Many real, exothermic systems involve more than one simultaneous reaction. Even when they are chemically independent, interactions must arise through their several responses to the collective generation of heat. A simple and unifying approach is possible to the behaviour of such systems below and up to criticality. It introduces a communal activation energy
E
as the basis for dimensionless quantities (
θ, δ, ϵ
and so on) but otherwise involves only familiar ideas from basic thermal explosion theory. The definition of
E
is
E
=
RT
2
d (In
Z
)/d
T
, where
Z
= Ʃ
Z
i
. Here,
Z
is the rate of energy release per unit volume (the power density) by the whole system and
Z
i
is the contribution of the constituent
i
. This enables us to define and use the conventional dimensionless parameter
δ
for the whole system and for its constituent reactions. We illustrate affairs by considering a pair of concurrent, exothermic reactions; heat is transferred solely by conduction towards the faces (temperature
T
a
) of an infinite slab of thickness 2
a
and conductivity
k
. For a constituent reaction (
i
= 1, 2 here)
δ
i
= (
Ea
2
/
k RT
2
a
)
Z
i
(
T
a
) and for the whole system
δ
=
δ
1
+
δ
2
(+...) for two (or more) reactions. We find that the condition
δ
>
δ
cr
guarantees instability, where
δ
cr
is always less than 0.878. The bounds 0.65 <
δ
cr
< 0.878 are good enough for a substantial range of relative sizes of activation energy 0.2 <
E
1
/
E
2
< 5. We also pursue the problem numerically and present solutions for critical
δ
and critical central temperature excess over the whole composition range for a pair of simultaneous exothermic reactions.
Influence of dimensionality on phase transition in VO2 nanocrystals
