Question

DDT (molar mass = 354.49 g/mol) was a widely used insecticide that was banned from use in the United States in 1973. This ban was brought about due to the persistence of DDT in many different ecosystems, leading to high accumulations of the substance in many birds of prey. The insecticide was shown to cause a thinning of egg shells, pushing many birds toward extinction. If a 20-L drum of DDT was spilled into a pond, resulting in a DDT concentration of how long would it take for the levels of DDT to reach a concentration of (a level that is generally assumed safe in mammals)? Assume the decomposition of DDT is a first-order process with a half-life of 56.0 days.

Short answer

The reaction time to attain the concentration is 142.61 s.

Step-by-step solution

Calculating the half-life for the reaction

DDT (molar mass = 354.49 g/mol)

$\begin{array}{l}{\left[\mathrm{A}\right]}_0=8.75\times {10}^{-5}\mathrm{M}\\ \left[\mathrm{A}\right]=1.41\times {10}^{-7}\mathrm{M}\\ {\mathrm{t}}_{\frac{1}{2}}=56.0\mathrm{days}\end{array}$

The first order reaction half-life t ½,

$\begin{array}{l}{\mathbf{t}}_{\frac{\mathbf{1}}{\mathbf{2}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{693}\mathbf{/}\mathbf{k}\\ \mathrm{k}=0.693/56.0\mathrm{days}\\ \mathrm{k}=0.0128\mathrm{day}{{\mathrm{s}}^{-1}}^{}\end{array}$

Calculating the time for the reaction at 1.41×10-7M

Using the first order reaction equation

$\begin{array}{l}\mathrm{In}\left[\mathrm{A}\right]=\mathrm{In}{\left[\mathrm{A}\right]}_0-\mathrm{kt}\\ \mathrm{In}\left(1.41\times {10}^{-7}\right)=\mathrm{In}\left(8.75\times {10}^{-5}\right)--0.0128\mathrm{t}\\ \mathrm{In}\left(0.001611\right)=-0.0128\mathrm{t}\\ -1.825=-0.0128\mathrm{t}\\ \mathrm{t}=142.61{\mathrm{s}}^{-1}\end{array}$

The reaction time to attain the concentration is 142.61 s.