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

Q: Why does a larger decay constant mean it decays faster?

The decay constant $\lambda$ represents the probability of decay per unit time for a single nucleus. A higher probability means more atoms will decay in the same time interval, leading to a faster overall decay rate.

Q: Is the rate of decay constant over time?

No, the rate of decay $R = \lambda N$ decreases exponentially over time because the number of active nuclei $N$ is continuously decreasing as they decay.

Q: What is the physical significance of mean life?

Mean life is the average time an atom survives before decaying. Mathematically, it is exactly the time required for the sample's active population to reduce to $1/e$ (about 37%) of its initial value.

Q: Why do we use $\ln 2$ for half-life?

The decay equation is $N = N_0 e^{-\lambda t}$. For half-life, the remaining atoms are $N = N_0/2$. Substituting this gives $1/2 = e^{-\lambda t}$. Taking the natural log on both sides gives $-\ln 2 = -\lambda t$, which simplifies to $t = \ln 2 / \lambda$.

Q: Will X ever decay faster than Y?

Yes! Because Y decays faster initially, its population $N_Y$ drops very rapidly. Eventually, the remaining number of atoms $N_X$ will be so much larger than $N_Y$ that the product $\lambda_X N_X$ will actually exceed $\lambda_Y N_Y$.

Question

The half-life period of a radioactive element $X$ is same as the mean life time of another radioactive element $Y$. Initially they have the same number of atoms. Then,

Options:

(A) $X$ will decay faster than $Y$

(B) $Y$ will decay faster than $X$

(C) $Y$ and $X$ have same decay rate initially

(D) $X$ and $Y$ decay at same rate always

Sigma Percentile · Accepted Answer

### The Initial Setup

Imagine we have two radioactive samples, $X$ and $Y$. The problem sets the stage by telling us that initially, they both contain the exact same number of atoms. Let's denote this initial population as $N_0$.

$$N_X(0) = N_Y(0) = N_0$$

This is a crucial starting point because the rate at which a radioactive sample decays depends heavily on how many atoms are present to begin with.

### Decoding the Lifetimes

Now, we are given a very specific and fascinating condition: the half-life of element $X$ is exactly equal to the mean life of element $Y$.

$$T_{1/2}(X) = 	au(Y)$$

To make sense of this, we need to translate these macroscopic timeframes into their microscopic drivers—the decay constants ($\lambda$). The decay constant is the true DNA of a radioactive element; it tells us the inherent probability of an atom decaying per unit time.

Let's recall our standard formulas. The half-life is the time it takes for half the sample to decay, given by:

$$T_{1/2} = \frac{\ln 2}{\lambda}$$

On the other hand, the mean life is the average survival time of an atom, given simply by the reciprocal of the decay constant:

$$	au = \frac{1}{\lambda}$$

### The Decay Constant Showdown

Let's substitute these formulas into our given condition.

$$\frac{\ln 2}{\lambda_X} = \frac{1}{\lambda_Y}$$

By rearranging this equation, we can find a direct relationship between their decay constants.

$$\lambda_Y = \frac{\lambda_X}{\ln 2}$$

We know that the natural logarithm of 2 is approximately $0.693$.

$$\lambda_Y \approx \frac{\lambda_X}{0.693}$$

Here is where the magic happens. Since the denominator ($0.693$) is strictly less than $1$, dividing $\lambda_X$ by it will yield a larger number.

**Therefore, $\lambda_Y > \lambda_X$.**

### The Physical Meaning: Rate of Decay

But what does a larger decay constant actually mean in the physical world? To answer this, we look at the radioactive decay law, which states that the rate of decay ($R$) is directly proportional to the decay constant and the number of active nuclei present.

$$R = \lambda N$$

Let's evaluate the initial decay rates for both samples. Since they both started with the same number of atoms ($N_0$), their initial rates are:

$$R_X = \lambda_X N_0$$

$$R_Y = \lambda_Y N_0$$

Because we have already established that $\lambda_Y$ is greater than $\lambda_X$, it mathematically guarantees that $R_Y$ must be greater than $R_X$.

**Conclusion: Sample $Y$ will have a higher initial decay rate, meaning it decays faster than $X$.**

Think of them as two runners starting at the same line. Runner $Y$ has a higher inherent speed (decay constant), so right out of the gate, $Y$ covers more ground (decays more atoms) than $X$.

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 they have the same number of atoms. Then,

Select Answer:

Visualized Solution

Initial State

Given Condition

Decay Formulas

Equating the Lifetimes

Comparing Decay Constants

The Inequality

Rate of Decay

Final Conclusion

The Way Forward

The Sigma Insight: Radioactivity

The Initial Setup

Decoding the Lifetimes

The Decay Constant Showdown

The Physical Meaning

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