A radioactive sample at any instant has its disintegration rate 5000 disintegrations per minute. After 5min, the rate is 1250 disintegrations per minute. Then, the decay constant (per minute) is?

To find the decay constant of a radioactive sample, we can use the relationship between the disintegration rate and the decay constant. The decay constant, often denoted by λ (lambda), is a measure of the probability of decay of a radioactive substance per unit time. We can derive it using the formula related to exponential decay.

Understanding the Decay Process

The disintegration rate of a radioactive sample decreases over time according to the equation:

N(t) = N0 * e^(-λt)

Where:

• N(t) is the disintegration rate at time t.
• N0 is the initial disintegration rate.
• λ is the decay constant.
• t is the time elapsed.

Given Values

From the problem, we know:

• Initial disintegration rate, N0 = 5000 disintegrations/minute
• Disintegration rate after 5 minutes, N(5) = 1250 disintegrations/minute
• Time, t = 5 minutes

Setting Up the Equation

We can substitute the known values into the exponential decay formula:

1250 = 5000 * e^(-λ * 5)

Solving for the Decay Constant

First, we can simplify the equation:

e^(-λ * 5) = 1250 / 5000

This simplifies to:

e^(-λ * 5) = 0.25

Next, we take the natural logarithm of both sides to solve for λ:

-λ * 5 = ln(0.25)

Now, we can calculate ln(0.25):

ln(0.25) = -1.3863 (approximately)

Substituting this back into the equation gives:

-λ * 5 = -1.3863

Dividing both sides by -5 results in:

λ = 1.3863 / 5

λ ≈ 0.2773 per minute

Final Result

Thus, the decay constant for the radioactive sample is approximately 0.2773 per minute. This value indicates how quickly the sample is decaying over time, with a higher decay constant representing a faster rate of disintegration.