Problem 55 - University Physics with Modern Physics

Question

BIO Radioactive Fallout. One of the problems of in-air testing of nuclear weapons (or, even worse, the use of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 \(\left(^{90} \mathrm{Sr}\right),\) which breaks down by \(\beta^{-}\) decay with a half-life of 28 years. It is chemically similar to calcium and therefore can be incorporated into bones and teeth, where, due to its rather long half-life, it remains for years as an internal source of radiation. (a) What is the daughter nucleus of the \(^{90}\) Sr decay? (b) What percentage of the original level of \(^{90}\) Sr is left after 56 years? (c) How long would you have to wait for the original level to be reduced to 6.25\(\%\) of its original value?

Step-by-Step Solution

Answer

(a) Yttrium-90; (b) 25% (c) 112 years.

Step 1: Identify the Daughter Nucleus

Strontium-90 (\(^{90}_{38}\mathrm{Sr}\)) undergoes \(\beta^{-}\) decay, meaning a neutron in the nucleus is converted into a proton, emitting an electron (\(\beta^{-}\) particle) and an antineutrino. As a result, the atomic number increases by 1, while the mass number remains the same. Therefore, the daughter nucleus is \(^{90}_{39}\mathrm{Y}\) (Yttrium-90).

Step 2: Calculate the Percentage Left After 56 Years

The half-life of \(^{90}\mathrm{Sr}\) is 28 years. To find out how many half-lives have passed in 56 years, divide the total time by the half-life: \(\frac{56}{28} = 2\) half-lives. After one half-life, 50% of the original material remains. After two half-lives, \(50\% \times 50\% = 25\%\) of the original material remains.

Step 3: Determine the Time for 6.25% Remaining

To find when 6.25% of \(^{90}\mathrm{Sr}\) remains, note that 6.25% is equivalent to \(\left(\frac{1}{2}\right)^4\). Therefore, \(^{90}\mathrm{Sr}\) must undergo 4 half-lives to reach this level. Since each half-life is 28 years, the total time is \(4 \times 28 = 112\) years.

Key Concepts

Beta Decay

Beta decay is a process by which an unstable nucleus releases energy by transforming a neutron into a proton. This transformation emits a beta particle, which is simply an electron (\(\beta^-\)), and an antineutrino. It's like a little switch happening in the nucleus. In the case of Strontium-90 (\(^{90}_{38}\mathrm{Sr}\)), beta decay increases the atomic number by one, turning Strontium into Yttrium-90 (\(^{90}_{39}\mathrm{Y}\)).

• Electrons are emitted during beta decay, which are high-energy particles.
• The atomic number changes, but the mass number stays the same.
• Beta decay is a common form of radioactive decay for many heavy elements.

Half-life

The concept of half-life is crucial in understanding radioactive decay. Half-life refers to the time it takes for half of a radioactive substance to decay. For Strontium-90, the half-life is 28 years. This means after 28 years, only half of the Strontium-90 remains. After two half-lives (56 years), only 25% of the original remains, and so forth.

• Half-life gives us a measurable and predictable way to track decay over time.
• Helps in calculating the time until a substance reaches a safe or minimal level.
• Provides insight into the longevity and potential risks of radioactive substances.

Strontium-90

Strontium-90 is a byproduct of nuclear reactions, often found in nuclear fallout. It is similar to calcium, which means it can settle in human bones and teeth if ingested or inhaled. This can be very harmful due to its long half-life and radioactive nature, potentially causing health issues.

• Commonly produced in nuclear explosions and reactor operations.
• Can replace calcium in bones, remaining there for many years.
• Poses long-term health risks due to internal radiation exposure.

Nuclear Fallout

Nuclear fallout refers to the residual radioactive material propelled into the upper atmosphere after a nuclear blast or accident. This material can descend back to earth, contaminating air, water, and soil. A significant concern is the presence of isotopes like Strontium-90, which can enter the food chain.

• Causes widespread environmental contamination.
• Can have long-lasting effects on ecosystems and human health.
• Requires significant cleanup efforts to reduce health risks.