1. Integrated Rate Law
Expresses the reactant concentrations as a function of time.
aA → products
Kinetics are first order in [A], and the rate law is
Rate = k[A]
Integrated first-order rate law is
ln[A] = -kt + ln[A]0
Equation shows how concentration of A depends on time. If the initial concentration of A and the rate constant k are known, the concentration of A can be calculated at any time.

2. ln[A] = -kt + ln[A]0
Equation is of the form y = mx + b, where a plot of y versus x is a straight line with slope m and intercept b.
y = ln[A] x = t m = -k b = ln[A]0
Thus, for a first-order reaction, plotting the natural logarithm of concentration vs. time always gives a straight line. For the reaction,
aA → products
the reaction is first order in A if a plot of ln[A] vs. t is a straight line.

3. ln[A] = -kt + ln[A]0
The integrated rate law for a first-order reaction also can be expressed in terms of a ratio of [A] and [A]0 as follows:

4. 2N2O5(g) → 4NO2(g) + O2(g)
Since the plot of ln[N2O5] vs. time is a straight line, it confirms that the reaction is first order in N2O5, since it follows the equation ln[N2O5] = -kt + ln[N2O5]0.

5. Half-Life of a First-Order Reaction
Half-life = the time required for a reactant to reach half its original concentration.
Designated by the symbol t1/2.
General equation for the half-life of a first order reaction is (derivation in textbook (p. 542):
Note for a first-order reaction, the half-life does not depend on concentration.

6. Second-Order Rate Laws
General reaction:
aA → products
That is second order in A, the rate law is:
Rate = k[A]2
The integrated second-order rate law has the form
A plot of 1/[A] vs. t will produce a straight line with a slope equal to k.

7. Equation shows how [A] depends on time and can be used to calculate [A] at any time t, provided k and [A]0 are known.
2C4H6(g) →C8H12(g) – second-order since a plot of 1/[C4H6] vs. t produces a straight line.
Expression for the half-life of a second order reaction:

8. Zero-Order Rate Laws The rate law for a zero-order reaction is:
Rate = k[A]0 = k(1) = k
For a zero-order reaction, the rate is constant.
It does not change with concentration as it does for first-order or second-order reactions.
Integrated rate law for a zero-order reaction is:
[A] = -kt + [A] Plot of [A] vs. t gives a straight line.
Half-Life equation: