Question

The half-life of a radioactive nuclide is \(0.693\) minutes. The time (in minutes) required for the disintegration of this nuclide from 10 grams to one gram is ........ (a) 1 (b) \(0.693\) (c) \(6.93\) (d) \(2.303\)

Short answer

The time required is approximately \(2.303\) minutes.

Step-by-step solution

Understand Half-life Concept

The half-life of a radioactive substance is the time it takes for half of the substance to decay. Here, the half-life is given as \(0.693\) minutes.

Apply the Half-life Formula

The half-life formula is \( N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \), where \(N\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the elapsed time, and \(T_{1/2}\) is the half-life.

Set Up the Equation

Here, \(N_0 = 10\) grams, \(N = 1\) gram, and \(T_{1/2} = 0.693\) minutes. Substitute into the formula: \( 1 = 10 \times \left(\frac{1}{2}\right)^{\frac{t}{0.693}} \).

Solve for \(t\)

First, divide both sides by \(10\) to isolate the power: \( \frac{1}{10} = \left(\frac{1}{2}\right)^{\frac{t}{0.693}} \). Convert \(\frac{1}{10}\) to base \(\frac{1}{2}\) as \(10\) is \( \left(\frac{1}{2}\right)^{-3.3219} \) approximately. So, \( -3.32192 = \frac{t}{0.693} \). Solve for \(t\) by multiplying by \(0.693\): \( t = -3.32192 \times 0.693 \).

Calculate Final Answer

Calculate \( t = -3.32192 \times 0.693 \approx 2.303 \). Therefore, the correct answer is approximately \(2.303\) minutes.

Key concepts

Half-life

The half-life of a substance is a fascinating concept in the study of radioactive decay. It refers to the amount of time it takes for half of the radioactive nuclei in a sample to decay into a different state or element. This period is constant for every particular nuclide and is independent of its initial quantity. For instance, if you start with 10 grams of a radioactive nuclide, after one half-life, you'll have 5 grams remaining; after another half-life, you'll have 2.5 grams, and so forth. It's worth noting that the half-life remains the same no matter how much of the substance you start with.
In the given exercise, the half-life is specified as 0.693 minutes. This sets the stage for calculating how quickly the substance decays over time. Knowing the half-life allows one to predict both the speed and extent of this decay. Calculations revolving around half-lives help in various scientific and practical aspects, such as medicine, archaeology, and nuclear physics.

Radioactive Nuclide

A radioactive nuclide, often simply known as a radionuclide, is an atom with an unstable nucleus. Its instability is due to an imbalance in the internal combination of protons and neutrons. Because of this imbalance, the nuclide undergoes radioactive decay to reach a more stable state.
During this decay, the nuclide emits particles or electromagnetic radiation, such as alpha, beta particles, or gamma rays. This process continues until a stable or less unstable nuclide is formed. The specific path a radioactive nuclide takes, including the types of emissions and intermediate nuclides, is determined by its particular atomic structure and decay pathway. In practical applications, radioactive nuclides are invaluable in fields ranging from medical diagnostics and treatment to carbon dating and energy production.

Disintegration Time

Disintegration time refers to the period required for a radioactive nuclide to decay to a specific fraction of its initial amount. When talking about radioactive decay, understanding how long it takes for a substance to reduce to a particular level is crucial. This is particularly critical in contexts such as nuclear medicine, where precise dosing is essential, or in nuclear reactors where fuel management is required.
In the exercise, the disintegration time was the period needed for the sample to decrease from 10 grams to 1 gram. By applying the half-life formula, one can determine that this interval is approximately 2.303 minutes. How quickly a particular nuclide decays provides valuable insights into its properties and potential applications.

Half-life Formula

The half-life formula is a key tool in understanding and calculating radioactive decay. It is expressed as follows:\[ N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]Here is what each symbol represents:

• \(N\) is the remaining quantity of the substance after time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(t\) is the time elapsed.
• \(T_{1/2}\) is the half-life of the substance.

By using this formula, you can calculate how much of a radioactive nuclide remains after a given period, or determine the time required for the substance to decay to a particular level. It's essential in scenarios where understanding radioactive decay patterns is crucial, such as radiocarbon dating or determining the dosage of radioactive substances for medical treatments. This formula provides a straightforward and reliable method for working with radioactive decay scenarios.