Animated Solution for Physics - Atoms and Nuclei: A radioactive sample at any instant has its disintegration rate 5000 disintegrations/min. After 5 min, the rate is 1250 disintegrations/min. Then, the decay constant (per minute) is

Let's begin by visualizing the physical reality of the problem. We are given a radioactive sample that is actively decaying. At our starting time, $t=0$, the disintegration rate—which we formally call the initial activity $A_0$—is $5000\text{ disintegrations/min}$.

Now, remember the fundamental law of radioactive decay? The activity of any radioactive sample decreases exponentially with time. The mathematical model governing this behavior is:

Here, $\lambda$ represents the decay constant, a unique signature of the radioactive material indicating how fast it decays.

The problem states that after $5\text{ minutes}$, the activity drops significantly to $1250\text{ disintegrations/min}$. Let's substitute these known values into our exponential equation. We get:

Let's simplify this raw setup. By dividing both sides by $5000$, we isolate the exponential term:

This fraction simplifies beautifully to $\frac{1}{4}$.

To make the exponent positive and easier to handle, we can take the reciprocal of both sides. This gives us:

Now, to bring the unknown $\lambda$ down from the exponent, we must apply the natural logarithm ($\ln$) on both sides. This yields:

Since $4$ is a perfect square ($2^2$), we can use the logarithmic property $\ln (x^y)=y\ln (x)$ to rewrite the right side as $2\ln (2)$.

Finally, we divide by $5$ to completely isolate $\lambda$:

Since $\frac{2}{5}$ is exactly $0.4$, our final decay constant is:

A Hidden Pattern

Here is a quick intuition check that bypasses heavy algebra. If you calculate the half-life using $T_{1/2}=\frac{\ln 2}{\lambda }$, it comes out to exactly $2.5\text{ minutes}$.

Notice that the given time of $5\text{ minutes}$ is exactly two half-lives! That is precisely why the activity dropped by a factor of $2^2=4$ (from $5000$ to $1250$). Always look for these hidden integer multiples of half-lives in exams like JEE; they can save you immense calculation time!