The decay constant, commonly represented by the Greek letter $\lambda$ (lambda), quantifies the rate at which a radioactive substance undergoes decay. It is intrinsically linked to the half-life of the substance, which is defined as the time required for half of the radioactive material to decay. The relationship between the decay constant and the half-life is expressed by the formula:

where $\ln(2)$ denotes the natural logarithm of 2, and $T_{1/2}$ represents the half-life.

To apply this formula, you first need to determine the half-life of the substance in question. This information can typically be found in reference materials or can be measured experimentally. Once you have obtained the half-life, you can substitute this value into the formula to calculate the decay constant. The natural logarithm of 2 is approximately $0.693$, allowing us to also express the formula as:

The decay constant is typically reported in units of inverse time, such as per second ($\text{s}^{-1}$), per minute ($\text{min}^{-1}$), per hour ($\text{h}^{-1}$), or per year ($\text{yr}^{-1}$), depending on the value of the half-life. For instance, if the half-life is measured in years, the decay constant will be expressed in units of per year.

This formula is derived from the exponential decay law, which asserts that the rate of decay of a radioactive substance is proportional to the quantity of the substance remaining. The decay constant serves as the proportionality constant in this law. It is a fundamental characteristic of the substance itself and remains unaffected by the amount of the substance present or by external conditions.

In summary, to calculate the decay constant from the half-life, use the formula:

Here, $\lambda$ represents the decay constant, while $T_{1/2}$ is the half-life measured in any time unit. This formula demonstrates the decay speed of a radioactive substance. Simply divide $0.693$ (the natural logarithm of 2) by the half-life to obtain the decay constant, which indicates the decay rate of the substance.