You are on a quest to develop the perfect meringue topping for a pie. The surface needs to become a golden brown, a process that results from caramelization of the sugars in the meringue. To form such a meringue, you bake the pie to set the meringue and as the final step, place it under a broiler element for a short time. A pie is roughly \(20 \mathrm{~cm}\) in diameter and we can approximate the pie as a hemisphere exposed to a small heating element of surface area \(2 \times 10^{-5} \mathrm{~m}^{2}\). The geometry of the situation is akin to Figure P15.30. The problem is quite complicated since the rate of heat transfer to the pie needs to be controlled to avoid burning the meringue.

a. If the heating element can deliver \(5 \mathrm{~W}\) at \(1800 \mathrm{~K}\) with an emissivity of 0.9 to the meringue surface and the meringue cannot be exposed to a temperature of more than \(500 \mathrm{~K}\), how far must the pie be from the heating element? Meringue, being porous, is a very good insulator and so will reradiate into space as a blackbody.

b. If the caramelization reaction is essentially a zero-order reaction obeying:

\[r_{c}\left(\frac{g}{s}\right)=k^{\prime \prime}\left(T_{\text {surface }}-422\right) \quad k^{\prime \prime}=1 \times 10^{-4} \frac{1}{K s}\]

and we need to react \(2.5 \mathrm{~g}\) of meringue to adequately produce the desired color, how long must the broiler be on?