Question

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 \(63 \mathrm{C}\), while in the yolk the proteins start to coagulate for temperatures above \(70 \mathrm{C}\). For a soft boiled egg, the white needs to have been heated long enough to coagulate at a temperature above \(63 \mathrm{C}\), but the yolk should not be heated above 70 C. For a hard boiled egg, the center of the yolk should be allowed to reach \(70 \mathrm{C}\). The following formula expresses the time \(t\) it takes (in seconds) for the center of the yolk to reach the temperature \(T_{y}\) (in Celsius degrees): $$ t=\frac{M^{2 / 3} c \rho^{1 / 3}}{K \pi^{2}(4 \pi / 3)^{2 / 3}} \ln \left[0.76 \frac{T_{o}-T_{w}}{T_{y}-T_{w}}\right] $$ Here, \(M, \rho, c\), and \(K\) are properties of the egg: \(M\) is the mass, \(\rho\) is the density, \(c\) is the specific heat capacity, and \(K\) is thermal conductivity. Relevant values are \(M=47 \mathrm{~g}\) for a small egg and \(M=67 \mathrm{~g}\) for a large egg, \(\rho=1.038 \mathrm{~g} \mathrm{~cm}^{-3}, c=3.7 \mathrm{Jg}^{-1} \mathrm{~K}^{-1}\), and \(K=5.4 \cdot 10^{-3} \mathrm{Wcm}^{-1} \mathrm{~K}^{-1}\). Furthermore, \(T_{w}\) is the temperature (in \(\mathrm{C}\) degrees) of the boiling water, and \(T_{o}\) is the original temperature (in \(\mathrm{C}\) degrees) of the egg before being put in the water. Implement the formula in a program, set \(T_{w}=100 \mathrm{C}\) and \(T_{y}=70 \mathrm{C}\), and compute \(t\) for a large egg taken from the fridge \(\left(T_{o}=4 \mathrm{C}\right)\) and from room temperature \(\left(T_{o}=20 \mathrm{C}\right)\). Name of program file: egg.py.

Short answer

Implement the formula in a Python program to calculate the time for two initial temperatures.

Step-by-step solution

Understanding the Problem

To solve this exercise, we need to calculate the time taken for the center of a large egg's yolk to reach 70°C using a provided formula. The problem gives us the necessary constants and asks us to evaluate this for two scenarios: an egg taken from the fridge (4°C) and an egg at room temperature (20°C).

Assigning Given Values

For the large egg, we are given: - Mass, \(M = 67\ g\) - Density, \(\rho = 1.038\ g\ cm^{-3}\)- Specific heat capacity, \(c = 3.7\ Jg^{-1} K^{-1}\)- Thermal conductivity, \(K = 5.4 \times 10^{-3}\ Wcm^{-1} K^{-1}\)- Initial boiling water temperature, \(T_{w} = 100°C\)- Final yolk temperature, \(T_{y} = 70°C\)We need to calculate for two initial temperatures: \(T_{o} = 4°C\) and \(T_{o} = 20°C\).

Writing the Formula in Python

The formula to calculate time \(t\) is:\[t = \frac{M^{2/3} c \rho^{1/3}}{K \pi^{2} (4 \pi/3)^{2/3}} \ln \left[0.76 \frac{T_{o}-T_{w}}{T_{y}-T_{w}}\right]\]Implement this in a Python program. Let's write a function in the file `egg.py` that computes this based on the specified parameters.

Implementing the Python Program

```python import math def boiling_time(M, T_o, T_w=100, T_y=70): rho = 1.038 c = 3.7 K = 5.4e-3 # Calculate the time using the provided formula t = ((M**(2/3)) * c * (rho**(1/3))) / (K * math.pi**2 * (4 * math.pi / 3)**(2/3)) t *= math.log(0.76 * (T_o - T_w) / (T_y - T_w)) return t # Conditions for a large egg M_large_egg = 67 # Mass in grams t_fridge = boiling_time(M_large_egg, 4) t_room_temp = boiling_time(M_large_egg, 20) print("Time for egg from fridge:", t_fridge, "seconds") print("Time for egg at room temperature:", t_room_temp, "seconds") ```

Running the Python Script

Save the program in a file named `egg.py`. Run the script to compute the time it takes for a large egg from the fridge (initially at 4°C) and from room temperature (initially at 20°C) to have its yolk's center reach 70°C. The program will print out these times.

Key concepts

Thermal Conductivity

Thermal conductivity is a material's ability to conduct heat. When you put an egg in boiling water, heat travels from the hot water into the egg. This transfer is influenced by the egg's thermal conductivity, denoted as \( K \) in the formula. It's a crucial property because it determines how easily heat can flow through the egg materials. In our example, the egg's thermal conductivity is given as \( K = 5.4 \times 10^{-3} \; \text{W cm}^{-1} \text{ K}^{-1} \). This means the egg has a relatively low thermal conductivity, explaining why it takes some time for the heat to penetrate into the yolk.

• High thermal conductivity means better heat transfer.
• Low thermal conductivity slows down the heating process.

Understanding this concept helps you appreciate why cooking times vary not only with different egg sizes but also with different temperatures at which you start the egg cooking process.

Specific Heat Capacity

Specific heat capacity \( c \) is a measure of how much heat energy a material needs to change its temperature. For our egg example, \( c \) is \( 3.7 \; \text{J g}^{-1} \text{ K}^{-1} \). This value indicates how much energy is required to raise the temperature of 1 gram of egg material by 1 degree Celsius.

• A higher specific heat capacity means the material can store more heat.
• Low specific heat capacity means the material heats up or cools down quickly.

When boiling eggs, knowing the specific heat helps us understand how much energy is needed from the boiling water to reach the desired yolk temperature of 70°C. This is why the formula considers \( c \) in calculating how long it takes for the yolk to heat completely.

Python Programming

Python programming can be a powerful tool to solve scientific problems, like calculating egg boiling times. In our example, we use Python to implement a formula that predicts how long it will take for an egg to cook. This involves several important steps: - Importing necessary libraries like `math` for complex calculations. - Defining functions to encapsulate the logic, enhancing code reuse and clarity. - Using pre-defined constants and input parameters to keep our calculations accurate. The formula is implemented into a function called `boiling_time`, making it flexible to compute times for different egg sizes or starting temperatures. Such scripts are a great way to see the immediate application of scientific formulas in real-world scenarios and allow for easy experimentation with different input values.

Temperature Calculation

Temperature calculation in this context refers to determining how long it takes for the yolk of an egg to reach a certain temperature when boiled. The formula considers initial temperatures, which in our problem were 4°C and 20°C. To understand the formula:- It involves logarithmic operations \( \ln \left[0.76 \frac{T_{o}-T_{w}}{T_{y}-T_{w}}\right] \), which handle nonlinear changes in temperature.- It accounts for mass, specific heat, and density factors through: \[ t = \frac{M^{2/3} c \rho^{1/3}}{K \pi^{2} (4 \pi/3)^{2/3}} \]These calculations aim to predict when the perfect conditions for protein coagulation in the yolk are reached without overcooking, which speaks to the importance of precision in scientific programming with Python. This type of analysis can be extended to countless applications, from food science to materials engineering.