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Lets say you have the reaction

$$\ce{$a$A + $b$B -> $c$C + $d$D}$$

The rate of the reaction is given by the change in initial concentration over the change in time.

$$-\frac{\Delta \left[\ce A \right]}{a\Delta t} = -\frac{\Delta \left[\ce B \right]}{b\Delta t} = \frac{\Delta \left[ \ce C \right]}{c\Delta t} = \frac{\Delta \left[\ce D \right]}{d\Delta t}$$

However, there is also the rate expression, which is of the form

$$\text{reaction rate} = k \left[\ce A \right]^{m}\left[\ce B \right]^{n}$$

What is the difference between these two, and what purpose does each one serve?

Further, what does the $k$ represent?

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Let's use the following gaseous chemical reaction as an example: $$\ce{H2 + I2 -> 2HI}$$ When we study reaction kinetics, we oftentimes can't observe the changes in concentration of each reactant and product. Therefore the rate expression helps us to relate the rates of appearance/disappearance of the species present: $$-\frac{1}{1}\frac{\Delta \left[ \ce{H2} \right]}{\Delta t}\; = \; -\frac{1}{1}\frac{\Delta \left[ \ce{I2} \right]}{\Delta t}\; =\; \frac{1}{2}\frac{\Delta \left[ \ce{HI} \right]}{\Delta t}\tag{rate expression}\label{rateex}$$ Note that each rate is modified by the inverse of the stoichiometric coefficient. I explicitly put $\frac{1}{1}$ in the expression just to emphasize this point. We can see that multiplying the coefficients of the reaction will not change the rate expression in the following example: let's multiply the coefficients in the chemical reaction by 2 and then write the corresponding rate expression. $$\ce{2H2 + 2I2 -> 4HI}$$ $$-\frac{1}{2}\frac{\Delta \left[ \ce{H2} \right]}{\Delta t}\; = \; -\frac{1}{2}\frac{\Delta \left[ \ce{I2} \right]}{\Delta t}\; =\; \frac{1}{4}\frac{\Delta \left[ \ce{HI} \right]}{\Delta t}\tag{new expression}\label{newex}$$ The $\ref{rateex}$ and $\ref{newex}$ are numerically equivalent.

Full disclosure here: the term rate expression is ambiguous and is also used as a synonym for the rate law. I've also seen the rate expression referred to as the rate relationship.

The rate law or differential rate law relates the rate of a reaction to the concentration (or pressure) of the reactants. The rate of a reaction is proportional to the concentration (or pressure) of the reactants modified by some experimentally determined number called the reaction order. $$\begin{align} -\frac{\Delta\left[ \ce{H2} \right]}{\Delta t} &\propto\left[\ce{H2}\right]^m\left[\ce{I2}\right]^n\\ -\frac{\Delta\left[ \ce{H2} \right]}{\Delta t} &= k\left[\ce{H2}\right]^m\left[\ce{I2}\right]^n\tag{rate law} \end{align}$$ Here, $m$ and $n$ are the reaction orders for $\ce{H2}$ and $\ce{I2}$, respectively, and the sum, $m + n$ yields the overall order of the reaction. The proportionality $a\propto b$ can be replaced with an equality $a = k b$. In kinetics, this constant of proportionality is referred to as the rate constant.


In summary: the rate expression tells us how the appearance/disappearance rates of the products and reactants relate to one another; the rate law tells us how the rate is related to the concentrations (or pressures) of reactants and the rate constant is a constant of proportionality that comes out of the rate law.

This answer is a summary of information found at the boundless website and from Tro's most recent general chemistry textbook.

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    $\begingroup$ How do you calculate the rate constant? Is Arrhenius the only equation for calculating it? $\endgroup$ – Physther Nov 21 '16 at 14:59
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    $\begingroup$ @Paul Have a look here for information on determining the rate constant. It is easy to do if you have a first-order reaction, but becomes more complicated for higher order systems. $\endgroup$ – bobthechemist Nov 22 '16 at 0:55
  • $\begingroup$ @bobthechemisWhich of the following formula describe the rate law (with explaining): $$ -\frac{1}{2}\frac{\Delta [ \ce{H2} ]}{\Delta t}= k[\ce{H2}]^m[\ce{I2}]^n$$ Or $$-\frac{\Delta[ \ce{H2} ]}{\Delta t}= k[\ce{H2}]^m[\ce{I2}]^n$$ $\endgroup$ – Adnan AL-Amleh Aug 31 at 8:19
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Reaction rate $\dfrac{dA}{dt}$ is the rate at a specific concentration and a specific time.

Rate law is an equation that shows how (velocity) a rate varies as concentration changes. It describes rates at ALL concentrations and NOT just one specific rate at one specific concentration.

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    $\begingroup$ I am sorry, but I seem to miss how this answers the question, could you add a little more explanation? $\endgroup$ – Martin - マーチン Sep 30 '16 at 5:38
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    $\begingroup$ The question is literally asking whats the difference between Reaction Rate and Rate Law $\endgroup$ – notorious Sep 30 '16 at 20:35

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