Understanding the law of radioactive decay is fundamental in nuclear physics. Radioactive decay is a spontaneous process where an unstable atomic nucleus loses energy by emitting radiation, transforming into a different nucleus or a lower energy state of the same nucleus. The rate at which this decay occurs is described by the law of radioactive decay.

Understanding the Radioactive Decay Law

The law of radioactive decay states that the rate of decay of radioactive nuclei at any instant is directly proportional to the number of radioactive nuclei present at that instant. Mathematically, this can be written as:

\begin{equation*}\frac{dN}{dt} \propto N\end{equation*}

Introducing a proportionality constant, \(\lambda\), which is known as the decay constant or disintegration constant, we get:

\begin{equation*}\frac{dN}{dt} = -\lambda N\end{equation*}

The negative sign indicates that the number of nuclei \(N\) decreases with time \(t\).

Deriving the Radioactive Decay Equation

We can solve this differential equation to find the number of undecayed nuclei \(N(t)\) remaining at any time \(t\). Separating the variables \(N\) and \(t\), we have:

\begin{equation*}\frac{dN}{N} = -\lambda dt\end{equation*}

Integrating both sides from time \(t=0\) to time \(t\) and from the initial number of nuclei \(N_0\) to \(N(t)\):

\begin{equation*}\int_{N_0}^{N(t)} \frac{dN}{N} = \int_{0}^{t} -\lambda dt\end{equation*}

This integration gives:

\begin{equation*}[\ln N]_{N_0}^{N(t)} = -\lambda [t]_0^{t}\end{equation*}

\begin{equation*}\ln N(t) - \ln N_0 = -\lambda (t - 0)\end{equation*}

\begin{equation*}\ln \left(\frac{N(t)}{N_0}\right) = -\lambda t\end{equation*}

To solve for \(N(t)\), we take the exponential of both sides:

\begin{equation*}\frac{N(t)}{N_0} = e^{-\lambda t}\end{equation*}

Thus, the equation for the law of radioactive decay is:

\begin{equation*}N(t) = N_0 e^{-\lambda t}\end{equation*}

Where:

• \(N(t)\) is the number of radioactive nuclei remaining at time \(t\).
• \(N_0\) is the initial number of radioactive nuclei at time \(t=0\).
• \(\lambda\) is the decay constant, a characteristic property of the radioactive isotope.
• \(t\) is the time elapsed.

Comparing Options for the Radioactive Decay Equation

Let's examine the given options in light of the derived law of radioactive decay equation:

• Option 1: \(N(t) = N_0 e^{-t/\lambda}\). This equation has \(t/\lambda\) in the exponent, which is incorrect. The time \(t\) should be multiplied by the decay constant \(\lambda\), not divided.
• Option 2: \(N(t) = N_0 e^{-\lambda t}\). This equation matches the derived equation for the law of radioactive decay, correctly showing the exponential decrease of nuclei over time with the decay constant \(\lambda\).
• Option 3: \(N(t) = N_0 e^{-\lambda/t}\). This equation has \(\lambda/t\) in the exponent, which is incorrect. The time variable should be in the numerator of the exponent, multiplied by \(\lambda\).
• Option 4: \(N(t) = N_0 e^{-2\lambda t}\). This equation has \(2\lambda\) in the exponent, suggesting a decay constant twice the actual value \(\lambda\). This does not represent the standard law of radioactive decay with decay constant \(\lambda\).

Based on the derivation, the correct equation for the law of radioactive decay is \(N(t) = N_0 e^{-\lambda t}\).

Revision Table: Key Radioactive Decay Concepts

Additional Information: Radioactive Decay and Activity

Related to the number of nuclei is the activity of a radioactive sample, which is the rate of decay. The activity \(A(t)\) at time \(t\) is given by:

\begin{equation*}A(t) = \left|\frac{dN}{dt}\right| = \lambda N(t)\end{equation*}

Substituting the expression for \(N(t)\):

\begin{equation*}A(t) = \lambda (N_0 e^{-\lambda t}) = (\lambda N_0) e^{-\lambda t}\end{equation*}

Let \(A_0 = \lambda N_0\) be the initial activity at \(t=0\). Then the activity equation is:

\begin{equation*}A(t) = A_0 e^{-\lambda t}\end{equation*}

This shows that the activity also decays exponentially with the same decay constant \(\lambda\). Common units for activity include Becquerel (Bq, 1 decay per second) and Curie (Ci, \(3.7 \times 10^{10}\) decays per second).

The radioactive decay process is independent of external factors like temperature, pressure, or chemical environment.