Chapter 20, Problem 46E

Interpretation Introduction

Interpretation:

The time required for the decay of technetium-99m from $0.050\mu \text{g}$ to $1.0\times {10}^{-3}\mu \text{g}$, having a half-life of 6.0 hours is to be determined.

Concept introduction:

The half-life of a substance is the numerical value in which the given radioactive substance is assumed to be reduced to half of its initial number. The half-life for a given substance is represented by t_1/2.

In case, the decay of a radioactive substance is exponential, it will remain constant for the life time of the substance.

After each half-life period, the amount of the substance is reduced to half of the initial number.

The time required for the decay of the substance to a given amount of substance can be calculated using the formula mentioned below:

$\ln \left(\frac{{\text{N}}_{\text{t}}}{{\text{N}}_{\text{0}}}\right)=-\text{kt}$

In the above equation, ‘N_t’ represents the mass of the radioactive substance after a certain time interval t, ‘N₀’ indicates the initial mass of the radioactive material, ‘k’ represents the decay constant and ‘t’ represents the time interval for the half-life (t_1/2).

The half-life of the element is determined by the following formula:

$t_{1/2}=\frac{0.693}{k}$