Question

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?

Short answer

The half-life of radium (\(_{88}^{224} \mathrm{Ra}\)) is approximately 109.35 hours.

Step-by-step solution

Identify Known Information

The decay rates (in counts per minute) at different times (in hours) of radium are given. They are: at \(t = 0h, rate = 1000cpm\); \(t = 1h, rate = 992cpm\); \(t = 10h, rate = 924cpm\); \(t = 100h, rate = 452cpm\); and \(t = 250h, rate = 138cpm\).

Identify Relevant Formula

The radioactive decay law is represented by \(N(t) = N_0 e^{-\lambda t}\), where \(N(t)\) is the number of atoms at time \(t\), \(N_0\) is the initial number of atoms, \(\lambda\) is the decay constant, and \(t\) is time. Furthermore, the decay constant is related to the half-life (\(T\)) through \(\lambda = \frac{ln(2)}{T}\)

Use the Decay Formulas

The decay constant can be calculated using the formula \(\lambda = -\frac{ln \frac{N(t)}{N_0}}{t}\). Consider two reading from the table, for instance, time \(t = 0h\) and \(t = 100h\). Plugging in the values, \(\lambda = -\frac{ln \frac{452}{1000}}{100}\)

Calculate the Decay Constant

By evaluating the expression, we get \(\lambda\) approximately equal to 0.00634 per hour.

Calculate the Half-Life

Applying the relationship between decay constant and half-life, \(T = \frac{ln(2)}{\lambda}\), we find \(T\) approximately equal to 109.35 hours.

Key concepts

Radioactive Decay

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.

Decay Constant

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.

First-Order Kinetics

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.