# Question #370a1

Dec 8, 2016

#### Explanation:

The equation for 1st order decay is:

$\textsf{{N}_{t} = {N}_{0} {e}^{- \lambda t}}$

$\textsf{{N}_{0}}$ is the initial number of undecayed atoms.

$\textsf{{N}_{t}}$ is the number of undecayed atoms after time t.

$\textsf{\lambda}$ is the decay constant.

We get this using:

$\textsf{\lambda = \frac{0.693}{t} _ \left(\frac{1}{2}\right) = \frac{0.693}{6} = 0.1155 \textcolor{w h i t e}{x} {\text{hr}}^{- 1}}$

Taking natural logs of both sides of the decay equation:

$\textsf{\ln {N}_{t} = \ln {N}_{0} - \lambda t}$

$\therefore$$\textsf{\lambda t = \ln {N}_{0} - \ln {N}_{t}}$

$\therefore$$\textsf{\lambda t = \ln \left(0.055\right) - \ln \left(0.0066\right) = - 2.9 - \left(- 5.02\right) = 2.12}$

$\therefore$$\textsf{t = \frac{2.12}{0.01155} = 18.3 \textcolor{w h i t e}{x} \text{hr}}$