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I am working through some reaction engineering problems, and something in particular has caught my attention related more to chemistry I think than reaction engineering.

Suppose we have a gas phase reaction

## A + 3B \rightarrow 2C ##

If this is an elementary reaction, the rate law is

##-r_{A} = k_{A}C_{A}C_{B}^{3}##

However, if species B is my limiting reactant, I wish to divide through to make the reaction

##\frac {1}{3} A + B \rightarrow \frac {2}{3} C##

Why is it that the rate law cannot be

##-r_{A} = k_{A}C_{A}^{1/3}C_{B}##?

It seems like all other parameters are allowed to (and must be changed) to calculate the concentration of a species as a function of conversion. For example, in a flow system that is isothermal and isobaric, the concentration of species ##i## is given as

[tex] C_{i} = C_{A0} \frac {\theta_{i} + \nu_{i} X}{1 + \epsilon X} [/tex]

where ##C_{i}## is the concentration of species ##i## at conversion ##X##, ##C_{A0}## is the initial concentration of species ##A##, ##\theta_{i} \equiv \frac {C_{i0}}{C_{A0}} = \frac {y_{i0}}{y_{A0}}##, and ##\epsilon \equiv y_{A0} \delta##, where ##y_{A0}## is the initial mole fraction of species ##A##, and ##\delta \equiv \sum_{i} \nu_{i}##, in this case ##\delta = \frac {2}{3} - \frac {1}{3} - 1 = - \frac {1}{3}##.

So my question can be summarized as this:

For the gas phase reaction,

## A + 3B \rightarrow 2C ##

t's perfectly fine to divide through to change the stoichiometric coefficients of the reaction to ##\frac {1}{3} A + B \rightarrow \frac {2}{3} C##, however the rate law must remain as ##-r_{A} = k_{A}C_{A}C_{B}^{3}##, and cannot be changed to ##-r_{A} = k_{A}C_{A}^{1/3}C_{B}##?

However, the ##\delta## term has to be based on the divided coefficients,

##\delta = \frac {2}{3} - \frac {1}{3} - 1 = - \frac {1}{3}##., and cannot be ##\delta = 2 - 3 - 1 = -2##.

This discrepancy is confusing to me because now I can't tell why I can just arbitrarily choose some coefficients to do some calculations, and other times I have to use different ones.

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# Rate law of elementary reaction

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