A radiactive has substance has life of 64.8h.A sample containing this isotope has an initial activity of 40 microcoulumb .Calculate the no.of nuclei that decay in the time interval between t1=10.0h and t2=12.0h.

To calculate the number of nuclei that decay between the time intervals of t1 = 10.0 hours and t2 = 12.0 hours for a radioactive substance with a half-life of 64.8 hours and an initial activity of 40 microcoulombs, we can follow a systematic approach. First, we need to understand the relationship between activity, decay constant, and the number of radioactive nuclei present.

Understanding Radioactive Decay

Radioactive decay is a random process where unstable nuclei lose energy by emitting radiation. The activity (A) of a radioactive sample is defined as the number of decays per unit time, which is proportional to the number of radioactive nuclei (N) present in the sample. The relationship can be expressed as:

A = λN

Here, λ (lambda) is the decay constant, which is related to the half-life (T½) of the substance by the formula:

λ = ln(2) / T½

Step 1: Calculate the Decay Constant

Given the half-life (T½) of the substance is 64.8 hours, we can calculate the decay constant:

λ = ln(2) / 64.8 h ≈ 0.0107 h⁻¹

Step 2: Determine the Initial Number of Nuclei

Next, we can find the initial number of nuclei (N₀) using the initial activity:

N₀ = A / λ

Substituting the values:

N₀ = 40 µC / 0.0107 h⁻¹ ≈ 3735.51 nuclei

Step 3: Calculate the Number of Nuclei at t1 and t2

Now, we need to find the number of nuclei remaining at t1 = 10.0 hours and t2 = 12.0 hours. The number of nuclei remaining at any time t can be calculated using the formula:

N(t) = N₀ * e^(-λt)

For t1 = 10.0 hours:

N(10) = 3735.51 * e^(-0.0107 * 10) ≈ 3735.51 * e^(-0.107) ≈ 3442.68 nuclei

For t2 = 12.0 hours:

N(12) = 3735.51 * e^(-0.0107 * 12) ≈ 3735.51 * e^(-0.1284) ≈ 3340.74 nuclei

Step 4: Calculate the Number of Decayed Nuclei

The number of nuclei that have decayed between t1 and t2 can be found by subtracting the number of remaining nuclei at t2 from the number at t1:

Decayed Nuclei = N(10) - N(12)

Decayed Nuclei = 3442.68 - 3340.74 ≈ 101.94 nuclei

Final Result

Thus, approximately 102 nuclei decay in the time interval between t1 = 10.0 hours and t2 = 12.0 hours. This calculation illustrates how we can use the principles of radioactive decay to determine the number of decayed nuclei over a specific time period, utilizing the decay constant and initial activity of the substance.