# The half-life of cobalt-60 is 5.2 years. Find the time it would take for a sample of 0.5 grams to decay to 0.1 gram?

Feb 16, 2017

$\textsf{12.8}$ years

#### Explanation:

The equation for 1st order decay is:

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

$\textsf{\lambda = \frac{0.693}{t} _ \left(\frac{1}{2}\right)}$

$\therefore$$\textsf{\lambda = \frac{0.693}{5.2} = 0.126 \textcolor{w h i t e}{x} {a}^{- 1}}$

Taking natural logs of both sides:

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

$\therefore$$\textsf{\ln \left[{M}_{t} / {M}_{0}\right] = - \lambda t}$

$\textsf{\ln \left[\frac{0.1}{0.5}\right] = - 0.126 \times t}$

$\therefore$$\textsf{- 1.609 = - 0.126 \times t}$

$\textsf{t = \frac{1.609}{0.126} = 12.77 \textcolor{w h i t e}{x} a}$