At a certain instant a piece of radioactive material contain 10¹² atom.the half life the materials is 30days . calculate the time of disintegration in the first second, what time will elapsed before 10⁴ atom remain , what is the count rate at this time

To tackle this problem, we need to understand a few key concepts related to radioactive decay, particularly the half-life of a substance and how to calculate the number of atoms remaining after a certain period. Let's break this down step by step.

Understanding Half-Life

The half-life of a radioactive material is the time it takes for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 30 days. This means that after 30 days, only half of the original atoms will remain.

Calculating Disintegration in the First Second

First, we need to determine how many atoms disintegrate in the first second. The decay of radioactive material can be modeled using the exponential decay formula:

• N(t) = N0 * e^(-λt)

Where:

• N(t) = number of atoms remaining at time t
• N0 = initial number of atoms
• λ = decay constant
• t = time elapsed

To find λ, we can use the relationship between half-life (T½) and λ:

• λ = ln(2) / T½

Substituting the half-life:

• λ = ln(2) / (30 days) ≈ 0.0231 days^-1

Next, we convert 30 days into seconds for our calculations:

• 30 days = 30 * 24 * 60 * 60 = 2,592,000 seconds

Now, we can calculate λ in seconds:

• λ ≈ 0.0231 / 2,592,000 ≈ 8.91 x 10^-9 seconds^-1

Now, we can find the number of atoms remaining after 1 second:

• N(1) = 10¹² * e^(-8.91 x 10^-9 * 1) ≈ 10¹² * 0.9999999911 ≈ 10¹² atoms

Thus, the number of disintegrated atoms in the first second is:

• Disintegrated = N0 - N(1) ≈ 10¹² - 10¹² = 0 atoms

Time Until 10⁴ Atoms Remain

Next, we want to find out how long it will take for the number of atoms to decrease to 10⁴. We can set up the equation:

• 10⁴ = 10¹² * e^(-λt)

Rearranging gives us:

• e^(-λt) = 10⁴ / 10¹² = 10^-8

Taking the natural logarithm of both sides:

• -λt = ln(10^-8)

Substituting λ:

• t = -ln(10^-8) / λ

Calculating this gives:

• t ≈ -(-18.42) / (8.91 x 10^-9) ≈ 2.07 x 10⁹ seconds

Converting this into days:

• t ≈ 2.07 x 10⁹ / (60 * 60 * 24) ≈ 24,000 days

Count Rate Calculation

The count rate can be calculated using the decay constant and the number of remaining atoms. The count rate (R) is given by:

• R = λ * N(t)

Substituting N(t) = 10⁴:

• R = (8.91 x 10^-9) * (10⁴) ≈ 8.91 x 10^-5 disintegrations per second

In summary, in the first second, no atoms disintegrate, it will take approximately 24,000 days for the sample to reduce to 10⁴ atoms, and the count rate at that time will be about 8.91 x 10^-5 disintegrations per second. This illustrates the slow nature of radioactive decay, especially when dealing with long half-lives.