Problem 4 - Applied Physics

Question

Find the remaining quantity of radon 222 from an original sample of \(75.0 \mathrm{~g}\) after \(10.0\) days. Its half-life is \(3.82\) days.

Step-by-Step Solution

Answer

After 10 days, approximately 12.525 grams of radon 222 remains.

Step 1: Understand the formula for exponential decay

The formula used to find the remaining quantity of a radioactive substance is given by:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{\frac{1}{2}}}} \]where:- \(N(t)\) is the remaining quantity after time \(t\).- \(N_0\) is the initial quantity.- \(t_{\frac{1}{2}}\) is the half-life of the substance.- \(t\) is the time elapsed.

Step 2: Substitute known values into the formula

Insert the given values into the exponential decay formula:- \(N_0 = 75.0\,\mathrm{g}\)- \(t = 10.0\,\mathrm{days}\)- \(t_{\frac{1}{2}} = 3.82\,\mathrm{days}\)Substitute these into the formula:\[N(10) = 75.0 \times \left( \frac{1}{2} \right)^{\frac{10.0}{3.82}}\]

Step 3: Calculate the exponent

First, compute the value of the exponent \(\frac{10.0}{3.82}\):- Divide \(10.0\) by \(3.82\) to get approximately \(2.62\).Thus, the exponent becomes:\[N(10) = 75.0 \times \left( \frac{1}{2} \right)^{2.62} \]

Step 4: Compute the power of one-half

Calculate \(\left( \frac{1}{2} \right)^{2.62}\):- Use a calculator or exponent tools to compute.- \(\left( \frac{1}{2} \right)^{2.62} \approx 0.167\).

Step 5: Calculate the remaining quantity

Now, compute \(N(10)\) by multiplying:\[N(10) = 75.0 \times 0.167 \approx 12.525 \text{ grams}\]So, the remaining quantity of radon 222 after 10 days is approximately \(12.525\,\mathrm{g}\).

Key Concepts

Radioactive Decay

Radioactive decay is a fundamental concept in nuclear chemistry and physics. It's the process by which an unstable atomic nucleus loses energy by emitting radiation.
This decay can happen in several different forms, such as alpha, beta, or gamma decay.

• Alpha decay: This involves the emission of alpha particles, which are helium nuclei composed of two protons and two neutrons.
• Beta decay: This involves the conversion of a neutron into a proton (or vice versa) in the nucleus, accompanied by the emission of a beta particle.
• Gamma decay: This occurs when the nucleus releases energy in the form of gamma rays without a change in the number of protons or neutrons.

Each of these processes reduces the energy of the nucleus and contributes to the transformation of elements from one type to another.
The important thing to remember is that radioactive decay is a random process at an individual atom level, but predictable over large collections of atoms.
Hence, formulas like the exponential decay formula we're using help predict how a large sample will behave over time.

Half-Life Calculations

Half-life is a core concept when studying radioactive decay, describing the time it takes for half of a given amount of a radioactive substance to decay.
It's a key factor in calculating how much of a substance remains after a certain period. In simple terms, during each half-life, half of the substance's atoms decay into different atoms, reducing the amount of the original material by half.

• The half-life remains constant, meaning no matter how much of the substance you have, its half-life will not change.
• This is why substances with shorter half-lives decay more quickly, while those with long half-lives decay much more slowly.

The formula for exponential decay incorporates half-life as a key variable:
\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{\frac{1}{2}}}} \]
where \(N(t)\) is the quantity remaining after time \(t\), \(N_0\) is the initial quantity, and \(t_{\frac{1}{2}}\) is the half-life. This mathematical approach makes it possible to accurately calculate the remaining quantity of any radioactive material, like in the case of radon 222, based on the time elapsed and its known half-life.

Radon 222

Radon 222 is a naturally occurring radioactive gas that forms from the decay of uranium in soil, rock, and water.
It is colorless, odorless, and tasteless, which means you can't detect it with human senses.
However, because radon 222 undergoes radioactive decay, it can contribute to health risks if inhaled over a long period.

• Radon 222 decays into other radioactive elements known as radon progeny or daughters, continuing the radioactive decay chain.
• Radon exposure is a known risk factor for lung cancer, making understanding its behavior vital for health and safety.

In the exercise, we calculated how much radon 222 remains after a certain period, highlighting its half-life of 3.82 days.
This information is crucial in environments or industries where radon levels need monitoring. Knowing how quickly it decays helps in assessing potential exposure risks and in taking the necessary measures to mitigate them.