First Order Reactions
First Order Reactions
First order reactions are a class of chemical reactions where the rate is directly proportional to the concentration of one reactant. This means that the rate of the reaction depends on the concentration of that single reactant raised to the first power.
Characteristics of First Order Reactions
- The rate of reaction is directly proportional to the concentration of one reactant.
- The half-life of a first-order reaction is constant and does not depend on the initial concentration of the reactant.
- The rate constant of the reaction has units of reciprocal time (s^-1, min^-1, etc.).
Mathematical Representation
The rate law for a first-order reaction can be written as:
[ \text{Rate} = k[A] ]
where:
- (\text{Rate}) is the rate of the reaction,
- (k) is the rate constant,
- ([A]) is the concentration of the reactant A.
The integrated rate law for a first-order reaction is:
[ \ln[A]_t = -kt + \ln[A]_0 ]
where:
- ([A]_t) is the concentration of A at time (t),
- ([A]_0) is the initial concentration of A,
- (k) is the rate constant,
- (t) is the time.
The half-life ((t_{1/2})) of a first-order reaction is given by:
[ t_{1/2} = \frac{0.693}{k} ]
Differences and Important Points
| Property | First Order Reaction |
|---|---|
| Rate Law | Rate = (k[A]) |
| Order | 1 |
| Units of Rate Constant ((k)) | Time^-1 (e.g., s^-1) |
| Integrated Rate Law | (\ln[A]_t = -kt + \ln[A]_0) |
| Half-Life ((t_{1/2})) | Constant ((t_{1/2} = \frac{0.693}{k})) |
| Concentration-Time Relationship | Exponential decay |
Examples
Example 1: Radioactive Decay
Radioactive decay is a classic example of a first-order reaction. The decay of a radioactive isotope is independent of its concentration and follows first-order kinetics.
For a radioactive isotope with a decay constant ((k)), the number of nuclei remaining at time (t) can be calculated using the integrated rate law:
[ N_t = N_0 e^{-kt} ]
where:
- (N_t) is the number of nuclei at time (t),
- (N_0) is the initial number of nuclei,
- (k) is the decay constant.
Example 2: Decomposition of Hydrogen Peroxide
The decomposition of hydrogen peroxide ((H_2O_2)) in aqueous solution is often catalyzed by iodide ion and follows first-order kinetics:
[ 2H_2O_2(aq) \rightarrow 2H_2O(l) + O_2(g) ]
The rate of decomposition at any time (t) can be determined using the rate law:
[ \text{Rate} = k[H_2O_2] ]
By measuring the concentration of (H_2O_2) at various times, one can determine the rate constant (k) and calculate the half-life of the reaction.
Conclusion
First-order reactions are an important class of chemical reactions that are characterized by a rate that is directly proportional to the concentration of one reactant. Understanding the rate laws, integrated rate laws, and half-life equations for first-order reactions is crucial for predicting the behavior of these reactions over time. Examples such as radioactive decay and the decomposition of hydrogen peroxide illustrate the practical applications of first-order kinetics in chemistry.