# Explosive reaction based on reaction rate

### Question

We have the following chemical reactions:

\begin{align} \ce{A + B^* &-> C + D^*}\tag{1}\\ \ce{D^* &-> B^* + B^*}\tag{2}\\ \ce{B^* &-> B}\tag{3} \end{align}

If the initial concentrations are $$[\ce{A}] = \pu{100e-6 mol cm-3}$$ and $$[\ce{B}] = \pu{1e-6 mol cm-3}$$, is the system explosive? We know that $$k_1 = \pu{1e6 cm3 mol-1 s-1},$$ $$k_2 = \pu{50 s-1}$$ and $$k_3 = \pu{25 s-1}.$$

### My solution

So I found the rate expressions for $$\ce{A}$$ and $$\ce{B},$$ by applying steady-state approximation (SSA) on $$[\ce{D^*}].$$ I get the following expressions:

\begin{align} \frac{\mathrm d[\ce{A}]}{\mathrm dt} &= -k_1 \cdot [\ce{A}] \cdot [\ce{B^*}]\\ \frac{\mathrm d[\ce{B}]}{\mathrm dt} &= (k_1 \cdot [\ce{A}] - k_3) \cdot [\ce{B^*}] \end{align}

Upon entering the numbers into the above expressions, I get the following numbers:

\begin{align} \frac{\mathrm d[\ce{A}]}{\mathrm dt} &= \pu{-\frac{1}{10000} mol cm-3 s-1}\\ \frac{\mathrm d[\ce{B}]}{\mathrm dt} &= \pu{\frac{3}{40000} mol cm-3 s-1} \end{align}

My conclusion was that the system is not explosive since the rate of formation of $$\ce{B}$$ is quite low. Since if the rate is too low, then the reaction can not be considered explosive.

But the problem with my explanation is that I would like to have a reference number. Is there a reference such as f.e. rate higher than $$10$$ is considered explosive, etc? I have tried looking everywhere but could not find any 'value.' Could anyone provide some insight into this:

• Are my explanation and conclusion correct?

• Is there a reference value for explosive reactions?

I have an additional question. What is the connection between an explosive system and the steady-state approximation? If the system is explosive, can I still use SSA or does it become invalid?

• The steady-state approximation only requires that the concentration of intermediates is low. As long as that is the case, it is valid. – Zhe Sep 27 '19 at 0:08
• @Zhe I'm not sure that what you write is necessarily true. What is important is that the rate of change is low, i.e. assumed to be zero, rather than the amount being low. – porphyrin Sep 27 '19 at 9:33
• Shouldn't be that if B• is formed faster than C + D than it is esplosive? Can't read the equation but my comment is general (and might be wrong). – Alchimista Sep 27 '19 at 10:40
• Taking your equation $d[B]^*/dt\cdots$ and integrating gives $[B^*]=[B_0^*]\exp((k_1A-k_3)t)$ and if $k_1A \gt k_3$ the exponential will grow and then one might expect an explosion. – porphyrin Sep 27 '19 at 11:10
• @porphyrin The steady state approximation argues that the rate of change is near zero by arguing that the concentration of intermediate is low. If it's always low, it can't be changing by much. – Zhe Sep 27 '19 at 12:00