Q_{10} – rubberstamp ‘thinking’.
‘Reaction rates double for every 10^{o}C 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 Q_{10}, 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 (everincreasing) 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 empiricallydetermined 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 temperaturedependence of
k is
described by the Arrhenius equation (S Arrhenius, 1889, building on earlier work of
J H van’t Hoff (1884)):
ln (k_{2}/k_{1}) = (E_{a}/R)
(1/T_{1}
– 1/T_{2})
where k_{1} and
k_{2} are the rate constants at
(absolute, i.e. Kelvin) temperatures T_{1} and
T_{2},
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 Q_{10} is k_{2}/k_{1}, and this is how
I shall refer to it from now on.
Two important points
What is not usually stated in A level kinetics:
 k_{2}/k_{1} depends on the value of
E_{a};
 k_{2}/k_{1} depends on temperature – Q_{10} is not
constant. Increasing the temperature decreases k_{2}/k_{1}
.
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^{o}C and 10^{o}C, i.e. for k_{2}/k_{1}
= 2. Then with this value of E_{a} we shall see what the effect is on
k_{2}/k_{1}
at two different 10^{o}C ranges. Remember that the temperatures in the Arrhenius
equation must be in K.
For the calculation of E_{a} we take
k_{2}/k_{1}
= 2, T_{1} =
273 K, T_{2} = 283 K,
R = 8.314 J K^{1}
mol^{1}:
Arrhenius: 
ln (k_{2}/k_{1})
= (E_{a}/R) (1/T_{1} – 1/T_{2}) 

Therefore: 
ln (k_{2}/k_{1})
/ (1/T_{1}
– 1/T_{2}) =
E_{a}/R 

Substituting the values given: 
(8.314
x ln 2) / (1/273
– 1/283) = E_{a} 

thus 
(8.314
x 0.693) / (1/273
– 1/283) = E_{a} 

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
k_{2}/k_{1}:
If we now use this value of E_{a} and find the value of
k_{2}/k_{1}
at different temperatures for this hypothetical reaction, i.e. evaluate
ln (k_{2}/k_{1}) = (44,500/8.314) (1/T_{1}
– 1/T_{2})
more arithmetic shows the following (try it):

T_{1}/K

T_{2}/K

k_{2}/k_{1} 


273 
283 
2.00 


373 
383 
1.45 


473 
483 
1.26 

Indeed the value of k_{2}/k_{1}, our old – but now
unreliable – friend, Q_{10}, falls as the temperature increases.
Would you need this in an A level exam? No. But at least you won’t fall into
rubberstamp 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 Q_{10}); 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.
