Introduction to Newton’s law of cooling

Newton’s law of cooling states that a body’s rate of loss of heat is directly proportional to the temperature difference between the body and its surroundings. When a hot body is placed at a lower temperature, the hot body slowly cools down by losing heat energy to its surroundings.

Isaac Newton found that the temperature of a hot object decreases at a rate proportional to the difference between it and the surrounding temperature. Obversely, an object colder than its surroundings warms at a rate proportional to the same difference.

Equation of Newton’s law of Cooling

The formula governing the law is dQ/dt = c (T - S)        (1)

where T is the object's temperature, S is the surrounding temperature, dQ is the quantity of heat lost in time dt, and c is a constant of proportionality.

Derivation of Equation

For a small temperature difference between a body and its surroundings, the body’s rate of cooling is directly proportional to the temperature difference. If a body of temperature T and surface area A is kept in a surrounding temperature T₀(T₀ < T), then net loss of thermal energy per unit time amounts to:

dQ/dt =εσ A(T⁰ - T₀⁰)

If the temperature difference is small, then:

     T = T₀ + ΔT

=>εσ A[(T₀ +Δ T)⁴] =εσ A[T₀⁴ (1 + ΔT/T₀ )⁴ - T₀⁴]

=>εσ AT₀⁴[1 + 4 ΔT/T₀ + higher powers of ΔT/T - 1]

= 4εσ AT₀³ ΔT

Now, the rate of loss of heat at temperature T is:

dQ/dt = -mc dT/dt

mc dT/dt = - 4εσ AT₀³[T - T₀]

dT/dt = - K[T - T₀]

K = 4εσAT_0³ /mc

dT/dt  α(T - T₀)

We know that the heat that a body loses depends on its heat capacity. If m is the mass of the body and s is the specific heat, then

                                     dQ/dt = m s dT/dt

With Newton’s law (eq. 1), this equation can be used to find the specific heat capacity of the liquid.
 

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Reference links:

•  http://www.hyperphysics.phy-str.gsu.edu/hbase/thermo/spht.html
•  http://www.physics.about.com/od/energyworkpower/f/KineticEnergy.htm
•  http://www.en.wikipedia.org/wiki/heat