Newton's Law of Cooling

8. Newton's law of cooling

According to Newton's law of cooling, if the temperature of a body, T, is not very different from the temperature of its surroundings, T₀, then the rate at which the body loses heat (rate of cooling) is proportional to the temperature difference between the body and the surroundings.

Thus, rate of cooling is proportional to (T - T₀).

To show this in differential form, we write the proportionality as an equality by introducing a positive constant of proportionality.

Assume the temperature of the body at time t is given by

T = T₀ + ΔT

Here ΔT = T - T₀, assumed small so that the linear approximation inherent in Newton's law is valid.

Expressing rate of change of temperature with respect to time, the law becomes

or, in standard differential form,

where the left-hand side represents the rate of fall of the temperature of the body and the right-hand side is proportional to the temperature difference. We therefore write

dT/dt = -k (T - T₀),

where k (> 0) is a constant depending on the nature of the body and the conditions of the surroundings. The minus sign indicates that when T > T₀ the temperature of the body decreases.

8.1 Variation of temperature of a body according to Newton's law

Suppose a body has initial temperature T_i at time t = 0 and it is placed in an atmosphere whose temperature is constant and equal to T₀. We seek the temperature of the body at any later time t, assuming Newton's law holds (i.e., temperature difference remains small or proportionality is valid).

The law gives

Writing the differential equation explicitly,

Here k is a positive constant characteristic of the cooling process.

Solving the differential equation by separation of variables and integration gives the temperature at time t as

From this expression we see that at t = 0, q = q_i, and as t → ∞, q → q₀. Thus the temperature of the body approaches the ambient temperature exponentially with time. The temperature versus time graph is a decreasing exponential curve from q_i toward q₀.

Note : If the body cools by thermal radiation and if temperatures involved are not very different, one often uses the approximation obtained by linearising the Stefan-Boltzmann law around the ambient temperature. Writing T = T₀ + ΔT and expanding T⁴ ≈ T₀⁴ + 4T₀³ΔT (neglecting higher order terms), the radiative heat-loss term becomes proportional to ΔT.

Using this approximation, the cooling equation

reduces to an exponential form equivalent to Newton's law:

This exponential form is the basis for solving numerical problems involving Newton's law of cooling.

8.2 Limitations of Newton's law of cooling

• The temperature difference between the body and surroundings must be small for the proportionality (linear approximation) to hold.
• The dominant mode of heat loss should be one that can be approximated as proportional to temperature difference (for example convective cooling or radiative cooling linearised around the ambient temperature); pure radiative cooling without linearisation follows T⁴ dependence and is not strictly Newtonian for large temperature differences.
• The temperature of the surroundings must remain essentially constant during the cooling of the body.

Ex.20 A body at temperature 40°C is kept in a surrounding of constant temperature 20°C. It is observed that its temperature falls to 35°C in 10 minutes. Find how much more time will it take for the body to attain a temperature of 30°C.

Solution:

Using Newton's law in the form T(t) = T₀ + (T_i - T₀) e^-kt,

for the interval in which temperature falls from 40°C to 35°C:

Here T₀ = 20°C, T_i = 40°C and after t = 10 min, T = 35°C.
Substitute into the exponential law and solve for k:

Now for the next interval when temperature falls from 35°C to 30°C, let the required additional time be t.

(30 - 20) = (35 - 20) e^-kt

Using the value of k obtained from the first interval (from the previous equations)

for the interval in which temperature falls from 35°C to 30°C we obtain

From the exponential relation (equation above),

Therefore the required additional time is

Alternate: (by approximate method)
Using the same exponential relation and the values from the first interval to get k, then applying to the second interval directly gives the same result as above.

9. NATURE OF THERMAL RADIATIONS : (WIEN'S DISPLACEMENT LAW)

From the energy distribution curve of black-body radiation (spectral energy distribution versus wavelength) the following conclusions can be drawn:

• The higher the temperature of a black body, the greater is the area under the spectral-energy curve; that is, a body emits more total energy per unit area at a higher temperature.
• The energy emitted by the body is not uniform across wavelengths; for both very long and very short wavelengths the emitted energy is small.
• For any given temperature there is a particular wavelength, denoted λ_m, at which the spectral emissive power is maximum.
• As the temperature of the black body increases, the wavelength λ_m corresponding to the maximum of the curve shifts to shorter wavelengths.

Experimental study of the black-body spectrum established that the wavelength λ_m at which emission is maximum is inversely proportional to the absolute temperature T of the black body. This relation is known as Wien's displacement law:

Or written explicitly:

Here the constant b is Wien's displacement constant. Numerically, b = 0.282 cm·K (or 2.82 × 10⁻³ m·K when expressed in SI units depending on the units used for wavelength).

Summary: Newton's law of cooling states that for small temperature differences the rate of cooling is proportional to the temperature difference, leading to an exponential decay of temperature toward the ambient temperature. The law is an approximation valid under specific conditions (small ΔT, approximate linearity of heat-loss mechanism and constant surroundings). Wien's displacement law describes how the wavelength of maximum emission from a black body shifts inversely with temperature.