanation. In aqueous solution, sodium chloride dissociates into sodium and chloride ions, so there are really two sets of solutes. Thus the total molality of the fully dissociated (ionised) solute will be double its undissociated value, and the freezing-point depression will be twice the expected amount.

By parallel reasoning, sodium sulfate dissociates into 3 aqueous ions (2 sodium ions and one sulfate ion) and in the case of aluminum sulfate there are 5 ions (2 aluminum ions and 3 sulfate ions).

It was compelling logic; the truth of Arrhenius’ thesis of ions in solution, and its implications for the understanding of chemical reactions and bonding, could no longer be denied.

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François-Marie Raoult’s work on solutions, and his discovery of the uncommon effect of common salt on the depression of freezing point, marked the start of a chain of circumstances that directly contributed to the founding of physical chemistry as a modern science. Not only did it provide affirmation of electrolytic dissociation and the existence of ions in solution, it also brought together the bright minds of Svante Arrhenius and Jacobus Henricus van ‘t Hoff, who with Wilhelm Ostwald were to propel physical chemistry into the modern age.

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The Nobel Prize in Chemistry 1901 was awarded to JH van ‘t Hoff “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”.

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The Nobel Prize in Chemistry 1903 was awarded to Svante Arrhenius “in recognition of the extraordinary services he has rendered to the advancement of chemistry by his electrolytic theory of dissociation”.

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The Nobel Prize in Chemistry 1909 was awarded to Wilhelm Ostwald “in recognition of his work on catalysis and for his investigations into the fundamental principles governing chemical equilibria and rates of reaction”.

photo credits: nobelprize.org

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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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I have always thought that calling the constant A the ‘pre-exponential factor’ is a singularly pointless label. One could equally write the equation as

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and call it the ‘post-exponential factor’. The position of A in relation to the exponential factor has no relevance.

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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