A sample containing \(_{88}^{224} \mathrm{Ra},\) which decays by \(\alpha\) -particle emission, disintegrates at the following rate, expressed as disintegrations per minute or counts per minute \((\mathrm{cpm}): t=0,1000 \mathrm{cpm} ; t=1 \mathrm{h}\) \(992 \mathrm{cpm} ; t=10 \mathrm{h}, 924 \mathrm{cpm} ; t=100 \mathrm{h}, 452 \mathrm{cpm}\) \(t=250 \mathrm{h}, 138 \mathrm{cpm} .\) What is the half-life of this nuclide?

Imagine you have a small piece of a substance that emits particles or radiation as it changes into a different element; this is known as radioactive decay. It's a spontaneous process where an unstable atomic nucleus loses energy by emitting radiation. In simple terms, it's like a ticking time bomb at the atomic level, but instead of an explosion, we get a release of particles or energy.

In the case of \( _{88}^{224} \mathrm{Ra} \), it decays by emitting an alpha particle, which is essentially a helium nucleus. This loss transforms the radium into another element and this transformation is what we measure in counts per minute (cpm). It’s an important concept as it helps us to determine the stability of an element and its potential uses in fields like medicine or energy.

The decay constant, symbolized by \( \lambda \), is a probability factor that quantifies the rate at which an atom of a radioactive substance decays. Think of it like a rating for how quickly a radioactive element 'falls apart'. The higher the decay constant, the more rapidly the substance decays.

Mathematically, we express it in an equation involving the natural logarithm. For instance, if you start with a certain number of atoms, \( N_0 \), and after some time, you're left with \( N(t) \) atoms, the decay constant tells you the pace at which those atoms are disappearing. In our exercise example, once you know how many atoms you had at the start and how many you're left with at a later time, you can calculate \( \lambda \) using \( \lambda = -\frac{ln \frac{N(t)}{N_0}}{t} \). This number is critical because it's directly related to the half-life of the substance -- the time it takes for half of the atoms in a sample to decay.

Radioactive decay follows a pattern known as 'first-order kinetics'. What does that mean for us? It signifies that the rate at which the substance decays is directly proportional to how much of it you have at any given moment. To put it differently, if we start with twice as much of a radioactive substance, it will decay twice as fast.

First-order kinetics is crucial in calculating the half-life as it tells us that regardless of the amount of substance we start with, the half-life will remain the same. This reliable predictability allows scientists to use mathematics to understand and calculate important properties of radioactive materials, such as their half-lives. Hence, using the first-order kinetics formula, one can comprehensively track the decay process over time and predict the behavior of radioactive substances.