First Order Concentration-Time Graphs
- As shown in the previous section, the concentration vs time graph for a first order reaction is not a straight line
- However, line equations are easier to interpret because they have a constant slope
- The rate equation for a first order reaction is shown below:
Rate = k[A]
- Since the rate is the change in concentration of the reactant per unit of time, the rate equation can be transformed in:
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- Remember that the change from the reactant must have a negative sign
- Using a mathematical operation called integration a new rate equation can be constructed
- The name of this new equation is the integrated rate law for a first order reaction
ln [A]t – ln [A]0 = – kt
- This equation links the concentration of A at any time (), with the initial concentration of A (), the rate constant (k), and the time (t)
- It can be used to calculate the concentration of A at any time, just by replacing the time
- The initial concentration of A and the rate constant are always the same and they, in some cases, given by the statement
- If the new rate equation is rearranged, the following equation is obtained:
ln [A]t = – kt + ln [A]0
- This new equation is equivalent to the equation of an straight line:
y = mx + c
- Comparing carefully both equations, if ln [A]t is plotted against t, an straight line going down is obtained
- The slope of the straight line is -k
- The y-intercept of the line is ln [A]0
The concentration-time graph and the integrated rate law graph for a first order reaction
Diagram showing the difference between a concentration-time graph and the graph after the integration to obtain the integrated rate law for a first order reaction. The elements of the integrated rate law graph are labeled