Question

A human body required \(0.01 \mu\) activity of radioactive substance after 24 hours. Half-life of radioactive substane is 6 hours. Then injection of maximum activity of a radioactive suhstance that can he injected is (a) \(0.08\) (b) \(0.04\) (c) \(0.16\) (d) \(0.32\)

Short answer

The maximum injection activity is 0.16 microcuries (option c).

Step-by-step solution

Understand The Problem

We need to find the initial activity of a radioactive substance to be injected, given that its remaining activity is 0.01 microcuries after 24 hours, and the half-life of the substance is 6 hours.

Use The Half-Life Formula

Radioactive decay follows the formula: \[ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \] where \( A \) is the activity after time \( t \), \( A_0 \) is the initial activity, and \( T_{1/2} \) is the half-life.

Substitute Given Values

We know \( A = 0.01 \ \mu \), \( t = 24 \) hours, and \( T_{1/2} = 6 \) hours.Substitute into the formula:\[ 0.01 = A_0 \times \left(\frac{1}{2}\right)^{\frac{24}{6}} \]

Simplify the Exponent

Calculate the exponent: \( \frac{24}{6} = 4 \). Therefore, the equation becomes:\[ 0.01 = A_0 \times \left(\frac{1}{2}\right)^4 = A_0 \times \frac{1}{16} \]

Solve for Initial Activity

Rearrange the equation to solve for \( A_0 \):\[ A_0 = 0.01 \times 16 = 0.16 \ \mu \]

Key concepts

Half-life Calculation

Understanding the concept of half-life is crucial for calculating how substances decay over time in radioactive decay problems. The half-life is the period it takes for a substance to reduce to half of its initial activity. Let's say you have a substance with a half-life of 6 hours. This means after 6 hours, only half of the original material remains active.

To find out how much of the substance remains after a specific period, you count how many half-lives fit into that time span. For instance, if 24 hours pass, and the half-life is 6 hours, you have gone through 4 half-lives (24 divided by 6 equals 4). At each half-life, the activity of the substance is halved:

• After the first half-life, you have 50% of the initial activity.
• After the second, 25% remains.
• After the third, 12.5% is left.
• And the fourth half-life leaves you with just 6.25%.

This sequential halving process gives insight into how quickly a radioactive substance diminishes over time.

Understanding these decreases helps determine how much of a substance was initially present.

Radioactive Activity

Radioactive activity refers to the rate at which a radioactive substance disintegrates. As we expose a sample to time, its activity decreases exponentially. In practical terms, activity is measured in units called microcuries (µCi), which quantify the disintegration rate. Tracking this activity is fundamental because it tells us the likelihood of decay occurring within a specific nucleus per second.

More activity means more nuclear disintegrations happening. When working with materials in radioactive decay problems, one often needs to decide on starting activity levels that ensure safety and efficiency. For example, when deciding on how much radioactive material to inject into a patient, doctors need to calculate how the activity level will decrease over time to arrive at a safe yet effective amount.

Once the target activity level is known, calculations can ensure the initial dose is adequate to maintain therapeutic effectiveness while avoiding unnecessary exposure.

Exponential Decay Formula

The exponential decay formula is vital for predicting how the quantity of a radioactive substance changes over time. This formula is given by: \[ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where:

• \( A \) is the activity after time \( t \).
• \( A_0 \) is the initial activity.
• \( T_{1/2} \) is the half-life of the substance.

By plugging the known values into this equation, you can calculate the remaining activity at any given point in time.

Let's apply this with an example: Suppose you need to have 0.01 µCi of a material after 24 hours, and its half-life is 6 hours. Using four half-lives in this time span (24 divided by 6), apply the formula to backtrack the necessary initial activity. The remaining activity becomes a simple fraction of the original: \( \frac{1}{16} \) of the original value.

Adjust the initial value accordingly by multiplying the desired activity by the inverse of the survival fraction to find the starting point. This calculation indicates how to distribute a substance timely and helps secure the desired end result effectively.