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Let's take the reaction $2A + B \rightarrow C$, with the assumption that this is an elementary reaction. If I'm only given this information, how would I derive the following forward rate law from scratch?

$$ \frac{dC}{dt} = k[A]^2 [B] $$

I tried to find an answer online but most sources jump straight to the above format for a rate law. I also understand that there are many experimental considerations which can affect what is actually observed (e.g. pseudo $0^{th}$rate law behavior) but I just want to know if there's a way to derive these rate laws given just the reaction. Thank you.

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    $\begingroup$ They're found experimentally. $\endgroup$ – Mithoron Jan 12 '18 at 1:41
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    $\begingroup$ I just read online that the form for an elementary rate equation can be derived from collision theory. Can anyone confirm that this is a common theoretical approach? $\endgroup$ – Cain Jan 12 '18 at 1:49
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    $\begingroup$ I think you may need to be more precise in wording your question. You're trying to ask whether or not the law of mass action can be derived, not whether reaction mechanisms can be derived, correct? I think a lot of these comments are talking about the latter. $\endgroup$ – a-cyclohexane-molecule Jan 12 '18 at 5:56
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    $\begingroup$ I mostly agree @a-cyclohexane-molecule's preceeding comment. The question has been misinterpreted; OP does say to assume an elementary reaction. The classical ways of deriving rate laws are collision theory (Trautz–Lewis), and Eyring transition state theory (also Evans–Polanyi). The question is quite broad, though. You (@OP) might include more of your background after following up on these two approaches. $\endgroup$ – Linear Christmas Jan 12 '18 at 15:36
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    $\begingroup$ I see! Thank you for your answers. I'm sorry if I worded my question poorly. $\endgroup$ – Cain Jan 12 '18 at 19:27
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I am not a professional or anything close, but I think the answer to your question is simply that $C$ is jointly proportional to $A$ and $B$. The following link explains joint-proportion (https://www.mathwords.com/jk/joint_variation.htm). Just in case, the link gets broken in the future: enter image description here

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Here is a source that directly answers the question "just want to know if there's a way to derive these rate laws given just the reaction". The answer is generally no except for single-step mechanism, but the exponents will equal to the stoichiometric coefficients for the rate-determining step. Reference link: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Kinetics/Rate_Laws/The_Rate_Law , to quote :

"For nearly all forward, irreversible reactions, the rate is proportional to the product of the concentrations of only the reactants, each raised to an exponent. For the general reaction

aA + bB → cC + dD (11)

the rate is proportional to [A]m[B]n :

rate = k[A]m[B]n (12)

This expression is the rate law for the general reaction above, where k is the rate constant....

The dependence of the rate of reaction on the reactant concentrations can often be expressed as a direct proportionality, in which the concentrations may be raised to be the zeroth, first, or second power. The exponent is known as the order of the reaction with respect to that substance. In the reaction above, the overall order of reaction is given by the following:

order = m+n (13)

The order of the chemical equation can only be determined experimentally, i.e., m and n cannot be determined from a balanced chemical equation alone (e.g., Equation 11). The overall order of a reaction is the sum of the orders with respect to the sum of the exponents (Equation 13). Furthermore, the order of a reaction is stated with respect to a named substance in the reaction. The exponents in the rate law are not equal to the stoichiometric coefficients unless the reaction actually occurs via a single step mechanism (an elementary step); however, the exponents are equal to the stoichiometric coefficients of the rate-determining step. In general, the rate law can calculate the rate of reaction from known concentrations for reactants and derive an equation that expresses a reactant as a function of time.

The proportionality factor k, called the rate constant, is a constant at a fixed temperature; nonetheless, the rate constant varies with temperature. "

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