Question

The half-life of indium- 111, a radioisotope used in studying the distribution of white blood cells, is \(t_{1 / 2}=2.805\) days. What is the decay constant of \({ }^{111}\) In?

Short answer

The decay constant of \( ^{111} \)In is approximately \( 0.247 \text{ day}^{-1} \).

Step-by-step solution

Identify the Known Value

The half-life, denoted as \( t_{1/2} \), of indium-111 is given as \( 2.805 \) days.

Use the Half-Life Formula for Decay Constant

The decay constant, \( \lambda \), is related to the half-life by the equation:\[\lambda = \frac{\ln(2)}{t_{1/2}}\]This equation arises because the half-life is the time required for half of the radioactive nuclei to decay.

Plug in the Known Half-Life

Substitute the given half-life \( t_{1/2} = 2.805 \) days into the formula:\[\lambda = \frac{\ln(2)}{2.805}\]

Calculate the Natural Logarithm

Calculate \( \ln(2) \), which is approximately \( 0.693 \).

Perform the Division

Divide \( 0.693 \) by \( 2.805 \) to find the decay constant:\[\lambda \approx \frac{0.693}{2.805} \approx 0.247 \text{ day}^{-1}\]

Key concepts

half-life

Half-life is a fundamental concept in the study of radioactive decay. It refers to the time it takes for half of the radioactive nuclei in a sample to decay into its products. In the context of indium-111, the half-life is given as 2.805 days. This value tells us how long it takes for half of the indium-111 atoms to transform.

Understanding half-life is crucial because it helps scientists predict how long a radioactive element will remain active. Knowing the half-life can also enable us to calculate the decay constant, a key parameter that describes the rate of decay.

• Half-life is specific to each radioactive isotope.
• It provides insights into how quickly a substance will lose its radioactivity.
• The half-life remains constant regardless of the starting quantity of the sample.

decay constant

The decay constant, denoted by the symbol \( \lambda \), is a measure that describes the speed of radioactive decay. It is related to the half-life of a substance. The smaller the decay constant, the slower the decay rate. For indium-111, we calculated the decay constant using its half-life.

To find the decay constant using half-life, we use the formula:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
This equation shows the relationship between decay constant and half-life, where \( \ln(2) \approx 0.693 \) is the natural logarithm of 2. The decay constant tells how quickly the indium-111 will reduce by half in its original quantity.

• The decay constant is expressed in units of inverse time, like \( \text{day}^{-1} \).
• It is a crucial factor in the exponential decay equation.
• Understanding \( \lambda \) helps in calculating the remaining quantity of a substance over time.

natural logarithm

The natural logarithm, represented by \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828. It is used extensively in exponential growth and decay problems, including radioactive decay. When computing the decay constant from half-life, the natural log of 2, \( \ln(2) \), plays a crucial role.

In our calculations for indium-111, we used \( \ln(2) \) because it is the factor that transforms the equation from describing half-life to expressing the decay constant. Mathematically, the value of \( \ln(2) \) is approximately 0.693, and it is used in the decay constant calculation formula:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]

• \( \ln \) helps convert complex exponential relationships into simpler linear ones.
• It allows easier manipulation in calculus and algebra.
• Understanding \( \ln(2) \) is essential for working with half-life and decay constant equations.