198^{Au} is an unstable isotope of standard gold (197^{Au}) with a half-life \( t^{1/2} = 2.7 \) days. It can be produced by bombarding a foil of gold with thermal neutrons, via the reaction \( 197^{Au} + n \rightarrow 198^{Au} + \gamma \). The cross-section for this reaction is \( \sigma = 97.8 \times 10^{-24} \) cm\(^2\).

A gold foil (initially 100% pure in \( 197^{Au} \)) with a thickness of \( \delta = 0.01 \) cm is irradiated with thermal neutrons at a rate of \( I = 10^{11} \) neutrons/cm\(^2\)/s. The density of gold is \( \rho = 19.3 \) g/cm\(^3\).

(i) Starting from the corresponding differential equations, show that the equations describing the number of \( 197^{Au} \) and \( 198^{Au} \) nuclei per unit surface (\( N_{1} \) and \( N_{2} \) respectively) as a function of time are:

\[ N_{1}(t) = N_{1}(0)e^{-\sigma It} \]

\[ N_{2}(t) = \frac{\sigma I}{\lambda - \sigma I} N_{1}(0) \left(e^{-\sigma It} - e^{-\lambda t} \right) \]

where \( \lambda = 1/\tau \) and \( \tau \) is the lifetime of \( 198^{Au} \).

Evaluate the numerical value of \( N_{1}(0) \).

(ii) Explain what is meant by secular equilibrium, and verify whether the conditions for that are met in this system.

(iii) Find the activity of \( 198^{Au} \) after 10 minutes of irradiation.

(iv) Is there a maximum value of \( N_{2} \) that can be obtained with this irradiation level? If yes, give a numerical value for it.