Chemistry Reference
In-Depth Information
E
ln
k
ln
A
--
RT
(9.9)
which is instantly recognisable as the equation of a straight line (
y
mx
). This means that if the reaction rate,
k
, is determined at a number of
temperatures, a graph of ln
k
against 1/
T
(
T
in kelvin) will yield a straight
line of slope
c
E
/
R
that intersects the vertical axis at ln
A
. The activation
energy,
E
, for the reaction may be determined from data like these.
Even more usefully, if the reaction rate
k
1
is determined at a tempera-
ture
T
1
, and the rate
k
2
is determined at a temperature
T
2
, then the two
forms of equation (9.9) may be subtracted to give
(
--
--
)
k
2
E
11
ln --
-
k
1
RT
2
T
1
(9.10)
This useful equation may be used to predict the reaction rate at any temper-
ature once
k
1
and
E
are known for temperature
T
1
. This type of calculation
is extremely important in pharmaceutical science since it is used to predict
shelf-life for medicines. Once a medicine has been manufactured, it is stored
under high-stress conditions (e.g. at elevated temperature, high humidity,
under strong lighting, etc.), the rates of decomposition are measured and
the activation energy is calculated. From these data, the value of
k
may be
predicted and the likely shelf-life for the medicine can be calculated for
room temperature (25
C). Another useful
point to notice is that since
k
enters into the graphs as ln
k
, and into the
equations as a ratio, any physical quantity that is proportional to
k
, such as
the actual reaction rates at fixed concentrations of reactants, may be used
in the equation instead of
k
.
Calculations using
Arrhenius plots
, such as those described above, are
carried out in the pharmaceutical industry every day. It should be made
clear, however, that they involve a number of assumptions. It is assumed
that the linearity of the graph obtained from equation (9.9) extends to room
temperature, or, mathematically, that
A
and
E
are independent of tempera-
ture. If the line cannot be extrapolated to room temperature, shelf-life
predictions are invalid. Second, it is assumed that the same chemical reac-
tion is occurring with decomposition at high temperature as at low temper-
ature. This is usually the case, but until proven it remains an assumption in
most calculations.
C) or refrigerator temperature (4
