Question

The time required for half of an original sample of a radioactive nuclide to decay is referred to as the of the nuclide.

Short answer

The half-life of a radioactive nuclide, denoted as \(t_{1/2}\), is the time required for half of an original sample to decay. It can be calculated using the formula \(t_{1/2} = \frac{ln(2)}{\lambda}\), where \(\lambda\) is the decay constant, representing the probability of decay per unit time. Given the decay constant, one can plug the values into the formula and solve for the half-life.

Step-by-step solution

(Step 1: Clarify the half-life concept)

(The half-life is a characteristic of radioactive decay. It represents the amount of time it takes for half of an initial amount of a radioactive substance to decay into another substance. In this process, the radioactive substance breaks down into stable or less radioactive substances.)

(Step 2: Determine the appropriate formula)

(We will use the following formula to calculate the half-life: \[ t_{1/2} = \frac{ln(2)}{\lambda} \] Where: - \(t_{1/2}\) is the half-life. - \(\lambda\) is the decay constant (probability of decay per unit time).)

(Step 3: Find the decay constant (\(\lambda\)))

(Given the half-life, we can also find the decay constant (\(\lambda\)). This is a parameter that determines the probability of decay for each nuclide per unit time. For example, different radioactive elements may have different decay constants. The decay constant can be found using the same formula, with the half-life value given: \[ \lambda = \frac{ln(2)}{t_{1/2}} \])

(Step 4: Calculate the half-life)

(In this problem, since we're not provided with specific values, we will describe the calculation process in general terms: 1. Determine the decay constant (\(\lambda\)) 2. Plug the known values (decay constant and the natural logarithm of 2) into the formula \(t_{1/2} = \frac{ln(2)}{\lambda}\) 3. Calculate the half-life of the nuclide by solving for \(t_{1/2}\)) As a result, we could solve for the half-life of a radioactive nuclide if given the specific decay constant, or vice versa.

Key concepts

Radioactive Decay

Radioactive decay is a fascinating process that involves the breakdown of unstable atomic nuclei. This natural phenomenon takes place in certain types of atoms which are referred to as radioactive nuclides. Over time, these atoms release energy and particles in order to become more stable. This is what we call radioactive decay.
Think of radioactive decay as the journey of a radioactive nuclide transforming itself into a more stable nuclide or sometimes a completely different element. It's a bit like a dance where the atom moves to find its most balanced form. The rate at which this transformation happens depends on the type of element and its intrinsic properties. Some elements decay quickly, while others may take thousands of years to show significant changes.

• The process is unpredictable on the scale of single atoms but very predictable in large numbers.
• It's crucial for applications such as radiocarbon dating and nuclear medicine.

Understanding radioactive decay helps us not only explore ancient artifacts but also develop medical technologies and manage nuclear waste effectively.

Decay Constant

The decay constant, usually symbolized by \( \lambda \), is an important parameter in nuclear physics. It describes the probability of a single radioactive atom decaying in a given time period. In essence, it tells us how quickly or slowly a radioactive isotope will decay.
The decay constant is intrinsic to each radioactive nuclide, meaning each element and isotope has its own unique value. This makes the decay constant an essential tool for scientists to understand and predict the behavior of different radioactive substances.
For example, if a substance has a high decay constant, it implies that it will decay rapidly, quickly transforming into another chemical form or isotope. Alternatively, a low decay constant indicates a slower process. The decay constant can be calculated if we know the half-life: \( \lambda = \frac{ln(2)}{t_{1/2}} \).

• It is measured in units of inverse time, such as per second or per year.
• A higher decay constant implies a greater likelihood of decay within the time frame.

This concept is pivotal to fields such as nuclear physics and radioactive waste management because it defines how long radioactive materials must be handled with caution before they decay to safe levels.

Exponential Decay

Exponential decay is a mathematical concept often used to describe the process of radioactive decay. When we talk about exponential decay, we're describing a situation where the quantity of material decreases at a rate proportional to its current value. In simpler terms, the rate at which something decreases keeps getting slower over time, much like a ball bouncing to lower and lower heights.
For radioactive substances, the number of radioactive atoms decreases exponentially over time. As each atom decays, fewer are left to decay, which means that the process gradually slows down.

• It's represented by the equation \( N(t) = N_0 e^{-\lambda t} \) where:
• \( N(t) \) is the quantity that has not yet decayed at time \( t \).
• \( N_0 \) is the initial quantity of the substance.
• \( \lambda \) is the decay constant.

This decay process is easily visualized as a smooth, downward-sloping curve when graphed over time. Understanding exponential decay is essential for accurately calculating how long a radioactive substance will remain hazardous, which is crucial for safety measures in fields like nuclear power and radiological health assessments.