Question

what Percent of original radioactive substance is left after 5 half life time (A) \(3 \%\) (B) \(5 \%\) (C) \(6 \%\) (D) \(12 \%\)

Short answer

The remaining percentage of the original radioactive substance after 5 half-lives is approximately \(3\%\), so the correct answer is (A) \(3\%\).

Step-by-step solution

Understand the half-life formula

Half-life (t1/2) is the time it takes for the initial quantity of a radioactive substance to decrease by half. The formula for calculating the remaining quantity after n half-lives have passed is given by: Remaining quantity = Initial quantity * (1/2)^n Where n is the number of half-lives and (1/2)^n is the fraction of the radioactive substance remaining.

Apply the formula for the given number of half-lives

In this exercise, we are given that n = 5, which means that 5 half-lives have passed. We will now apply the formula to find the remaining percentage of the initial radioactive substance: Remaining percentage = (1/2)^5

Calculate the remaining percentage

To find the remaining percentage, raise 1/2 to the power of 5: Remaining percentage = (1/2)^5 = 1/32 = 0.03125 Now, to convert this fraction into a percentage, we multiply by 100: Remaining percentage = 0.03125 * 100 = 3.125% Since the given answer choices are all integers, we must round this value to the nearest integer. In this case, the closest integer value is 3%.

Find the answer

The remaining percentage of the original radioactive substance after 5 half-lives is approximately 3%, so the correct answer is: (A) 3%

Key concepts

Half-life calculation

Half-life is a crucial concept when it comes to understanding radioactive decay. It refers to the time taken for half of the radioactive atoms in a substance to decay. This means, after one half-life, only 50% of the initial radioactive material remains. Here's a simplified way to understand this:

• First half-life: 50% remains
• Second half-life: 25% remains
• Third half-life: 12.5% remains
• Fourth half-life: 6.25% remains
• Fifth half-life: approximately 3.125% remains

Calculating how much of a substance remains after multiple half-lives involves repeated halving. This is expressed by the formula:
\[\text{Remaining Fraction} = \left(\frac{1}{2}\right)^n\]
where \(n\) is the number of half-lives that have passed. This formula tells us the fraction of the original radioactive substance that remains.

Radioactive substance

Radioactive substances are materials that emit radiation as they decay. This decay process involves the transformation of radioactive atoms into different atoms or elements. Here's a bit more to understand about them:

• Every element has isotopes, which are versions of the element with different numbers of neutrons. Some isotopes are stable, while others are radioactive.
• Radioactive elements naturally decay over time, releasing energy in the form of radiation.
• This decay continues until a stable isotope is reached.

Radioactivity is measured by how quickly the substance decays, often represented by its half-life. The rate of decay is unique to each radioactive isotope, helping scientists determine the age of objects and understand the behavior of these elements.

Exponential decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the rate at which a substance decreases is directly proportional to the amount of substance present.
The formula for exponential decay is
\[N(t) = N_0 e^{-\lambda t}\]where:

• \(N(t)\) is the remaining quantity of the substance at time \(t\)
• \(N_0\) is the initial quantity
• e is the base of the natural logarithm
• \(\lambda\) is the decay constant specific to the substance

This formula helps describe the gradual reduction of a radioactive substance over time. Exponential decay appears in various natural and scientific processes, illustrating an orderly decline that happens in predictable steps.