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Chapter 18: Problem 2503

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

Expert verified
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

01

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.
02

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
03

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
04

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.
05

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).

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