Newton's Law of Cooling

Newton's Law of Cooling describes the rate at which an object changes temperature through radiation. It states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings, provided the temperature difference is small.

The Law

Mathematically, Newton's Law of Cooling is expressed as:

$$ \frac{dT}{dt} = -k(T - T_{\text{env}}) $$

where:

• $\frac{dT}{dt}$ is the rate of change of temperature of the object with time.
• $T$ is the temperature of the object.
• $T_{\text{env}}$ is the temperature of the environment.
• $k$ is a positive constant that depends on the characteristics of the object (such as surface area, nature of the surface, and surrounding medium).

Assumptions

Newton's Law of Cooling is based on several assumptions:

• The temperature difference between the object and the environment is relatively small.
• The heat transfer is solely due to radiation.
• The properties of the object and the environment remain constant during the cooling process.
• The object is a perfect black body (ideal emitter and absorber of radiation).

Important Points and Differences

Aspect	Description
Temperature Difference	The temperature difference between the object and the environment should be small for the law to hold accurately.
Heat Transfer Mechanism	The law assumes that heat transfer occurs only through radiation, not considering conduction or convection.
Surrounding Environment	The temperature of the environment is considered constant during the cooling process.
Object Characteristics	The constant $k$ is specific to the object's properties and the nature of its surroundings.

Formulas and Examples

Cooling Rate Formula

The cooling rate formula derived from Newton's Law of Cooling is:

$$ \frac{dT}{dt} = -k(T - T_{\text{env}}) $$

Example 1: Calculating the Cooling Rate

Suppose a cup of coffee at $80^\circ C$ is placed in a room with a constant temperature of $20^\circ C$. If the constant $k$ is found to be $0.07 \, \text{min}^{-1}$, what is the rate of cooling at the moment the coffee is placed in the room?

Using the formula:

$$ \frac{dT}{dt} = -k(T - T_{\text{env}}) $$

$$ \frac{dT}{dt} = -0.07(80 - 20) $$

$$ \frac{dT}{dt} = -0.07(60) $$

$$ \frac{dT}{dt} = -4.2^\circ C/\text{min} $$

The negative sign indicates that the temperature is decreasing.

Example 2: Temperature After a Certain Time

If you want to know the temperature of the coffee after 5 minutes, you can integrate the cooling rate formula to get the exponential decay formula:

$$ T(t) = T_{\text{env}} + (T_0 - T_{\text{env}})e^{-kt} $$

where $T_0$ is the initial temperature of the object.

Given $T_0 = 80^\circ C$, $T_{\text{env}} = 20^\circ C$, $k = 0.07 \, \text{min}^{-1}$, and $t = 5 \, \text{min}$, we can calculate:

$$ T(5) = 20 + (80 - 20)e^{-0.07 \times 5} $$

$$ T(5) = 20 + 60e^{-0.35} $$

$$ T(5) = 20 + 60 \times 0.7048 $$

$$ T(5) = 20 + 42.29 $$

$$ T(5) = 62.29^\circ C $$

The coffee will be approximately $62.29^\circ C$ after 5 minutes.

Conclusion

Newton's Law of Cooling is a useful principle in thermodynamics for predicting the cooling behavior of objects. However, it is important to consider its limitations and the assumptions on which it is based. The law is most accurate when the temperature difference is small and when heat transfer by radiation is the dominant mechanism.