Svante Arrhenius (1859-1927)

On the Reaction Velocity of the Inversion of Cane Sugar by Acids

...

In his notable book Studies in Chemical Dynamicsvan't Hoff gives a theoretically-based formulation of the influence of temperature on the rate of reaction. Thus, in the case of an equilibrium between four substances A, B, E and D (of which A and B can be formed from E and D, and vice versa) we have

(2)[1]

k1CACB = k2CECD

in which equation C_A, etc., signify the concentrations of the substances A, etc. This relationship also says that the amounts of E and D produced in unit time from A and B are equal to the amounts of A and B formed in the same time from E and D. That is to say, the first quantity is represented by the term k₁C_AC_B where k₁ is the rate constant (the specific reaction velocity): in the same way k₂C_EC_D represents the quantity of A and B newly formed in unit time. For the system at equilibrium van't Hoff gives

(3)	d lognatk1/dT - d lognatk2/dT = q/2T2

an equation that can be derived from the mechanical theory of heat and van't Hoff's law. In this equation T is the absolute temperature and q the amount of heat (in calories) which is set free when 1 gram-molecule each of A and B are converted into E and D.

The two velocity constants for two reciprocal reactions are thus related by this equation (3). "This equation", says van't Hoff, "does not provide the desired relationship between the value of k (the reaction velocity) and the temperature; it shows, however, that this relationship has the following form:

(4)	d lognatk/dT = A/T2 + B."

In equation (4) A and B are treated as constants. It is, however, easily seen that B can be any function, F(T) of the temperature, and equation (3) can only be obtained from (4) if the F(T) belonging to the two reciprocal reactions are the same. I am of the opinion, with regard to this, that there is no definite solution of this problem, since F(T) can be anything at all.

It is not possible to proceed further without introducing a new hypothesis, which is in a certain sense a paraphrase of the observed facts. In order to reach such a hypothesis, which will be used throughout, the following considerations must be taken into account. The influence of temperature on the specific reaction rate is very large in that, at ordinary temperatures, the rate increases by 10 to 15 per cent for each one-degree rise in temperature. It cannot be assumed, therefore, that the increasing reaction velocity comes from the increasing frequency of collisions of the reacting molecules. According to the kinetic theory of gases, the velocity of the gas molecules changes only by about 1/6 per cent of its value for each one-degree rise in temperature and the frequency of collisions increases in the same ratio. It is difficult to say with certainty how large is the corresponding change in the case of liquids, but it is suggested that it is similar to that in gases. Even if the assumption that the velocity of the solute molecules changes by 1/6 per cent per degree is not accepted, it must at least be agreed that the difference between this value and the observed 10 to 15 per cent per degree is much too large for it to be assumed that the increase in the number of collisions of the reacting molecules is the reason for the increase in the reaction velocity with the temperature. In the same way it can not very well be assumed that the decrease of viscosity with increasing temperature is the reason for the phenomenon under consideration, for the viscosity decreases[2] by only some 2 per cent per degree. Apart from this, the increase in reaction velocity differs in a very important way from the increase with temperature in most physical quantities. Thus the change of these quantities is approximately equal for an increase in temperature of 1°, even for very different values of the temperature (e.g. 0° and 50°). On the other hand the increase (in absolute value) per degree of the reaction velocity is perhaps twice as large at 6° as at 0°, at 12° four times as large as at 0°, at 30° perhaps thirty (2⁵) times as large as at 0°, etc. This fact indicates that the increase in reaction velocity with the temperature cannot be explained in terms of the change in physical properties of the reacting substances. There remains one way out. A similar extraordinarily large change in specific reaction velocity (k) is to be found in another sphere. This is the change in rates of reactions which are brought about by weak bases or acids, by the addition of insignificant amounts of neutral salts (e.g. by addition of NH₄Cl to NH₃ acting upon ethyl acetate). In this case the facts are explained thus: although the amount of the actual base (NH₃) or acid does not undergo a change on the addition of neutral salt (NH₄Cl ), by its addition there is a very rapid removal of free OH^- ions, which actually cause the reaction. Can it not be assumed that the reaction velocity (for example the inversion of cane sugar) also originates in this way, that the amounts of the actual reacting substances rapidly increase with temperature? One actual reacting substance is the H⁺ ion of the acid employed. The concentration of this ion (for a constant amount of acid) changes only very little with temperature for strong acids like HCl (HNO₃ or HBr), increasing (by something like 0.05 per cent per degree at 25°) when the temperature rises. The explanation therefore can not lie here. It must therefore be assumed, to be consistent, that the other actual reacting substance is not cane sugar, since the amount of sugar does not change with temperature, but is another hypothetical substance which is regenerated from cane sugar as soon as it is removed through the inversion.

This hypothetical substance, which we call "active cane sugar", must rapidly increase in quantity with increasing temperature (by about 12 per cent per degree) at the expense of the ordinary "inactive" cane sugar. It must be formed from cane sugar at the expense of heat (of q calories). It is further known that cane sugar in solution at high temperatures has no important properties that it does not possess at lower temperatures, as would probably be the case if the absolute amount of the hypothetical active cane sugar were increased considerably; it may therefore be assumed that the absolute amount (although it increases by some 12 per cent per degree) is still, at the highest temperatures employed, exceedingly small in comparison with the amount of the "inactive cane sugar". The amount of the latter is therefore independent of temperature. Furthermore at constant temperature the reaction velocity is approximately proportional to the amount of cane sugar. Under these conditions (constant temperature), therefore, the amount of "active cane sugar", M_a, is approximately proportional to the amount of inactive cane sugar, M_i. The equilibrium condition is thus:

(5)

Ma = k Mi

The form of this equation shows us that a molecule of "active cane sugar" is formed from a molecule of inactive cane sugar either by a displacement of the atoms or by addition of water. In the equilibrium equation there is always on each side the product of the amounts (concentrations) of the reacting substances. Since, however, the amount of water is constant (the amount of cane sugar varying) the water can also be active in the reaction without its concentration appearing in the equation. Thus for the constant k (or what is the same thing M_a/M_i) we have the equation

(6)	d lognatk/dT = q/2T2

which integrates to

(7)	kT1 = kT0exp[q(T1 - T0)/2T0T1].

If it is further assumed that the reaction velocity is proportional only to the amounts of the reacting substances, a similar equation can be written down for the reaction velocity corresponding to a constant amount of reacting acid:

ρT1 = ρT0exp[q(T1 - T0)/2T0T1].

We have given this equation above as equation (1) and have found it to be in agreement with experiment.

[2]The original paper says "increases" (nimmt zu) but this must be a mistake. [Back & Laidler]