Derive Newton's Law of Cooling

In summary, Newton's Law of Cooling states that the rate of change of an object's temperature is directly proportional to the difference between its temperature and the temperature of its surroundings. This can be represented by the equation dT/dt ∝ T-Ts, where T is the temperature of the object and t is time. The constant of proportionality, k, represents the specific rate of cooling for the object.
  • #1
Junior Newton
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Homework Statement



Mathematically prove how Newton's Law of Cooling is derived.

Homework Equations


Newton’s Law of Cooling states that the rate of change of a temperature of an object is directly proportional to the difference between its own temperature and the temperature of its surroundings (Smith, 2010). Therefore, from this information it can be seen that the following relationship can be interpreted.

dT/dt ∝ T-Ts

Where T = temperature of the object and t = time.

The Attempt at a Solution



not sure how the k comes into it?
 
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  • #2
What does the term "directly proportional" mean in mathematical terms? That should answer your question.
 

1. What is Newton's Law of Cooling?

Newton's Law of Cooling is a mathematical equation that describes the rate at which an object cools down in a surrounding environment. It states that the rate of change of temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings.

2. How do you derive Newton's Law of Cooling?

To derive Newton's Law of Cooling, we start with the basic equation for heat transfer, Q = mcΔT, where Q is the amount of heat transferred, m is the mass of the object, c is its specific heat capacity, and ΔT is the change in temperature. We then use the concept of heat flux, which is the rate of heat transfer per unit area, and apply it to the different surfaces of the object to obtain the final equation.

3. What are the assumptions made in deriving Newton's Law of Cooling?

The assumptions made in deriving Newton's Law of Cooling are that the object is small enough to have uniform temperature throughout, there are no internal heat sources, and the surrounding environment is large enough to maintain a constant temperature.

4. Can Newton's Law of Cooling be applied to all objects?

No, Newton's Law of Cooling is only applicable to objects that can be considered as ideal or have a uniform temperature throughout. It is not suitable for objects with complex shapes or those that experience significant fluctuations in temperature.

5. How is Newton's Law of Cooling used in real life?

Newton's Law of Cooling has various applications in everyday life, such as in meteorology, cooking, and food preservation. It is also used in industries that involve temperature control, such as refrigeration, air conditioning, and chemical reactions. Additionally, it is used in forensic science to determine the time of death of a body by measuring its temperature.

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