Question

If a substance with half-life of 3 days is taken to another place in 12 days. What is the amount of substance left now? (a) \(1 / 8\) (b) \(1 / 32\) (c) \(1 / 4\) (d) \(1 / 16\)

Short answer

The correct answer is (d) \(1 / 16\).

Step-by-step solution

Identify Half-Life Details

We know the half-life of the substance is 3 days, which means every 3 days, the amount of the substance is reduced by half.

Determine the Number of Half-Life Periods

Determine how many half-life periods have passed in the given 12 days. Divide the total time passed (12 days) by the half-life (3 days):\[\text{Number of half-lives} = \frac{12}{3} = 4\]

Calculate Remaining Substance

With every half-life period, the substance quantity is halved. Since there are 4 half-life periods, we will apply this halving process 4 times:After 1st half-life: \(\frac{1}{2}\)After 2nd half-life: \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)After 3rd half-life: \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\)After 4th half-life: \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\)

Finalize the Solution

After 4 half-life periods, only \(\frac{1}{16}\) of the original substance remains.

Key concepts

Half-life

When we talk about the half-life of a substance, we refer to the time it takes for half of that substance to decay or reduce to half of its initial amount. This concept is crucial in understanding how certain materials, especially radioactive ones, decrease over time.
If you have a substance with a half-life of 3 days, every 3 days, the material left is only half of what was there before. So, in 6 days, a quarter is left, in 9 days, an eighth, and this pattern continues. Knowing the half-life helps in calculating how much of a substance remains after a specific period.
Consider our previous example, if 12 days have passed for a material with a half-life of 3 days, it means four half-life periods have gone by. In each period, the substance halves:

• 1st period: 1/2 of the original amount.
• 2nd period: 1/4 of the original amount.
• 3rd period: 1/8 of the original amount.
• 4th period: 1/16 of the original amount.

Ultimately, understanding half-life helps in various fields, from archaeology to medicine, where decay measurements offer insights into timings and ages.

Exponential decay

Exponential decay is a process where the quantity of an item decreases at a rate proportional to its current value. This concept is fundamental in many scientific and mathematical analyses, as it describes how certain quantities fade over time.
The essence of exponential decay lies in its consistent and proportional reduction. For example, in the halving process like radioactive decay, you could express the remaining substance after 'n' half-lives as a mathematical expression:\[\text{Remaining quantity} = \left(\frac{1}{2}\right)^n\]where 'n' represents the number of half-life periods that have passed. This method of decay is predictable, and the process always leads to the original quantity becoming negligible over time.
Exponential decay is visually represented by a continually downward-sloping curve. At each stage of decay, the amount lessens more rapidly at first and then gradually slows down as it decreases. Familiar examples in life include not only radioactive materials decaying over time but also things like the cooling of a hot object or the discharge of a capacitor in electronics.

Nuclear chemistry

Nuclear chemistry focuses on the radioactivity and changes that occur in the nucleus of an atom. It’s a branch of chemistry particularly vital in energy production, medical treatments, and understanding cosmic phenomena.
In nuclear chemistry, understanding radioactive decay is significant, as it pertains to how unstable atoms shed energy in the form of particles or rays, achieving a more stable state. Such decay includes processes like alpha and beta decay.
A fascinating aspect of nuclear chemistry is how it utilizes half-life and exponential decay to monitor and use radioisotopes. In medicine, specific isotopes help diagnose diseases by tracking their decay patterns. For energy, nuclear reactors use radioactive materials to generate significant amounts of power, depending heavily on precise nuclear reactions controlled by the principles of decay.
Moreover, nuclear chemistry aids in dating ancient artifacts. By measuring remaining radioactive elements, scientists can estimate how long an item has been around. Thus, nuclear chemistry not only powers our homes but also unravels the history of the earth and the universe.