Detailed Summary of First Order Reactions
First-order reactions are defined as those where the rate of reaction is directly proportional to the concentration of one reactant raised to the power of one. Mathematically, this is represented as:
\[ Rate = -\frac{d[R]}{dt} = k[R] \]
where \( k \) is the rate constant. The integrated rate law for a first-order reaction can be derived and is given by:
\[ \ln [R] = -kt + \ln [R]_0 \]
This integrated form allows us to determine the concentration of reactants over time and can be rearranged to express the rate constant \( k \) as:
\[ k = \frac{1}{t} \ln \frac{[R]_0}{[R]} \]
This is particularly useful in applications such as the decay of radioactive isotopes, where the time taken for half of a substance to decay (the half-life) is a constant that does not depend on its initial concentration. The half-life for first-order reactions is calculated as:
\[ t_{1/2} = \frac{0.693}{k} \]
In summary, understanding first-order reactions involves recognizing their unique characteristics and their dependence on concentration through these derived equations.