Q₁₀ – rubber-stamp ‘thinking’.

‘Reaction rates double for every 10^oC rise in temperature’ – I expect that you’ve heard this statement. It is, for some reason, very popular with those studying biology; it is there that you find Q₁₀, the ratio of reaction rate at temperature T to that at T + 10. The doubling is usually offered as a fact of chemical (or biochemical) life; such a pity, then, that it isn’t true.

As originally promulgated in university texts it was ‘Reaction rates roughly double or triple…’; but in the (ever-increasing) sanitising found in A level texts a number of fairly important words were jettisoned. Like ‘roughly’, ‘double or triple’. (Yes, guilty; if you’ve read my Philip Allan texts there are numerous omissions there, and no, I don’t like that.) A number of other important caveats disappeared too, so let’s examine the ideas further.

The Arrhenius equation.

The rate of a chemical reaction is described by an empirically-determined equation of the form

rate = k [A]^x [B]^y

where k is the rate constant, [A] and [B] are the molar concentrations of the species A and B, and x and y are the orders of reaction with respect to A and to B. There may, of course, be only A, or there may be more than two species involved.

The concentrations do not depend significantly upon temperature; the term that changes with a change in temperature  is k. That is why all experiments on reaction rates must be done at constant temperature. The temperature-dependence of k is described by the Arrhenius equation (S Arrhenius, 1889,  building on earlier work of J H van’t Hoff (1884)):

ln (k₂/k₁) = (E_a/R) (1/T₁ – 1/T₂)

where k₁ and k₂ are the rate constants at (absolute, i.e. Kelvin) temperatures T₁ and T₂, E_a is the activation energy for the reaction in J mol^-1, and R is the gas constant, 8.314 J K^-1 mol^-1.

The term that is Q₁₀ is k₂/k₁, and this is how I shall refer to it from now on.

 

Two important points

What is not usually stated in A level kinetics:

• k₂/k₁ depends on the value of E_a;
• k₂/k₁ depends on temperature – Q₁₀ is not constant. Increasing the temperature decreases k₂/k₁ .

 

A few calculations.

A value for E_a:
Firstly we shall find the value of E_a that would give a doubling of reaction rate between 0^oC and 10^oC, i.e. for k₂/k₁ = 2. Then with this value of E_a we shall see what the effect is on k₂/k₁ at two different 10^oC ranges. Remember that the temperatures in the Arrhenius equation must be in K.

For the calculation of E_a we take k₂/k₁ = 2, T₁ = 273 K, T₂ = 283 K, R = 8.314 J K^-1 mol^-1:

Arrhenius:	ln (k2/k1) = (Ea/R) (1/T1 – 1/T2)
Therefore:	ln (k2/k1) / (1/T1 – 1/T2)  = Ea/R
Substituting the values given:	(8.314 x ln 2) / (1/273 – 1/283)  = Ea
thus	(8.314 x 0.693) / (1/273 – 1/283) = Ea

and it is only arithmetic to show that E_a = 44,500 J mol^-1 = 44.5 kJ mol^-1.

 

The effect of increasing the temperature on k₂/k₁:
If we now use this value of E_a and find the value of k₂/k₁ at different temperatures for this hypothetical reaction, i.e. evaluate

ln (k₂/k₁) = (44,500/8.314) (1/T₁ – 1/T₂)

more arithmetic shows the following (try it):

	T1/K	T2/K	k2/k1
	273	283	2.00
	373	383	1.45
	473	483	1.26

Indeed the value of k₂/k₁, our old – but now unreliable – friend, Q₁₀, falls as the temperature increases.

 

Would you need this in an A level exam? No. But at least you won’t fall into rubber-stamp thinking on this topic again.

Lastly – the rate of a reaction seems to be determined by E_a. This is true to a good approximation (unlike Q₁₀); but in fact it is the free energy of activation that is the important factor. But that’s for another day. E_a is pretty good until then.