Two radioactive substance X and Y have a life period of 50 mins and 100 mins respectively.Initially both of contain equal no.of atoms.Find the ratio of atoms left Nx/Ny after 200 minutes.

To determine the ratio of the remaining atoms of two radioactive substances, X and Y, after 200 minutes, we can use the concept of half-life. The half-life is the time required for half of the radioactive atoms in a sample to decay. In this case, substance X has a half-life of 50 minutes, while substance Y has a half-life of 100 minutes. Let's break this down step by step.

Understanding Half-Life

The half-life tells us how many times a substance will halve over a given period. For substance X, which has a half-life of 50 minutes, we can calculate how many half-lives fit into 200 minutes:

• 200 minutes / 50 minutes (half-life of X) = 4 half-lives

For substance Y, with a half-life of 100 minutes:

• 200 minutes / 100 minutes (half-life of Y) = 2 half-lives

Calculating Remaining Atoms

Let’s denote the initial number of atoms of both substances as N₀. After each half-life, the number of atoms remaining can be calculated using the formula:

N = N₀ * (1/2)^(t/T_1/2)

Where:

• N is the number of atoms remaining.
• N₀ is the initial number of atoms.
• t is the total time elapsed.
• T_1/2 is the half-life of the substance.

Atoms Remaining for Substance X

For substance X, after 200 minutes:

• N_X = N₀ * (1/2)^(200/50)
• N_X = N₀ * (1/2)^4
• N_X = N₀ * (1/16)

Atoms Remaining for Substance Y

For substance Y, after 200 minutes:

• N_Y = N₀ * (1/2)^(200/100)
• N_Y = N₀ * (1/2)^2
• N_Y = N₀ * (1/4)

Finding the Ratio

Now that we have the remaining atoms for both substances, we can find the ratio of atoms left:

• Ratio N_X/N_Y = (N₀ * (1/16)) / (N₀ * (1/4))
• Ratio N_X/N_Y = (1/16) / (1/4)
• Ratio N_X/N_Y = (1/16) * (4/1) = 4/16 = 1/4

Final Result

The ratio of the number of atoms left of substance X to substance Y after 200 minutes is:

This means that for every atom of substance Y remaining, there is only one-fourth of an atom of substance X left. This illustrates how the differing half-lives significantly affect the decay of radioactive substances over time.