Understanding Newton's Law of Cooling

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

Mathematically, Newton's Law of Cooling can be expressed as:

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

Where:

• $\frac{dQ}{dt}$ is the rate of heat loss.
• $k$ is a positive constant that depends on the properties of the object and its surroundings.
• $T$ is the temperature of the object.
• $T_s$ is the temperature of the surroundings.

The negative sign indicates that heat is lost when the object's temperature is higher than the surroundings ($T > T_s$). This law is widely used in various fields to analyze Heat Transfer processes.

Relating Newton's Law of Cooling to Stefan's Law

While Newton's Law of Cooling provides a simple approximation, its theoretical basis comes from a more fundamental law governing thermal Radiation: Stefan-Boltzmann's Law (often referred to as Stefan's Law).

Stefan's Law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of the black body's thermodynamic temperature ($T$). For a real object (not a perfect black body) and considering Radiation exchange with the surroundings, the net rate of heat loss by Radiation from an object at temperature $T$ in surroundings at temperature $T_s$ is given by:

$\frac{dQ}{dt} = \epsilon \sigma A (T^4 - T_s^4)$

Where:

• $\epsilon$ is the emissivity of the object.
• $\sigma$ is the Stefan-Boltzmann constant.
• $A$ is the surface area of the object.
• $T$ is the absolute temperature of the object.
• $T_s$ is the absolute temperature of the surroundings.

Newton's Law as an Approximation of Stefan's Law

Newton's Law of Cooling is an Approximation of Stefan's Law under specific conditions, particularly when the temperature difference between the object and its surroundings is small. Let's see how this Approximation works.

The term $(T^4 - T_s^4)$ can be rewritten using algebraic identities:

$T^4 - T_s^4 = (T^2 - T_s^2)(T^2 + T_s^2) = (T - T_s)(T + T_s)(T^2 + T_s^2)$

So, the rate of heat loss is:

$\frac{dQ}{dt} = \epsilon \sigma A (T - T_s)(T + T_s)(T^2 + T_s^2)$

If the temperature difference $(T - T_s)$ is small, then $T$ is approximately equal to $T_s$. Let's say $T \approx T_s \approx T_{avg}$, where $T_{avg}$ is an average temperature of the object and surroundings.

In this case, the terms $(T + T_s)$ and $(T^2 + T_s^2)$ can be approximated:

• $(T + T_s) \approx (T_{avg} + T_{avg}) = 2T_{avg}$
• $(T^2 + T_s^2) \approx (T_{avg}^2 + T_{avg}^2) = 2T_{avg}^2$

Substituting these into the equation for $\frac{dQ}{dt}$:

$\frac{dQ}{dt} \approx \epsilon \sigma A (T - T_s)(2T_{avg})(2T_{avg}^2)$

$\frac{dQ}{dt} \approx \epsilon \sigma A (T - T_s)(4T_{avg}^3)$

We can group the terms $\epsilon \sigma A (4T_{avg}^3)$ into a single constant, say $k'$, which is approximately constant if the temperature difference is small (meaning $T_{avg}$ doesn't change much):

$\frac{dQ}{dt} \approx k'(T - T_s)$

This equation has the same form as Newton's Law of Cooling, where $k'$ corresponds to the constant $k$ in Newton's Law. This demonstrates that Newton's Law is indeed an Approximation of Stefan's Law under conditions of small Temperature Difference.

Considering Other Options

• Wien's Displacement Law: This law relates the peak wavelength of emitted Radiation by a black body to its temperature. It is related to Radiation but does not directly give the total rate of heat loss needed for Newton's Law of Cooling.
• Kirchhoff's Law of Thermal Radiation: This law states that for an object in thermal equilibrium with its surroundings, its emissivity is equal to its absorptivity. It is important for understanding Radiation properties but isn't the fundamental law from which Newton's Law of Cooling is derived as an Approximation.
• Jean's Law (Rayleigh-Jeans Law): This law is an early attempt to describe the spectral radiance of thermal Radiation from a black body based on classical ideas. It works well for long wavelengths but fails at short wavelengths (the "ultraviolet catastrophe"). It's part of the history leading to Planck's Law but is not the basis for Newton's Law of Cooling.

Therefore, Newton's Law of Cooling is best understood as an Approximation derived from Stefan's Law when the temperature difference between the object and its surroundings is small, simplifying the calculation of Heat Transfer by Radiation.