Why is reaction rate proportional to the order of reactants? [duplicate]

I know that for some elementary reaction $$m\text{A}+n\text{B}\rightarrow\text{C}$$, the rate of reaction is given by $$-\frac{\text{dA}}{\text{dt}}=k[\text{A}]^m[\text{B}]^n$$.

Now if we consider the case $$m=1$$ and $$n=1$$ and where A and B are gas molecules, then, from collision theory we know that the rate of reaction can also be written as collision frequency $$\cdot$$ fraction of molecules with sufficient energy $$\cdot$$ steric coefficient: $$\sigma\left(\frac{8\text{RT}}{\pi\mu}\right)^{1/2}\text{N}_\text{A}^2[\text{A}][\text{B}]\cdot e^{\frac{-E_{min}}{\text{RT}}}\cdot P$$

Where $$\mu=\frac{\text{M}_A\text{M}_{B}}{\text{M}_A+\text{M}_B}$$ ($$\text{M}_A$$ and $$\text{M}_B$$ are molar masses of A and B, respectively), $$N_\text{A}$$ is Avogadro's number, and $$P$$ is the steric coefficient. With this equation, if we let $$k=\sigma\left(\frac{8\text{RT}}{\pi\mu}\right)^{1/2}\text{N}_\text{A}^2\cdot e^{\frac{-E_{min}}{\text{RT}}}\cdot P$$ then we get the general equation for the rate of reaction, rate = $$k[A][B]$$, and we also understand the relationship between the rate constant, temperature, activation energy, and the pre-exponential factor.

I am wondering if there is any way to explain how the equation can be modified to explain the general form for rates of elementary reactions ($$\text{rate}=k[\text{A}]^m[\text{B}]^n$$), and why it would be the case that concentrations of A and B are raised to the their stoichiometric coefficients.

Equation from: Atkins, P., & Jones, L. (2009). Collision Theory. In Chemical Principles: The Quest For Insight (5th ed., pp. 591–594). essay, W. H. Freeman and Company.

• Put another A instead of B in your equations and you'd see why. If you got so far, what's the point of asking this? Feb 26 at 17:09
• BTW what's the point of using \text in Mathjax, when there's \pu, \mathrm and \ce? Feb 26 at 17:12
• I figured as much, but I just wanted some confirmation. Also would this be sufficient in explaining why the rate of a reaction is proportional (rather than related by some other function) to the order of the reactants? Side note: I just picked up \text first when I was learning Mathjax, not other reason tbh. Feb 26 at 17:36
• Chem+Math MathJax formatting reference: Basics / Expressions/formulas/equations / Upright vs Italics / Math SE Mathjax tutorial // MathJax is preferred not to be used in CH SE Q titles. Feb 26 at 18:50
• chemistry.stackexchange.com/questions/131105/… Feb 27 at 0:31