Understanding the Law of Radioactive Decay

The law of radioactive decay describes how unstable atomic nuclei lose energy by emitting radiation and transform into more stable forms. This law governs the rate at which the number of radioactive nuclei decreases over time and is fundamental in nuclear physics, with applications in radiometric dating, medicine, and nuclear energy.

Nature of Radioactive Decay

Radioactive decay is a spontaneous and random process in which an unstable nucleus changes into a different nucleus, usually accompanied by the emission of particles or radiation. The probability that a given nucleus will decay in a certain interval of time is constant and independent of the nucleus's age or history.

This characteristic randomness means it is impossible to predict exactly when a particular nucleus will decay, but the overall decay behavior for large samples can be described statistically and mathematically.

Types of Radioactive Decay

Radioactive decay can occur in several ways, primarily as alpha decay, beta decay, and gamma decay. Alpha decay involves the emission of a helium nucleus, beta decay involves the emission of electrons or positrons, and gamma decay involves the release of high-energy photons. Detailed explanations of these mechanisms are provided in the article on Alpha, Beta, and Gamma Decay.

Fundamental Law of Radioactive Decay

The law of radioactive decay states that the rate of disintegration of nuclei at any instant is directly proportional to the number of undecayed nuclei present at that instant. Mathematically:

$\dfrac{dN}{dt} = -\lambda N$

Here, $N$ is the number of undecayed nuclei at time $t$, and $\lambda$ is the decay constant, representing the probability per unit time that a nucleus will decay.

Derivation of the Radioactive Decay Law

To derive the decay law, separate variables and integrate:

$\dfrac{dN}{N} = -\lambda \, dt$

On integrating both sides from $N_0$ at $t=0$ to $N$ at time $t$:

$\int_{N_0}^{N} \dfrac{dN}{N} = -\lambda \int_{0}^{t} dt$

$\ln \left( \dfrac{N}{N_0} \right) = -\lambda t$

$N = N_0 e^{-\lambda t}$

This equation represents an exponential decrease in the number of undecayed nuclei with time.

Decay Constant and Its Significance

The decay constant $\lambda$ is a unique property of each radioactive element and determines the probability that a nucleus will decay per unit time. A higher value of $\lambda$ implies a faster decay rate and a shorter average lifetime for the nucleus.

The decay constant connects to other nuclear properties, such as half-life and mean life, which are widely used to characterize radioactive substances in nuclear reactions and Atom and Nuclei.

Decay Rate and Activity

The decay rate, often called the activity ($A$ or $R$), is defined as the number of decays occurring per unit time. It is given by:

$R = -\dfrac{dN}{dt} = \lambda N$

The SI unit of activity is the Becquerel (Bq), where 1 Bq equals one decay per second. For freshly prepared samples, the initial activity is $R_0 = \lambda N_0$, and it decreases exponentially with time.

Half-Life of a Radioactive Substance

Half-life ($T_{1/2}$) is the time required for half of the original number of radioactive nuclei to decay. It is a key characteristic for comparing different isotopes and their stability.

From the decay law:

$N = N_0 e^{-\lambda t}$

Set $N = N_0/2$ when $t = T_{1/2}$:

$\dfrac{N_0}{2} = N_0 e^{-\lambda T_{1/2}}$

Solving for $T_{1/2}$ gives:

$T_{1/2} = \dfrac{\ln 2}{\lambda} \approx \dfrac{0.693}{\lambda}$

Mean Life or Average Lifetime

Mean life ($\tau$) is defined as the average lifetime of a radioactive nucleus before decay. It is given by:

$\tau = \dfrac{1}{\lambda}$

The mean life always exceeds the half-life because some nuclei persist much longer than the mean value, as described by the exponential nature of the decay law.

Exponential Decay Graph

The law of radioactive decay yields an exponential graph when plotting the number of undecayed nuclei $N$ versus time $t$. The number of nuclei drops by half in every successive half-life interval, never reaching zero but decreasing continuously. The shape and features of this graph are described quantitatively by the decay law equation.

Applications and Significance

The law of radioactive decay is crucial in nuclear physics, radiometric age determination, medical imaging, radiation therapy, and nuclear energy. Proper use of decay laws enables calculation of activity, age of samples, and assessment of nuclear stability. Further applications can be found in Nuclear Fission and Fusion as well as Ionization Energy.

Key Features of Radioactive Decay

• Radioactive decay is spontaneous and random
• Rate is proportional to remaining undecayed nuclei
• Decay law is exponential in form
• Each isotope has a characteristic decay constant
• Half-life and mean life are fundamental parameters
• Activity decreases exponentially with time

Related Concepts

The principles behind the law of radioactive decay are integral to topics such as Atomic Structure and Dual Nature of Matter, and provide a foundational understanding essential for advanced studies in nuclear physics and quantum mechanics.