# What are some first order half life practice problems?

Feb 14, 2017

Cobalt-60 has a half-life of 5.26 years.

a) Calculate the first-order rate constant.

b) If 20% decays, how much time did it take?

c) Write the rate law that describes this process.

d) Determine the initial rate of the process if one starts with $\text{1.00 M}$ of cobalt-60.

a) From the first-order half-life:

${t}_{\text{1/2}} = \ln \frac{2}{k}$

=> color(blue)(k) = ln2/t_"1/2" = ln2/"5.26 yrs" = color(blue)("0.1318 yrs"^(-1))

b) From the integrated rate law:

$\ln \left(\frac{\left[A\right]}{{\left[A\right]}_{0}}\right) = - k t$

$\textcolor{b l u e}{t} = - \ln \frac{\frac{\left[A\right]}{{\left[A\right]}_{0}}}{k} = - \ln \frac{\frac{0.2 \cancel{{\left[A\right]}_{0}}}{\cancel{{\left[A\right]}_{0}}}}{k}$

$= - \ln \frac{0.2}{\text{0.1318 yrs"^(-1)) = color(blue)("3.268 yrs}}$

c) $\textcolor{b l u e}{r \left(t\right) = k \left[A\right]}$

d) color(blue)(r(t)) = "0.1318 yrs"^(-1)("1.00 M") = color(blue)("0.1318 M/yr")