A thermal explosion occurs when a chemical system undergoes an exothermic reaction during which insufficient heat is removed from the system so that the reaction process becomes self heating. Since the reaction rate increases with temperature, the reaction rapidly runs away and the system explodes[^1] Consider a schematic reaction mechanism focusing only on the intermediates or radicals ($R$). All intermediates are denote by the same letter. $\ce{X->[k_1]}R\tag{chain initiation}$ $\ce{R->[k_2]}\alpha R\tag{chain branching}$ More radicals are formed if $\alpha > 1$ $\ce{R->[k_3]}P + R\tag{chain propagation}$ In this reaction a radical is consumed to produce a product and another radical. It is similar to $H_2 +OH$ reaction in the [[H2-O2 reaction]] system which is an important exothermic reaction. $\ce{R->[k_4]}S\tag{chain termination on surface}$ $\ce{R->[k_5]}X\tag{chain termination on gas}$ To consider the conditions at which this reaction will be explosive, let us first write the rate of production of the product $\frac{dC_p}{dt}=k_3C_R... = f_pC_R$ To write out a [[Steady State Approximation]] for the radical R just prior to an explosion, let us write out individual components of $\frac{dC_R}{dt}$ From chain initiation step $\omega_i = \frac{dC_R}{dt}$ From chain branching $\omega_b = \frac{dC_R}{dt}=(\alpha-1)k_2C_R....=f_b(\alpha-1)C_R$ $f_b$ is a simplified notation for the rate equation since this reaction is missing some stable species. For chain termination ( surface and gas based termination steps) $\omega_s = -\frac{dC_R}{dt}=f_sC_R$ $\omega_g = -\frac{dC_R}{dt}=f_gC_R$ Rate of production of intermediates is $\frac{dC_R}{dt} = \omega_i + f_b(\alpha -1)C_R - f_sC_R - f_gC_R$ Applying steady state approximation $\frac{dC_R}{dt}=0$ $ C_R=\frac{\omega_i}{(f_s+f_g )-f_b(\alpha-1)}$ Rate of production of product $\frac{dC_p}{dt}=f_pC_R = \frac{f_pw_i}{(f_s+f_g)-f_b(\alpha-1)}$ First two terms in the denominator are rate of termination. Last term is rate of branching and the numerator is rate of initiation of intermediates Chemical explosion occurs when denominator is zero and rate of production of product increases to a very high value. This only happens when $\alpha > 1$ . i.e intermediates get branched. At low pressures $f_s$ predominates over $f_g$ and vice versa at high pressures. Note that the reaction rate of the chain initiation steps determines the rate of production of P but does not affect the condition of explosion. ## 1. Explosion limit Explosion limits are pressure temperature boundaries for a specific [[Mixture Ratio]] that separates the regions of slow and fast reaction. i.e explosive and non-explosive conditions. - at low pressure we as we increase pressure we can reach a limit where the radical generation just exceeds termination. This is called lower (1st) explosion limit - As pressure increases further $f_g$ increases and $f_g$ decreases. Eventually $f_g$ increases a lot and termination catches up with production and we get upper (2nd) explosion limit At lower explosion limit, the intermediates are terminated at surface more and at upper the intermediates are terminated in gas more ![[explosion_limit.png]] [^1]: Combustion, Glassman