The half life of a radioactive isotope ' \(X\) ' is 20 years. It decays into another stable element \(\mathrm{Y}\) '. The two elements ' \(X\) ' and ' \(Y\) ' were found to be in the ratio \(1: 7\) in a sample of a given rock. Then age of the rock is estimated to be : (1) 60 years (2) 80 years (3) 100 years (4) 40 years

The concept of half-life can seem tricky at first, but it's really quite simple. The half-life of a radioactive isotope is the time required for half of the original sample to decay. For example, if you start with 100 grams of a substance with a half-life of 20 years, after 20 years you'll have 50 grams remaining. After another 20 years (40 years total), you will have 25 grams left, and so on. The formula for half-life is generally represented as \[ N(t) = N_0 \times \frac{1}{2}^\frac{t}{t_{1/2}} \] where:

• \(N(t)\) is the remaining quantity of substance after time \(t\)
• \(N_0\) is the initial quantity of the substance
• \(t_{1/2}\) is the half-life

Understanding this formula helps you predict how a substance will decay over time.

Now, let’s go deeper into the decay process over time. In the problem, you start with an initial quantity of the isotope 'X'. Over time, this isotope decays into a stable element 'Y'. According to the half-life principle, if you start with a certain amount of 'X', after one half-life, 50% of 'X' will remain. After two half-lives, only 25% remains. For this specific exercise, you are told that the ratio of 'X' to 'Y' in the rock is 1:7. This means for every one unit of 'X', there are seven units of 'Y' formed from the decayed 'X'. Therefore, if 1 part 'X' remains, it implies that 7 parts 'Y' have been created from that decayed 'X'. Given this 1:7 ratio, the remaining fraction of 'X' would be \( \frac{1}{8} \) of the original amount.

When dealing with ratios in radioactive decay, the ratio of the remaining isotope to the newly formed element can tell us a lot. Consider the decay process in our exercise: after a certain time, the original isotope 'X' ratio to the stable element 'Y' is 1:7. This ratio helps us figure out how many half-lives have passed. We know that \( \frac{1}{8} \) of the original 'X' remains (since 1 part 'X' to 7 parts 'Y' means out of 8 parts, 1 is the remaining 'X'). Using the half-life formula, we find that: \[ \frac{1}{8} = \frac{1}{2}^3 \] This indicates that three half-lives have elapsed. Finally, as each half-life is 20 years, the total time that has passed since the rock started to decay can be calculated by multiplying the number of half-lives by the duration of each half-life: \[ 3 \times 20 = 60 \text{ years} \] Therefore, the age of the rock is 60 years.