How does a termolecular elementary reaction happen under the law of mass action? My physical chemistry textbook only says details about bimolecular and unimolecular reactions with collision theory. I have searched some documents, but almost all of them didn't elaborate anything crucial, I mean, anything quantitative about its rate constant.

There is only one mentioning a result done by R.C.Tolman. It says we need to introduce a variable $\delta$ to measure the distance between the molecules (To be honest, I'm not sure what he really means). Anyway, the formula mentioned in the paper is: $$k=4N^2_A\delta\sigma_{AB}\sigma_{BC}\sqrt{\frac{8RT}{\pi}}(\frac{1}{\sqrt{\mu_{AB}}}+\frac{1}{\sqrt{\mu_{BC}}})Pe^{-\frac{E_a}{RT}}$$ in which
$N_A$ is Avogadro number;
$\sigma_{AB}=\pi(r_A+r_B)^2$ and $\sigma_{BC}=\pi(r_B+r_C)^2$ are collision cross sections;
$R$ is universial gas constant;
$T$ is the temperature;
$\mu_{AB}=\frac{1}{\frac{1}{M(A)}\ +\frac{1}{M(B)}}$ and $\mu_{BC}=\frac{1}{\frac{1}{M(B)}\ +\frac{1}{M(C)}}$ are reduced molar masses;
$P$ is a probability factor;
$E_a$ is the activiation energy.

So $v=kc(A)c(B)c(C)$, according to the law of mass action.

But why? I don't have any idea to demonstrate it, and I cannot find the original papers of Tolman. I'd appreciate it if anyone professional at theory could lend me a hand. Any helpful or valuable thought is welcome.


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