Determining reaction orders

In a nutshell

Reaction orders show the relationship between the concentration of the reactant and rate of reaction. The rate constant $$k$$ states how fast or slow a reaction proceeds and has the units, $$mol\ dm^{-3}$$. The most common orders of a reaction are $$0$$​, $$1$$​ and $$2$$​.

Equations

This is the general equations that you need to know and apply in specific situations.

Reaction order and concentration

The order of a reaction tells you how the reactant concentration affects the reaction rate.

​$$0^{th}$$ order ​

If $$[X]$$​ changes, the rate of reaction stays the same. If $$[X]$$​ is doubled the reaction rate would still be the same. The order of reaction with respect to $$[X]$$​ is $$0$$.​

​$$1^{st}$$ order​

If $$[X]$$​ is doubled and the rate is doubled, or $$[X]$$​ is tripled and the reaction rate triples, the reaction order with respect to $$[X]$$​ is $$1$$. 

 $$rate \propto [X]$$​ ​

​$$2^{nd}$$ order​

If $$[X]$$​ is doubled and the rate quadruples, or if $$[X]$$​ is tripled and the reaction rate increases by nine times, it suggests the reaction order with respect to $$[X]$$​ is $$2$$. 

$$rate \propto [X]^2$$​

The reaction orders cannot be calculated from chemical equations, they can only be found from experimental studies.

The overall order of the reaction is the sum of the orders of all the reactants.

Note: $$\propto$$ shows proportionality​

Rate-concentration graphs shape

The data used in concentration-time graphs can also be used to make rate-concentration graphs. These can tell you the order of a reaction. 

Concentration-time graphs

Using the time concentration graphs the gradient of the graphs needs to be found throughout the extent of the reaction.

Rate-concentration graphs

	$$0^{th}$$  order		​ $$1^{st}$$ order		​ $$2^{nd}$$ order
Concentration-time graphs
Rate-concentration graphs
Description	Regardless of the change in reactant concentration, the rate does not change.		The rate is proportional to $$[X]$$.		The rate is proportional to $$[X]^2$$.

Half-life

The half-life of a reactant is the time it takes for half of the reactant to be used up. 

To calculate the half-life $$(t_{\frac{1}{2}})$$ of a reactant, the concentration ($$y-axis$$​) must be plotted against the time ($$x-axis$$​). Then, every time the concentration is halved read off the time taken.

Example

Upon heating calcium carbonate decomposes to form calcium oxide and carbon dioxide. 

$$CaCO_{3(s)} \ \rightarrow\ CaO_{(s)}\ +\ CO_{2(g)}$$ 

The mass of calcium carbonate $$(CaCO_3)$$ halves every $$144$$​ seconds:

 $$4 g$$ to $$2g\ = \ 144\ sec$$ ​

$$2g$$ to $$1g\ =\ 144\ sec$$ 

$$1g\$$to $$0.5g\ =\ 144\ sec$$​​

​

Since this is a solid, the mass is being measured. In a liquid, the concentration​ of the solution would be measured to keep track of the rate of decomposition.

Half-lives can be used to identify a first-order reaction without having to draw a rate-concentration graph. The half-life $$(t_{\frac{1}{2}})$$​ is always constant for a first-order reaction. 

Rate constant $$(k)$$​​

Rate constants state how the concentration of reactants affects the rate of reaction. 

For equation $$A\ +\ B\ \rightarrow\ C\ +\ D$$, the rate equation would be:

​$$rate\ =\ k[A]^m[B]^n$$ 

Note: m and n are the order of reaction and $$k$$​ is the rate constant given in $$mol\ dm^{-3}$$.

The temperature has an effect on the rate constant. As the temperature increases, so does the rate constant and vice versa.