Problem 25 - Chemistry An Atoms First Approach

Question

The rate constant for a certain radioactive nuclide is \(1.0 \times\) \(10^{-3} \mathrm{h}^{-1} .\) What is the half-life of this nuclide?

Step-by-Step Solution

Answer

The half-life (\(t_{1/2}\)) of a radioactive nuclide is related to its rate constant (\(k\)) by the formula \(t_{1/2} = \frac{0.693}{k}\). Given the rate constant \(k = 1.0 \times 10^{-3} h^{-1}\), we can calculate the half-life as \(t_{1/2} = \frac{0.693}{1.0 \times 10^{-3} h^{-1}} = 693 \, h\). Therefore, the half-life of the radioactive nuclide is 693 hours.

Step 1: Recall the half-life and rate constant relation

In a radioactive decay process, the half-life (\(t_{1/2}\)) is related to the rate constant (\(k\)) by the following formula: \[ t_{1/2} = \frac{0.693}{k} \]

Step 2: Substitute the rate constant

We are given the rate constant for the radioactive nuclide, \(k = 1.0 \times 10^{-3} h^{-1}\). Plug the given value of \(k\) into the half-life formula: \[ t_{1/2} = \frac{0.693}{1.0 \times 10^{-3} h^{-1}} \]

Step 3: Calculate the half-life

Now, we need to calculate the half-life by dividing 0.693 by the rate constant. \[ t_{1/2} = \frac{0.693}{1.0 \times 10^{-3} h^{-1}} = 693 \, h \] The half-life of the radioactive nuclide is 693 hours.

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear physics and chemistry describing the process whereby an unstable atomic nucleus loses energy by emitting radiation. This emission can come in several forms, including alpha particles, beta particles, gamma rays, and conversion electrons. The decay process is random and spontaneous, meaning it occurs without any external influence and at a time that cannot be predicted for individual atoms.

Radioactive decay is characterized by the decay rate, which is the number of decays per unit time. The units of this rate are disintegrations per second (Becquerel, Bq) or disintegrations per minute (Curie, Ci). The decay rate is proportional to the number of radioactive nuclei in the sample, leading to an exponential decrease in the number of radioactive nuclei over time.

Understanding this process is crucial in many fields, from medicine, where it's used in diagnosis and treatment (for example, PET scans and radiation therapy), to archaeology, for dating artifacts through techniques like carbon dating.

Rate Constant

The rate constant, symbolized as 'k,' plays a pivotal role in characterizing the kinetics of radioactive decay. It's an inherent property of each radioactive isotope and indicates the likelihood of decay of a single nucleus per unit time.

The rate constant is a critical factor in the various mathematical models of decay such as the equation: \( N(t) = N_0 e^{-kt} \), where \( N_0 \) is the original number of undecayed nuclei, \( N(t) \) is the number of undecayed nuclei after time 't,' and 'k' is the rate constant. This constant allows us to predict how quickly a sample will decay and is instrumental in a range of applications, including determining the safety protocols for handling radioactive materials and calculating the dosage of radioactive tracers in medical applications.

Half-life Formula

Half-life, commonly denoted as \( t_{1/2} \), is the time required for half of the radioactive nuclei in a sample to undergo decay. It's a measure of the stability of a radionuclide and an essential tool in understanding the long-term behavior of radioactive substances. Regardless of the amount of material, every radioactive isotope has a constant half-life.

The mathematical relationship between the half-life and the rate constant is described by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
This formula suggests that the half-life is inversely proportional to the rate constant—a higher rate constant means a shorter half-life, implying that the radioactive atoms decay more quickly. The half-life concept is extensively used in fields such as environmental science, nuclear power, and pharmacology to ensure the safe handling, storage, and disposal of radioactive materials.