Question

${}^{90}Kr(t_{1/2}=32s)$ is used to study respiration. How soon after being made must a sample be administered to the patient if the activity must be at least 90% of the original activity?

Short answer

The time after which a sample must be administered to the patient is t = 4.8641 s.

Step-by-step solution

Concept Introduction

Respiration is a metabolic process that all living things go through. It is a biological reaction that takes place within the cells of living organisms. The energy (ATP-Adenosine triphosphate) is produced during this process by the breakdown of glucose, which is then utilised by cells to execute various functions. Respiration is performed by all living organisms, from single-celled organisms to dominant multicellular animals.

Calculation for Time

Identify the decay constant (k) using the formula for half-life.

$$\begin{gathered} t_{\frac{1}{2}}=\frac{\ln 2}{k}\to k=\frac{\ln 2}{t_{\frac{1}{2}}} \\ k=\frac{\ln 2}{t_{\frac{1}{2}}}=\frac{\ln 2}{32s}=\frac{0.0216608}{s} \end{gathered}$$

Using the expression for finding the number of nuclei remaining, solve for the value of t .

$$\begin{gathered} \ln \frac{N_t}{N_0}=-kt \\ \ln \frac{0.9}{1}=\frac{0.0216608}{s}\left(t\right) \\ t=4.8641s \end{gathered}$$

Therefore, the value for time is obtained as t = 4.8641 s.