The half-life period of a radioactive element x is same as the mean life time of another radioactive

Q: Why does $y$ decay faster than $x$ initially?

The rate of decay depends on both the decay constant and the number of active nuclei. Since both have the same initial number of nuclei, the one with the larger decay constant (in this case, $y$) will decay faster initially.

Q: Will $y$ always decay faster than $x$?

Not necessarily! As $y$ decays faster, its number of active nuclei decreases more rapidly than $x$'s. Eventually, the number of active nuclei of $y$ will be so small that its decay rate ($R = \lambda N$) will drop below that of $x$.

Q: What is the physical meaning of mean life?

Mean life is the average lifetime of all the nuclei in a radioactive sample. It is mathematically equal to the reciprocal of the decay constant ($1/\lambda$) and is the time required for the sample to reduce to $1/e$ (about 37%) of its initial value.

Q: How is half-life different from mean life?

Half-life is the time required for exactly half of the radioactive nuclei in a sample to decay. It is given by $\ln(2)/\lambda$, which is approximately $0.693$ times the mean life. Thus, half-life is always shorter than mean life.

Q: Can the decay rate be constant?

No, the decay rate of a radioactive sample decreases exponentially over time because the number of active nuclei available to decay is continuously decreasing.

Question

The half-life period of a radioactive element $x$ is same as the mean life time of another radioactive element $y$. Initially both of them have the same number of atoms. Then

Options:

(A) $x$ and $y$ have the same decay rate initially

(B) $x$ and $y$ decay at the same rate always

(C) $y$ will decay at a faster rate than $x$

(D) $x$ will decay at a faster rate than $y$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you have two radioactive samples, $x$ and $y$, sitting on your desk. Both start with the exact same number of active atoms, $N_0$. However, they have different "personalities" when it comes to decaying.

The problem gives us a very specific clue: the **half-life** of element $x$ is exactly equal to the **mean life** of element $y$.

Let's translate this into the language of math. The half-life of an element is the time it takes for half of its atoms to decay, given by $T_{1/2} = \frac{\ln 2}{\lambda}$. The mean life, on the other hand, is the average lifetime of an atom, given by $	au = \frac{1}{\lambda}$.

So, according to the problem:
$$ (T_{1/2})_x = (	au)_y $$
$$ \frac{\ln 2}{\lambda_x} = \frac{1}{\lambda_y} $$

### The Master Equation

Now, let's substitute the value of $\ln 2$, which is approximately $0.693$.

$$ \frac{0.693}{\lambda_x} = \frac{1}{\lambda_y} $$

Rearranging this to compare the decay constants, we get:
$$ \lambda_x = 0.693 \lambda_y $$

What does this tell us? Since $0.693$ is less than $1$, it means that $\lambda_x$ is strictly less than $\lambda_y$. In other words, element $y$ has a larger decay constant than element $x$ ($\lambda_y > \lambda_x$).

### Final Calculation

The question asks about the decay rate. The rate of decay, or activity $R$, is directly proportional to the decay constant and the number of active nuclei present at that instant:
$$ R = \lambda N $$

Initially, both elements have the same number of atoms, so $N_x = N_y = N_0$. 
Let's compare their initial decay rates:
$$ R_x = \lambda_x N_0 $$
$$ R_y = \lambda_y N_0 $$

Since we've already established that $\lambda_y > \lambda_x$, it mathematically follows that:
$$ R_y > R_x $$

This means that, initially, element $y$ is decaying at a faster rate than element $x$. The red curve for $y$ drops much more steeply than the blue curve for $x$ right from the start!

Animated Solution for Physics - Atoms and Nuclei: The half-life period of a radioactive element $x$ is same as the mean life time of another radioactive element $y$ . Initially both of them have the same number of atoms. Then

Select Answer:

Visualized Solution

$Given Condition$

$Comparing Decay Constants$

$Comparing Initial Decay Rates$

$The Way Forward$

The Sigma Insight: Radioactivity

Analyzing the Setup

The Master Equation

Final Calculation

Similar Questions