Integrated rate law for first order

For a reaction $\text{A} \rightarrow \text{Products}$, the rate of disappearance of reactant A is proportional to its concentration:

$$\frac{d[\text{A}]}{dt} = -k[\text{A}]$$

where $k$ is the rate constant.

$$\frac{d[\text{A}]}{[\text{A}]} = -k\,dt$$

Integrate from time $t=0$ (concentration $[\text{A}]_0$) to time $t$ (concentration $[\text{A}]_t$):

$$\int_{[\text{A}]_0}^{[\text{A}]_t} \frac{d[\text{A}]}{[\text{A}]} = \int_{0}^{t} -k\,dt$$ $$[\ln[\text{A}]]_{[\text{A}]_0}^{[\text{A}]_t} = -k[t]_{0}^{t}$$ $$\ln[\text{A}]_t - \ln[\text{A}]_0 = -kt$$

$$\ln\left(\frac{[\text{A}]_t}{[\text{A}]_0}\right) = -kt$$

This can also be written as:

$$\ln[\text{A}]_t = -kt + \ln[\text{A}]_0$$

Or, in exponential form:

$$[\text{A}]_t = [\text{A}]_0 e^{-kt}$$

Where:

• $[\text{A}]_t$ is the concentration of reactant A at time $t$.
• $[\text{A}]_0$ is the initial concentration of reactant A at $t=0$.
• $k$ is the rate constant.
• $t$ is the time.