# Differential equation for the rate of unimolecular reaction

Given in my book is the following reaction mechanism for the unimolecular $$\ce{A -> B + C}$$,

$$\ce{A + M ->[k_1] A^* + M} \\ \ce{A^* + M ->[k_2] A + M} \\ \ce{A^* ->[k_3] B + C}$$

Here $$A^*$$ is an $$A$$ molecule with enough vibrational energy to isomerize or decompose. In other words, part of kinetic energy of a bimolecular collision (between $$A$$ and $$M$$) has been used to raise an $$A$$ molecule to higher vibrational energy.

Since $$A^*$$ is never present at very high concentration we can use steady state appoximination to obtain an expression for rate of reaction in terms of $$[A]$$ and $$[M]$$.

$$\dfrac{d[A^*]}{dt} = k_1[A][M] - [A^*](k_2[M] + k_3) = 0\tag{1}$$ $$-\dfrac{d[A]}{dt} = k_3[A^*]\tag{2}$$

I don't understand how the $$(2)$$ equation is obtained, in my opinion it should be $$-\dfrac{d[A]}{dt} = k_1 [A][M]- k_2[A^*][M].$$

I did not understand how we get $$k_3$$ instead of $$k_1$$ and why not $$[M]$$ in the rate law ?

• Your equation for the rate of change of $[A]$ is incorrect. It should be $$\frac{d[A]}{dt}=-k_{1}[A][M]+k_{2}[A^*][M].$$ – Argon Jul 22 '17 at 16:41

To start with, the equation you wrote for the rate of change of $[A]$ is not correct. It should instead be

$$-\frac{d[A]}{dt}=k_{1}[A][M]-k_{2}[A^*][M].$$

Note that $[A]$ is consumed in the first reaction step and produced in the second. Now, rearrange equation $(1)$ to get

$$k_{1}[A][M]=[A^*](k_{2}[M]+k_{3}).$$

Plug in the above equation into the actual expression for $-d[A]/dt$ to get

$$-\frac{d[A]}{dt}=[A^*](k_{2}[M]+k_{3})-k_{2}[A^*][M].$$

This, of course, can be simplified to yield the desired equation $(2)$:

$$-\frac{d[A]}{dt}=k_{3}[A^*].$$

• Sorry, that was a big typo on my part :). – user8277998 Jul 22 '17 at 17:02
• I see you edited your original post. It's still not entirely accurate. Species $A$ is consuned when it reacts with $M$. As such, it should be $k_{1}[A][M]$ for that step if you're describing $-d[A]/dt$. Also, it is not usually suggested to change the content of your post in a way that changes the original question. If you want to update an equation you suggested, you can clearly note the change without overwriting the original. Just for future reference! – Argon Jul 22 '17 at 17:04
• I will take care from next time. Since I think it does not affect the original question in any way I will leave as edited. – user8277998 Jul 22 '17 at 17:13