Problem 66 - University Physics with Modern Physics

Question

\(10^{30}\) Atoms of a radioactive sample remain after 10 half-lives. How many atoms remain after 20 half-lives?

Step-by-Step Solution

Answer

Answer: After 20 half-lives, there will be N_{20} = 10^{30} * 2^{-10} atoms remaining.

Step 1: Understand half-life

A half-life is the time it takes for half of a radioactive sample to decay. After each half-life, the number of remaining radioactive atoms is halved.

Step 2: Find the initial number of atoms

We are given that 10^{30} atoms remain after 10 half-lives. Relating to the half-life concept, we can find the initial number of atoms (N_0) by multiplying the remaining atoms by 2, 10 times since it has gone through 10 half lives: N_0 = 10^{30} * 2^{10}

Step 3: Calculate number of atoms after 20 half-lives

Knowing the initial number of atoms, we can now find the number of atoms remaining after 20 half-lives. After each half-life, the number of remaining atoms is halved. Thus, after 20 half-lives, the remaining atoms can be calculated by dividing the initial number of atoms by 2, 20 times: N_{20} = N_{0} * 2^{-20}

Step 4: Substitute and solve

Replace N_0 with the expression we found in step 2 and solve for N_{20}: N_{20} = (10^{30} * 2^{10}) * 2^{-20} Simplify the expression: N_{20} = 10^{30} * 2^{-10}

Step 5: Final answer

After 20 half-lives, there will be: N_{20} = 10^{30} * 2^{-10} atoms remaining.

Key Concepts

Half-life

The concept of half-life is like a ticking clock for radioactive substances. It tells us how fast the material is decaying.
Imagine you are watching a cake that keeps getting smaller and smaller. With each slice, half of it disappears. This is similar to what happens with radioactive substances. The half-life is the time it takes for half of the radioactive atoms in a sample to decay away.
This means, with every passing half-life:

• The original number of radioactive atoms is reduced by half.
• If you start with 100 atoms, after one half-life, you'll have 50 remaining.
• After the second half-life, only 25 will be left, and so on.

Recognizing and understanding half-lives allows scientists and students to predict how long a radioactive material will remain active. This is crucial in fields like archaeology, medicine, and nuclear physics.

Exponential Decay

Exponential decay is a fancy way of saying something decreases quickly at first, but then slowly over time.
If you ever watched a balloon deflate slowly after someone let go of the string, you've got a hint of what exponential decay is like. In the world of physics and chemistry, it's used to describe how radioactive materials lose their atoms.

• At the start, the number of atoms decreases rapidly, each half-life noticeably reducing the amount.
• As time goes on, there are fewer atoms left to decay, so the reduction slows down.

Formally, exponential decay is modeled using the formula:\[ N(t) = N_0 \times (\frac{1}{2})^{t/t_{1/2}} \]where:

• \(N(t)\) is the quantity of the substance that still remains and has not decayed after time \(t\).
• \(N_0\) is the initial amount of the substance.
• \(t_{1/2}\) is the half-life.

This mathematical framework is crucial for accurate predictions in situations like dating ancient artifacts or determining the safety of materials in waste disposal.

Radioactive Atoms

Radioactive atoms are like small fireworks, but on a microscopic scale. They constantly break down and release energy in the form of radiation. This breaking down is called decay. Radioactive atoms are unstable because their nuclei want to reach a more balanced state.
Imagine each radioactive atom as a tiny ticking bomb ready to burst. When it does, it transforms into a more stable atom, releasing energy in the process. Here's what you should know:

• A radioactive atom has an unsteady nucleus that likes to stabilize itself by shedding particles or energy.
• These atoms are key in various scientific and medical fields, providing insights into ages of objects or acting as tracer elements in medical scanning.
• The decay of these atoms is fundamental to nuclear power generation and many technologies we depend on daily.

Understanding radioactive atoms is essential not just for solving textbook exercises but for comprehending many modern technologies and processes.