# Exercise 3

## cooking the perfect egg

As an egg cooks, the proteins first denature and then coagulate. When the temperature exceeds a critical point, reactions begin and proceed faster as the temperature increases. In the egg white the proteins start to coagulate for temperatures above 63 C, while in the yolk the proteins start to coagulate for temperatures above 70 C. For a soft-boiled egg, the white needs to have been heated long enough to coagulate at a temperature above 63 C, but the yolk should not be heated above 70 C. For a hard-boiled egg, the centre of the yolk should be allowed to reach 70 C.

The following equation gives the time \(t\) it takes (in seconds) for the centre of the yolk to reach the temperature \(T_{y}\) (Celsius):

\begin{equation} t = \frac{M^{2/3} c \rho^{1/3}}{K \pi^{2}(4\pi/3)^{2/3}} \ln \left[ 0.76 \frac{T_{o}-T_{w}}{T_{y}-T_{w}} \right] \end{equation}where \(M\), \(\rho\), \(c\) and \(K\) are properties of the egg: \(M\) is mass, \(\rho\) is the density, \(c\) is the specific heat capacity, and \(K\) is the thermal conductivity. Relevant values are \(M=47\) g for a small egg and \(M=67\) g for a large egg, \(\rho=1.038\) g cm\(^{-3}\), \(c=3.7\) J g\(^{-1}\) K\(^{-1}\), and \(K=0.0054\) W cm\(^{-1}\) K\(^{-1}\). The parameter \(T_{w}\) is the temperature (in Celsius) of the boiling water, and \(T_{o}\) is the original temperature of the egg before being put in the water.

Implement the equation in a program, set \(T_{w}=100\) C and \(T_{y}=70\) C, and compute \(t\) for a large egg taken from the fridge (\(T_{o}=4\) C) and from room temperature (\(T_{o}=20\) C).

Test your program with this example:

Is the egg large (1) or small (0)? 1 enter the initial temperature of the egg reminder 4.0 for fridge, 20.0 for room: 15.0 time taken to cook the egg is: 342.271 seconds (5 minutes, 42 seconds)