Step 1: Definition. For a first order change \(R \rightarrow\) products, the speed is proportional to how much R is present: \(-\dfrac{d[R]}{dt} = k[R]\).
Step 2: Rearrange and integrate as an indefinite integral. \(\dfrac{d[R]}{[R]} = -k\,dt\). Integrating gives \(\ln[R] = -kt + C\), where \(C\) is the integration constant.
Step 3: Fix the constant. At \(t = 0\), \([R] = [R]_0\), so \(C = \ln[R]_0\). Putting this back: \(\ln[R] = -kt + \ln[R]_0\), that is \(\ln\dfrac{[R]_0}{[R]} = kt\), or in log form \(k = \dfrac{2.303}{t}\log\dfrac{[R]_0}{[R]}\).
Step 4: Half-life from the same relation. Half-life is reached when \([R]\) has dropped to \(\dfrac{[R]_0}{2}\). Then \(\dfrac{[R]_0}{[R]} = 2\), so \(k = \dfrac{2.303}{t_{1/2}}\log 2\).
Step 5: Solve for the time and interpret. \(t_{1/2} = \dfrac{2.303 \times 0.3010}{k} = \dfrac{0.693}{k}\).
\[\boxed{t_{1/2} = \dfrac{0.693}{k}}\]
Since the starting concentration never appears in this result, each successive half-life takes the same time for a first order reaction, no matter how much reactant we begin with. This constant half-life is a signature test for first order kinetics (for example, radioactive decay).