The total decay rate R of a sample is related to the decay rate R0 at t = 0 and the disintegration constant or decay constant λ as:
In nuclear physics, radioactive decay is a fundamental process where an unstable atomic nucleus spontaneously transforms into a more stable nucleus by emitting radiation. This transformation occurs at a specific rate, known as the decay rate or activity. While the decay of individual nuclei is a random event, the overall decay rate of a large sample can be precisely described by an exponential law.
Radioactive Decay Rate: Understanding the Formula
The total decay rate (R) of a radioactive sample at any given time (t) is directly linked to its initial decay rate (R0) and the characteristic disintegration constant (λ) of the radioactive isotope. This relationship is a cornerstone of radioactive decay studies and is mathematically expressed as an exponential decay function.
The correct formula describing this relationship is:
\({R = {R_0}{e^{ - \lambda t}}}\)
Decay Rate: Understanding Key Terms
To fully grasp the decay rate formula, it's essential to understand each of its components:
- Total Decay Rate (R): This is the instantaneous rate at which nuclei in a sample are decaying at time 't'. It is also referred to as the activity of the sample and is typically measured in units like Becquerel (Bq) or Curie (Ci).
- Initial Decay Rate (R0): This represents the decay rate of the radioactive sample at the very beginning of the observation or experiment, i.e., at time \(t = 0\).
- Disintegration Constant (λ): Also known as the decay constant, this is a unique constant for each specific radioactive isotope. It quantifies the probability per unit time that an individual nucleus will decay. A higher value of λ indicates that the isotope decays more rapidly.
- Time (t): This variable denotes the elapsed time since the initial measurement (when the decay rate was R0).
- e: This is Euler's number, the base of the natural logarithm, an irrational mathematical constant approximately equal to 2.71828.
Exponential Decay: Derivation and Significance
The exponential nature of radioactive decay stems from the fact that the rate of decay of radioactive nuclei is directly proportional to the number of undecayed nuclei present at any given moment. As nuclei decay, their number decreases, which in turn causes the decay rate to decrease exponentially over time.
If \(N\) represents the number of radioactive nuclei remaining at time \(t\), and \(N_0\) is the initial number of nuclei at \(t = 0\), their relationship is given by:
\({N = {N_0}{e^{ - \lambda t}}}\)
The decay rate (R) is fundamentally defined as the product of the decay constant (λ) and the number of radioactive nuclei (N) present:
\({R = \lambda N}\)
By substituting the expression for \(N\) into the equation for \(R\), we obtain:
\({R = \lambda \left( {{N_0}{e^{ - \lambda t}}} \right)}\)
Since the initial decay rate \(R_0\) is defined as \(\lambda N_0\) (the decay rate at \(t = 0\)), we can simplify the equation to the well-known form:
\({R = {R_0}{e^{ - \lambda t}}}\)
This formula is crucial for predicting the future activity of a radioactive source or for calculating its initial activity if measured at a later time, provided the decay constant is known.
Decay Rate Options: A Detailed Analysis
Let's examine the provided options and compare them with the fundamental exponential decay rate formula:
| Option | Formula | Evaluation Against \(R = {R_0}{e^{ - \lambda t}}\) |
|---|---|---|
| 1 | \(R = {R_0}{e^{ - \left( {\frac{\lambda }{4}} \right)t}}\) | This formula is incorrect because the exponent contains \(- \frac{\lambda}{4} t\) instead of the correct \(- \lambda t\). It suggests a much slower decay than the actual decay constant dictates. |
| 2 | \(R = {R_0}{e^{ - \left( {\frac{\lambda }{2}} \right)t}}\) | This option is also incorrect. The exponent should be \(- \lambda t\), not \(- \frac{\lambda}{2} t\). While factors of 1/2 are often seen in half-life calculations, this is not the general form for the decay rate. |
| 3 | \(R = {R_0}{e^{ - \left( {2\lambda } \right)t}}\) | This formula is incorrect as well. The exponent is \(- 2\lambda t\), which implies a decay rate twice as fast as the actual decay constant (λ) would dictate. |
| 4 | \(R = {R_0}{e^{ - \left( \lambda \right)t}}\) | This option presents the correct and widely accepted formula for radioactive decay rate. It accurately describes the exponential decrease in activity over time, dependent on the initial rate and the disintegration constant. |
The exponential term \(e^{ - \lambda t}\) indicates that the decay rate diminishes over time, approaching zero asymptotically but never actually reaching it. This explains why radioactive materials continue to exhibit some level of activity, albeit decreasing, over extremely long periods.
Important Questions from Radioactivity
- The energies of the 3 lowest states of an atom are E 0 = −14 eV, E 1 = −9 eV and E 2 = −7 eV. The Einstein coefficients are A 10 = 3 × 10 8 s −1 , A 20 = 1.2 × 10 8 s −1 and A 21 = 8 × 10 7 s −1 . If a large number of atoms are in the energy level E 2 , the mean radiative lifetime of this excited state is
The nuclei of 137 Cs decay by the emission of β - particles with a half life of 30.08 years. The activity (in units of disintegrations per second or Bq) of a 1 mg source of 137 Cs, prepared on January 1, 1980, as measured on January 1, 2021 is closest to
The Q - value of the α - decay of 232 Th to the ground state of 228 Ra is 4082 keV. The maximum possible kinetic energy of the α - particle is closest to
Radioactivity is the characteristic of which of the following?
A free neutron decays spontaneously into a proton, an electron and ___________ .
