The number of atoms in a radioactive sample reduces from 1000 to 100 in two days. The number of atoms remaining at the end of four days will be?

To determine the number of atoms remaining in a radioactive sample after four days, we first need to understand the concept of half-life, which is the time required for half of the radioactive atoms in a sample to decay. In your case, the sample starts with 1000 atoms and reduces to 100 atoms in two days. Let's break this down step by step.

Calculating the Half-Life

From the information provided, we can see that the sample decreases from 1000 to 100 atoms in two days. This means that the sample has gone through two half-lives during this period. Here's how we can confirm that:

• After the first half-life (1 day), the number of atoms would be half of 1000, which is 500.
• After the second half-life (another day), the number of atoms would again be halved: 500 becomes 250.
• However, we see that after two days, the sample has only 100 atoms left, indicating that the decay is more rapid than a simple two half-lives.

To find the actual half-life, we can set up the equation based on the exponential decay formula:

Exponential Decay Formula

The general formula for radioactive decay is:

N(t) = N₀ * (1/2)^(t/T₁/₂)

Where:

• N(t) = remaining quantity of the substance after time t
• N₀ = initial quantity of the substance
• T₁/₂ = half-life of the substance

In this case, we know:

• N₀ = 1000
• N(t) = 100
• t = 2 days

Plugging these values into the formula, we have:

100 = 1000 * (1/2)^(2/T₁/₂)

Dividing both sides by 1000 gives:

0.1 = (1/2)^(2/T₁/₂)

Solving for Half-Life

To solve for T₁/₂, we can take the logarithm of both sides:

log(0.1) = (2/T₁/₂) * log(1/2)

Now, we can isolate T₁/₂:

T₁/₂ = 2 / (log(0.1) / log(0.5))

Calculating the logs gives us:

log(0.1) ≈ -1 and log(0.5) ≈ -0.301

Thus, T₁/₂ ≈ 2 / ( -1 / -0.301) ≈ 2 * 0.301 ≈ 0.602 days.

Finding Remaining Atoms After Four Days

Now that we have the half-life, we can calculate how many half-lives fit into four days:

Number of half-lives in four days = 4 days / 0.602 days ≈ 6.64 half-lives.

Since we can only consider whole half-lives for decay, we will consider 6 half-lives:

After 6 half-lives, the remaining quantity of atoms can be calculated as follows:

N(t) = 1000 * (1/2)^6 = 1000 * 1/64 = 15.625 atoms.

Since we cannot have a fraction of an atom, we round down to the nearest whole number, which gives us approximately 15 atoms remaining after four days.

Summary

In summary, after four days, the number of atoms remaining in the radioactive sample would be about 15. This example illustrates the exponential nature of radioactive decay and how to apply the concept of half-life effectively.