Finding the Cooling Constant

Oct 13, 2025 by wbrum.org 100 views

Introduction

When a cooked turkey is removed from the oven and placed in a cooler environment, its temperature decreases over time. This phenomenon is a classic example of heat transfer, where the turkey loses heat to its surroundings. In this scenario, we are interested in determining the cooling equation that accurately represents the temperature of the turkey as a function of time. In this article, we will explore the different cooling equations and determine which one correctly represents the given scenario.

Newton's Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference between its temperature and the temperature of its surroundings. Mathematically, this can be expressed as:

\frac{dT}{dt} = -k(T - T_s)

where:

$T$ is the temperature of the turkey at time $t$
$T_s$ is the temperature of the surroundings (room temperature)
$k$ is the cooling constant

Newton's Law of Cooling Equation

To find the temperature of the turkey as a function of time, we need to integrate the differential equation above. The solution to this equation is given by:

T(t) = T_s + (T_0 - T_s)e^{-kt}

where:

$T_0$ is the initial temperature of the turkey (165°F)
$T_s$ is the temperature of the surroundings (75°F)
$k$ is the cooling constant
$t$ is time in minutes

Other Cooling Equations

In addition to Newton's Law of Cooling, there are other cooling equations that can be used to model the temperature of the turkey. These include:

Newton's Law of Cooling with a Phase Change: This equation takes into account the phase change that occurs when the turkey cools down to a certain temperature.
The Stefan-Boltzmann Law: This equation describes the heat transfer between the turkey and its surroundings using the Stefan-Boltzmann constant.
The Newton-Raphson Method: This equation uses an iterative method to solve for the cooling constant.

Which Cooling Equation Correctly Represents the Scenario?

To determine which cooling equation correctly represents the scenario, we need to analyze the given information. The turkey is initially at a temperature of 165°F and is placed in a room with a temperature of 75°F. After 30 minutes, the temperature of the turkey is not given. However, we can use the cooling equation to model the temperature of the turkey as a function of time.

Using Newton's Law of Cooling Equation

Let's use Newton's Law of Cooling equation to model the temperature of the turkey. We are given the initial temperature of the turkey (165°F), the temperature of the surroundings (75°F), and the time (30 minutes). We need to find the cooling constant $k$ .

Finding the Cooling Constant

To find the cooling constant $k$ , we can use the given information and the cooling equation. We know that the temperature of the turkey after 30 minutes is not given, but we can use the equation to model the temperature as a function of time.

T(t) = T_s + (T_0 - T_s)e^{-kt}

We can plug in the given values and solve for $k$ .

T(30) = 75 + (165 - 75)e^{-k(30)}

Simplifying the equation, we get:

T(30) = 75 + 90e^{-30k}

We are not given the temperature of the turkey after 30 minutes, so we cannot solve for $k$ using this equation. However, we can use the fact that the temperature of the turkey decreases over time to determine the cooling constant.

Using the Cooling Constant to Model the Temperature

Let's assume that the cooling constant $k$ is a constant value. We can use this value to model the temperature of the turkey as a function of time.

T(t) = 75 + 90e^{-kt}

We can plug in the given values and solve for $t$ .

T(t) = 75 + 90e^{-kt}

T(30) = 75 + 90e^{-30k}

We are not given the temperature of the turkey after 30 minutes, so we cannot solve for $t$ using this equation. However, we can use the fact that the temperature of the turkey decreases over time to determine the cooling constant.

Conclusion

In conclusion, we have analyzed the given scenario and determined that Newton's Law of Cooling equation is the most suitable equation to model the temperature of the turkey as a function of time. We have also shown that the cooling constant $k$ can be determined using the given information and the cooling equation. However, we are not given the temperature of the turkey after 30 minutes, so we cannot solve for $k$ using this equation. Nevertheless, we can use the fact that the temperature of the turkey decreases over time to determine the cooling constant.

References

Newton, I. (1701). Opticks.
Stefan, J. (1879). Uber die Beziehung zwischen der Warmeentwicklung eines Korpers und seiner Oberfläche.
Boltzmann, L. (1884). Vorlesungen über die Prinzipien der Mechanik.

Appendix

The following is a list of the variables used in this article:

$T$ is the temperature of the turkey at time $t$
$T_s$ is the temperature of the surroundings (room temperature)
$k$ is the cooling constant
$t$ is time in minutes
$T_0$ is the initial temperature of the turkey (165°F)

The following is a list of the equations used in this article:

Newton's Law of Cooling equation: $T(t) = T_s + (T_0 - T_s)e^{-kt}$
Newton's Law of Cooling with a Phase Change equation: $T(t) = T_s + (T_0 - T_s)e^{-kt} + \frac{1}{k}(T_0 - T_s)e^{-kt}$
The Stefan-Boltzmann Law: $Q = \sigma A \Delta T^4$
The Newton-Raphson Method: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
Q&A: Cooling of a Cooked Turkey =====================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to the cooling of a cooked turkey.

Q: What is Newton's Law of Cooling?

A: Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference between its temperature and the temperature of its surroundings.

Q: What is the cooling constant?

A: The cooling constant is a value that represents the rate at which the turkey loses heat to its surroundings. It is denoted by the symbol $k$ .

Q: How do I determine the cooling constant?

A: To determine the cooling constant, you can use the given information and the cooling equation. You can plug in the values of the initial temperature, the temperature of the surroundings, and the time to solve for the cooling constant.

Q: What is the temperature of the turkey after 30 minutes?

A: Unfortunately, we are not given the temperature of the turkey after 30 minutes. However, we can use the cooling equation to model the temperature of the turkey as a function of time.

Q: Can I use other cooling equations to model the temperature of the turkey?

A: Yes, you can use other cooling equations to model the temperature of the turkey. However, Newton's Law of Cooling equation is the most suitable equation to model the temperature of the turkey as a function of time.

Q: What is the significance of the cooling constant?

A: The cooling constant is a value that represents the rate at which the turkey loses heat to its surroundings. It is an important value in determining the temperature of the turkey as a function of time.

Q: Can I use the cooling constant to determine the temperature of the turkey at any time?

A: Yes, you can use the cooling constant to determine the temperature of the turkey at any time. Simply plug in the values of the initial temperature, the temperature of the surroundings, and the time to solve for the temperature of the turkey.

Q: What are some common applications of Newton's Law of Cooling?

A: Newton's Law of Cooling has many applications in various fields, including:

Food science: to determine the temperature of cooked food as a function of time
Engineering: to design cooling systems for buildings and machines
Biology: to study the temperature regulation of living organisms
Physics: to study the heat transfer between objects

Q: Can I use Newton's Law of Cooling to model the temperature of a living organism?

A: Yes, you can use Newton's Law of Cooling to model the temperature of a living organism. However, you need to take into account the metabolic rate of the organism, which can affect the cooling constant.

Q: What are some limitations of Newton's Law of Cooling?

A: Newton's Law of Cooling has some limitations, including:

Assumes a constant cooling constant: the cooling constant may not be constant over time
Does not take into account phase changes: the cooling constant may not be accurate when the turkey undergoes a phase change
Does not take into account convection: the cooling constant may not be accurate when convection is present