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205
Another common use of the Arrhenius equation is just to calculate the rate
constant (k
2
) at another temperature (T
2
) given that it is known (k
1
) at a
specific temperature (T
1
). Thus:
k
2
=
Ae
−
E
A
RT
2
⎛
⎝⎜
⎞
⎠⎟
k
1
=
Ae
−
E
A
RT
1
⎛
⎝⎜
⎞
⎠⎟
⇓
k
2
k
1
=
e
−
E
A
RT
2
⎛
⎝⎜
⎞
⎠⎟ − −
E
A
RT
1
⎛
⎝⎜
⎞
⎠⎟
=
e
−
E
A
R
⎛
⎝⎜
⎞
⎠⎟
1
T
1
−
1
T
2
⎛
⎝⎜
⎞
⎠⎟
To take an example: Provided E
A
equals 120 kJ mol
-1
K
-1
, how much faster
will a reaction go at 100
°
C than at 80
°
C? Inserting we obtain:
k
100
k
80
==
e
−
120 kJ mol
−
1
K
−
1
8.314 J mol
−
1
K
−
1
⎛
⎝⎜
⎞
⎠⎟
1
(273
+
80)
−
1
(273
+
100)
⎛
⎝⎜
⎞
⎠⎟
=
8.9
Thus, the reaction rate is 8.9 times higher at 100
°
C compared to 80
°
C.
If two reactions have the same activation energies then the ratio between the
rate constants are independent of the temperature. Thus, the result is the
same at low temperature as for high temperature, less the fact you have to
wait longer at low temperatures to obtain the same extent of reaction, for
example 50% conversion.
Another example: You find that it takes 48 hours (t
1
) to degrade a
polysaccharide to obtain the desired molecular weight (M
w,2
) starting from the
initial molecular weight (M
w,0
) and using a specific temperature (T
1
). You
wonder how much faster it goes by elevating the temperature to T
2
. We first
use the general equation for random depolymerization: