Meat left exposed to the air slowly turns brown due to reaction with atmospheric oxygen. You have purchased a steak and inadvertently left it out on the counter for \(3 \mathrm{hrs}\). Upon cutting the steak in half you discover the brown color extends to a depth of \(3 \mathrm{~mm}\) from the surface (the penetration depth). If the diffusivity of oxygen in the meat is \(10^{-12} \mathrm{~m}^{2} / \mathrm{s}\), what is the reaction rate constant for the meat/oxygen reaction? The solution for the oxygen concentration profile is:

\[\frac{c}{c_{o}}=\frac{1}{2} \exp \left(-x \sqrt{\frac{k^{\prime \prime}}{D}}\right)\left[\operatorname{erfc}\left\{\frac{x}{\sqrt{4 D t}}-\sqrt{k^{\prime \prime} t}\right\}+\operatorname{erfc}\left\{\frac{x}{\sqrt{4 D t}}+\sqrt{k^{\prime \prime} t}\right\}\right]\]