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Iodine \({ }_{53}^{131} \mathrm{I}\) is used in diagnostic and therapeutic techniques in the treatment of thyroid disorders. This isotope has a half-life of 8.04 days. What percentage of an initial sample of \({ }_{53}^{131}\)I remains after 30.0 days?

Short Answer

Expert verified
Approximately 8.35% of the initial sample remains after 30 days.

Step by step solution

01

Identify the Given Information

The problem provides the following details: the initial isotope is \\( \mathrm{Iodine}\) \\( ({}_{53}^{131}\mathrm{I})\), the half-life is 8.04 days, and the time after which we need to find the remaining percentage is 30.0 days.
02

Define the Formula for Radioactive Decay

To solve the problem, we use the formula for the decay of a substance: \\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] \where \\( N(t) \) is the remaining quantity of the substance after time \\( t\), \\( N_0\) is the initial quantity, and \\( T_{1/2} \) is the half-life.
03

Calculate the Number of Half-Lives

Determine how many half-lives \\( 30.0 \) days correspond to by dividing the total time by the half-life: \\[ \text{Number of half-lives} = \frac{30.0 \text{ days}}{8.04 \text{ days/half-life}} = 3.73 \]
04

Determine Remaining Percentage

Substitute the number of half-lives into the decay formula to find the remaining quantity as a percentage: \\[ N(t) = N_0 \left( \frac{1}{2} \right)^{3.73} \] \Calculate \\( \left( \frac{1}{2} \right)^{3.73} \approx 0.0835 \), \so about 8.35% of the initial sample remains.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-life
Half-life is a crucial concept in understanding radioactive decay. It refers to the amount of time it takes for half of the radioisotope material to decay. In other words, it's the time taken for half of the atoms in a radioactive sample to transform through decay. This transformation occurs due to the radioactivity of the isotope.
The half-life can vary dramatically between different isotopes. Some might decay in seconds, while others can take thousands or even millions of years. For instance, in our example, Iodine-131 has a half-life of 8.04 days, meaning that after 8.04 days, only half of any given amount remains radioactive. After another 8.04 days, half of that amount will decay again, and so on.
Understanding half-life is essential for applications in medicine, as it helps predict how long a radioactive isotope will remain active and effective.
Isotope
Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons. This difference in neutrons results in a change of the atomic mass but not the chemical properties. Isotopes can be stable or unstable.
Unstable isotopes are known as radioisotopes, and they decay over time, emitting radiation in the process. Each element on the periodic table can have multiple isotopes, each with unique properties.
For instance, Iodine-127 is a stable isotope and non-radioactive, while Iodine-131 is unstable and radioactive. This decay property of Iodine-131 is what makes it useful in various diagnostic and therapeutic medical procedures, such as treating thyroid disorders.
Iodine-131
Iodine-131 is a radioactive isotope of iodine. It is commonly used in the field of nuclear medicine, especially in the diagnosis and treatment of thyroid disorders. This isotope is a beta and gamma emitter, meaning it emits beta particles and gamma rays as it decays.
Iodine-131 has a relatively short half-life of 8.04 days, making it suitable for medical applications as its radioactivity diminishes quickly.
When used in medical treatments, such as radioactive iodine therapy, its emission of radiation can target specific cells, like overactive thyroid cells, reducing their function or eliminating them. This characteristic is why it's widely used in treating conditions like hyperthyroidism and certain types of thyroid cancer.
Thyroid Disorders
Thyroid disorders are medical conditions affecting the thyroid gland, an essential organ in the regulation of metabolism and energy usage in the body. Common disorders include hyperthyroidism (overactive thyroid) and hypothyroidism (underactive thyroid).
Symptoms of thyroid disorders can vary but often include fatigue, weight changes, and changes in heart rate. Treating these disorders effectively is crucial for maintaining overall health.
Iodine-131 plays a significant role in treating these disorders due to its ability to selectively target thyroid tissue. In cases of hyperthyroidism, radioactive iodine therapy can help reduce the thyroid gland's activity by ablating thyroid cells which take up iodine. This targeted approach minimizes damage to surrounding tissues and helps in restoring normal thyroid function.

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Most popular questions from this chapter

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