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the units for rates of reaction are usually $mol\;dm^{-3}\;s^{-1}$, for gasseous reactions, $Pa\;s^{-1}$ is used
the rate of reaction can be seen as the change in either the product or reactant over a certain period of time. mathematically we use ditferecniantion to find the gradient of the line that tracks the species concentration over time. this gradient is equal to the rate of reaction that point in the reaction
$$ \text{rate of reaction}=+\frac{\text{d[product]}}{\text{d}t}=-\frac{\text{d[reactant]}}{\text{d}t} $$
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elementary reactions involve no more than 2 species
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true elementary reaction can always be represented with one equation, ie, the reaction is not made up of multiple more simple steps
$$ \ce{A +B->C} $$
we can then use that chemical equation to write a rate equation for the reaction to realise the rate of reaction in terms of the concentrations fo the reacts and the rate constant, $k$
$$ \text{rate of reaction}= k\ce{[A][B]} $$
the rate constant is dependant on temperature
in theory we should be able to monitor the increase or decrease of any of the reactant in an elementary reaction at a given point to find the rate of reaction at said point. for this to be possible we must consider the stoichiometry of the reaction and make each term relative its stoichiometric contribution. we can do this by dividing the rate by its stoichiometric number. this is equal to multiplying it by the inverse of the stoichiometric number
$$ a\;\ce{A} + b\;\ce{B} \;\ce{->}\;c\;\ce{C}\; + g\;\ce{G} $$
$$ \text{rate of reaction}=-\frac{1}{a}\frac{d\ce{[A]}}{d\text t}=-\frac{1}{b}\frac{d\ce{[B]}}{d\text t}=+\frac{1}{c}\frac{d\ce{[C]}}{d\text t}=-\frac{1}{g}\frac{d\ce{[G]}}{d\text t} $$
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the overall order of a reaction is given by a + b (finding the sum of all the orders of the reactants)
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elementary reaction can only be up to 2nd order because there are no more than 2 reactants and only one step is involved.
$$ \text{rate of reaction}= k\ce{[A]}^a\ce{[B]}^b $$
using the concentration at a given time as well as the rate constant and the orders with respect to the reactants, we are able to find the rate of reaction also
these equations find the rate of reaction form the change in either the product or the reactant over time. they can be formed from the linear rate equations but do not rely on the rate constant, $k$