Question

The half life time of a radioactive elements of \(\mathrm{x}\) is the same as the mean life of another radioactive element \(\mathrm{y}\). Initially they have same number of atoms, then (A) \(\mathrm{y}\) will decay faster then \(\mathrm{x}\) (B) \(\mathrm{x}\) will decay faster then \(\mathrm{y}\) (C) \(\mathrm{x}\) and \(\mathrm{y}\) will decay at the same rate at all time (D) \(\mathrm{x}\) and \(\mathrm{y}\) will decay at the same rate initially.

Short answer

The short answer is: (A) $\mathrm{y}$ will decay faster than $\mathrm{x}$. This is because the mean life of $\mathrm{y}$ (T_m(y)) is equal to the half-life of $\mathrm{x}$ (T_h(x)), and from the relationship $\mathrm{ln(2)} / \lambda_x = 1 / \lambda_y$, it is evident that $\lambda_y$ will be greater than $\lambda_x$, meaning $\mathrm{y}$ will have a higher decay rate as compared to $\mathrm{x}$.

Step-by-step solution

Define half-life and mean life of radioactive elements

Half-life is the time taken for half of the initial number of radioactive atoms to decay. Mathematically, it is the time when the fraction of undecayed atoms in a sample is 0.5 (50%). The mean life of a radioactive element is the average time taken for an atom to decay.

Establish radioactive decay formulas

For a radioactive element, the decay equation can be written as: N(t) = N0 * e^(-λt) Where N(t) is the number of radioactive atoms at time t, N0 is the initial number of radioactive atoms, λ is the decay constant, and t is the time passed. We know that the half-life can be determined using the following equation: Half-life (T_h) = ln(2) / λ For a radioactive element with a mean life (T_m), the relationship with the decay constant is: Mean life (T_m) = 1 / λ

Express mean life of y in terms of half-life of x

Given that half-life of element x is equal to the mean life of element y, we have: T_h(x) = T_m(y) From the decay equations, we can rewrite this relationship as: ln(2) / λ_x = 1 / λ_y

Compare decay rates

We know that the radioactive decay equation for an element is given by: N(t) = N0 * e^(-λt) For x, this equation is: N_x(t) = N0 * e^(-λ_xt) Similarly, for y: N_y(t) = N0 * e^(-λ_yt) From the relationship established in Step 3, we can rewrite the decay equation for y in terms of λ_x: N_y(t) = N0 * e^(-(ln(2) / T_h(x))t) Now, we can compare the decay rates by analyzing the decay constants.

Analyze the decay rates

From the previous step, we see that y's decay equation involves λ_x in its decay constant. The decay constant determines how quickly a radioactive element decays. Since λ_y = ln(2) / T_h(x), it is evident that λ_y will be greater than λ_x, meaning y will have a higher decay rate as compared to x. Thus, we can conclude that y will decay faster than x, which corresponds to option (A).

Key concepts

Half-Life

Half-life is an essential concept in understanding radioactive decay. It refers to the time required for half of the radioactive atoms in a sample to decay.
This means that after one half-life, only 50% of the original radioactive atoms remain. The half-life is symbolic for many because it neatly describes how radioactive materials decrease over time.Key Points about Half-Life:

• It is a fixed property for any given radioactive isotope.
• It does not depend on how many atoms are present initially.
• Used to determine the age of fossils and archaeological findings.

Mathematically, we express half-life (T_h) using the formula:\[T_h = \frac{\ln(2)}{\lambda}\]where \(\lambda\) is the decay constant, an important factor influencing how rapidly a radioactive substance decays.

Mean Life

The mean life, also known as the average life, gives us more insight into the life expectancy of radioactive atoms. It's the average time a single atom is expected to remain undecayed.
Unlike half-life, which only centers around a 50% decay point, mean life provides a more comprehensive duration reference.Key Points about Mean Life:

• Mean life is longer than half-life for any given isotope.
• It's mathematically related to the decay constant \(\lambda\).
• Averages all possible life spans of atoms in a sample, providing a broader view compared to half-life.

The formula for calculating mean life \(T_m\) is:\[T_m = \frac{1}{\lambda}\]This relationship shows that mean life directly depends on the decay constant, making it a crucial factor in studies of radioactive elements.

Decay Constant

The decay constant, denoted by \(\lambda\), is a probability factor that indicates the fraction of atoms decaying per unit time.
It is pivotal in modeling the behavior of radioactive substances.Key Points about the Decay Constant:

• Higher \(\lambda\) indicates a faster decay rate.
• Directly affects both half-life and mean life of a radioactive material.
• Intrinsically connected to the statistical behavior of decay processes.

Every radioactive isotope has its own unique decay constant, portraying how stable or unstable it is. The formula involving decay constant in radioactive decay is given by:\[N(t) = N_0 e^{-\lambda t}\]Where \(N(t)\) is the number of atoms remaining after time \(t\), and \(N_0\) is the initial number of atoms. Understanding \(\lambda\) arms us with the knowledge to predict and comprehend the progression and endurance of radioactive materials.