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Half-life is a crucial concept in radioactive decay. It is the time required for half of a sample of a radioactive substance to decay into non-radioactive material. The half-life remains constant, meaning it doesn't depend on the amount of substance you have. The formula to calculate the half-life, denoted as \(T_{1/2}\), is derived from the decay formula. The key equation we'll use here is: \[N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\]
This equation helps us understand how much of the substance remains at any time \(t\). Here’s how to identify the half-life step-by-step:

- Given: In 320 days, the substance’s radioactivity decreases by 58%, so 42% remains.
- Set up the equation: \(N(320) = 0.42 N_0\)
- Use the formula: \(0.42 = e^{-\frac{320}{\tau}}\), then solve for \(\tau\)
- Convert the mean lifetime \(\tau\) to half-life using: \(T_{1/2} = \tau ln(2)\)

With these steps, you can find the exact half-life of the substance in question.

The decay constant, denoted as \(\lambda\), is another essential parameter in radioactive decay. It represents the probability per unit time that a given nucleus will decay. This is related to the mean lifetime \(\tau\) of the radioactive substance via: \[\lambda = \frac{1}{\tau}\]

The smaller the decay constant, the longer it takes for the substance to decay. To find \(\lambda\), use the relationship between the decay constant and the half-life:
- Firstly, calculate the mean lifetime \(\tau\) from the given data using \(\tau = -\frac{320}{ln(0.42)}\)
- Then, use \(\lambda = \frac{1}{\tau}\)

By knowing \(\lambda\), we can better understand the decay process and calculate how quickly or slowly it occurs. This constant gives a more intuitive grasp of the radioactive decay rate.

Exponential decay describes the process where the quantity of a substance decreases at a rate proportional to its current value. This is a fundamental principle observed in radioactive decay. The underlying equation reflects an exponential relationship:

\[N(t) = N_0 e^{-\lambda t}\]

Here, \(N_0\) is the initial amount, \(\lambda\) is the decay constant, and \(t\) is time. This equation implies that the substance decays faster at first, and then the rate slows down over time. To understand it better:

- Assume we start with 100 mg. If it decays to 88 mg over time, we'd use the equation in a rearranged form:
\(88 = 100 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\)
- Take the natural logarithm of both sides, solve for \(t\), and with the known \(T_{1/2}\), get the exact time.

Exponential decay covers various real-world scenarios beyond radioactivity, highlighting its importance in both academics and research.