Initial ratio of active nuclei in two different samples is 2:3. Their half lives are 2 hr and 3 hr respectively. Ratio of their activities at the end of 12 hr is ?

To find the ratio of the activities of two samples after a certain period, we need to consider both the initial ratio of active nuclei and the half-lives of the samples. Let's break this down step by step.

Understanding the Concepts

First, we need to clarify a few terms:

• Active nuclei: These are the radioactive atoms that are capable of undergoing decay.
• Half-life: This is the time required for half of the radioactive nuclei in a sample to decay.
• Activity: This refers to the rate at which a sample of radioactive material decays, typically measured in disintegrations per second.

Initial Setup

We start with two samples, A and B, with an initial ratio of active nuclei of 2:3. Let's denote:

• Initial active nuclei in sample A = 2x
• Initial active nuclei in sample B = 3x

Decay Calculation

Next, we need to calculate how many active nuclei remain after 12 hours for each sample, using the formula for radioactive decay:

N(t) = N0 * (1/2)^(t/T)

Where:

• N(t): Remaining nuclei after time t
• N0: Initial number of nuclei
• T: Half-life of the substance

Sample A

For sample A, with a half-life of 2 hours:

• Time elapsed = 12 hours
• Number of half-lives = 12 hr / 2 hr = 6

Applying the decay formula:

N_A(12) = 2x * (1/2)^6 = 2x * 1/64 = 2x/64 = x/32

Sample B

For sample B, with a half-life of 3 hours:

• Time elapsed = 12 hours
• Number of half-lives = 12 hr / 3 hr = 4

Using the decay formula again:

N_B(12) = 3x * (1/2)^4 = 3x * 1/16 = 3x/16

Calculating Activities

The activity of a sample is directly proportional to the number of remaining active nuclei. Therefore, we can express the activities of samples A and B as:

• Activity of A (A_A) = λ_A * N_A(12)
• Activity of B (A_B) = λ_B * N_B(12)

Where λ (lambda) is the decay constant, which can be calculated as:

λ = ln(2) / T

Decay Constants

For sample A (T = 2 hr):

λ_A = ln(2) / 2

For sample B (T = 3 hr):

λ_B = ln(2) / 3

Final Activity Ratio

Now, we can find the ratio of the activities:

Activity Ratio = A_A / A_B = (λ_A * N_A(12)) / (λ_B * N_B(12))

Substituting the values:

Activity Ratio = [(ln(2)/2) * (x/32)] / [(ln(2)/3) * (3x/16)]

Notice that the x and ln(2) terms will cancel out:

Activity Ratio = [(1/2) * (1/32)] / [(1/3) * (3/16)]

Now simplifying this gives:

Activity Ratio = (1/64) / (3/16) = (1/64) * (16/3) = 16/192 = 1/12

Final Result

The ratio of the activities of the two samples at the end of 12 hours is 1:12. This means that after 12 hours, sample A has significantly less activity compared to sample B, reflecting the differences in their half-lives and initial quantities.