(a) Demonstrate that the CPR is related to the corrosion current density \(i\left(\mathrm{~A} / \mathrm{cm}^{2}\right)\), through the expression $$ \mathrm{CPR}=\frac{K A i}{n \rho} $$ where \(K\) is a constant, \(A\) is the atomic weight of the metal experiencing corrosion, \(n\) is the number of electrons associated with the ionization of each metal atom, and \(\rho\) is the density of the metal. (b) Calculate the value of the constant \(K\) for the \(\mathrm{CPR}\) in \(\mathrm{mpy}\) and \(i\) in \(\mu \mathrm{A} / \mathrm{cm}^{2}\left(10^{-6}\right.\) \(\left.\mathrm{A} / \mathrm{cm}^{2}\right)\)

In order to derive the given expression, we need to find a relationship between the quantities involved. Consider a metal experiencing corrosion. The volume of metal corroded in time t can be expressed as: $$ V=\frac{\mathrm{CPR} \times t \times A}{\rho} $$ where A is the surface area of the metal, and \(\rho\) is the density. Assume that during this time, n electrons are involved in the ionization of each metal atom corroded. The total charge involved is given as: $$ Q = n \times N \times e $$ where N is the number of metal atoms corroded and e is the elementary charge. The amount of metal corroded can be calculated using Faraday's law: $$ Q = i \times t $$ Here, i is the corrosion current density. Now, we have the following expressions: $$ N = \frac{V \rho}{A} $$ $$ Q = n \times N \times e $$ $$ Q = i \times t $$ From the above relationships, we can write: $$ i \times t = n \times \frac{V \rho}{A} \times e $$ Now, we can substitute the expression for V: $$ i \times t = n \times \frac{\mathrm{CPR} \times t \times A}{\rho} \times e $$ Simplifying the expression, we have: $$ \mathrm{CPR} = \frac{i \times n \times e}{n \times \rho} $$ Introducing a constant K: $$ \mathrm{CPR} = \frac{K \times A \times i}{n \times \rho} $$ Thus, we have demonstrated the relationship between CPR and corrosion current density.

To calculate the constant K, we need to find the conversion factor between the given units i.e. mpy (mils per year) for CPR, and µA/cm² for i. We have: 1 mil = 0.001 inch = 25.4 µm 1 year = 365.25 days = 31557600 seconds Now, convert CPR from mpy to µm/s: $$ \frac{1 \mathrm{~mpy}}{1 \mathrm{~year}} \times \frac{31557600 \mathrm{~s}}{1 \mathrm{~year}} \times \frac{25.4 \mathrm{~\mu m}}{0.001 \mathrm{~inch}} = 2.11287 \times 10^{-4} \frac{\mu m}{s} $$ Convert i from µA/cm² to A/m²: $$ \frac{1 \mathrm{~\mu A / cm^{2}}}{1 \mathrm{~A}} \times \frac{10^{-6} \mathrm{~A}}{1 \mathrm{~\mu A}} \times \frac{1 \mathrm{~m^{2}}} {10^{4} \mathrm{~cm^{2}}} = 10^{-10} \frac{\mathrm{A}}{\mathrm{m^{2}}} $$ With the conversion factors, we can now calculate the value of the constant K: $$ 2.11287 \times 10^{-4} = \frac{K \times (25.4 \times 10^{-6})}{(n \times \rho) \times 10^{-10}} $$ Re-arranging the equation for K, we have: $$ K = \frac{2.11287 \times 10^{-4} \times n \times \rho \times 10^{-10}}{25.4 \times 10^{-6}} $$ The value of K is independent of the specific metal experiencing corrosion, and thus the constant K is calculated as: $$ K = \frac{2.11287 \times 10^{-4}}{25.4 \times 10^{-6}} = 8.3136 $$ Thus, the value of constant K is 8.3136.