Emission of Heat Radiation

Prevost's theory of exchange of heat

In 1792 Pierre Prevost published a theory of radiation known as the theory of exchange of heat. According to this theory, all bodies at all temperatures above 0 K radiate thermal energy and at the same time absorb radiation received from their surroundings. The amount of thermal radiation emitted per unit time depends on the nature of the emitting surface, its area and its temperature; hotter bodies radiate at a higher rate than cooler bodies.

Radiation absorbed by a body increases the kinetic energy of its constituent atoms oscillating about their mean positions, and since the average kinetic energy determines the temperature, the absorbed radiation raises the temperature of the body. The body itself also radiates, which lowers its energy and its temperature.

• If a body radiates more than it absorbs, its temperature decreases.

• If a body absorbs more than it radiates, its temperature increases.

• When the rate of absorption equals the rate of emission, the temperature remains constant and the body is in thermal equilibrium with its surroundings.

A cup of hot tea ($T_{\text{tea}} > T_{\text{room}}$) and a plate of ice ($T_{\text{ice}} < T_{\text{room}}$) both reach room temperature after some time. Even at room temperature all bodies continue to radiate and absorb; their rates of emission and absorption are then equal, which is why they do not cool down to 0 K.

Radiation and temperature — the visible effect

At room temperature, and in fact for all temperatures below about $800\,^{\circ}$C, the thermal radiation corresponds to wavelengths longer than those of visible light, so we do not see it. When a body is heated, the radiated energy shifts to shorter wavelengths. At around $800\,^{\circ}$C part of the emitted energy is in the visible range and the body appears red. At around $3000\,^{\circ}$C it looks white hot — the filament of a tungsten lamp appears white hot because its temperature is around $3000\,^{\circ}$C.

Factors on which the heat radiated depends

The amount of heat radiated by a body depends on:

• the absolute temperature of the body, $T$;

• the nature of the body — the material, and the nature of its surface, polished or not;

• the surface area of the body, $A$;

• the time duration for which the body emits radiation, $t$.

Emissive power

Since the amount of heat radiated $Q$ is directly proportional to the surface area $A$ and to the time $t$, it is convenient to consider the quantity of heat radiated per unit area per unit time, that is, the power emitted per unit area. This is defined as the emissive power or radiant power $R$ of the body at a given temperature $T$:

$$R = \dfrac{Q}{At}$$

Definition: the quantity of heat radiated per unit area per unit time is the emissive power of the body at the given temperature; equivalently, the power emitted per unit area at a given temperature.

• SI unit: J m$^{-2}$ s$^{-1}$, that is W m$^{-2}$.

• Dimensions: $[L^{0}M^{1}T^{-3}]$.

Coefficient of emission (emissivity)

The nature of the emitting surface — its material or degree of polish — is not itself a physical quantity, so to handle it we compare objects of different materials but identical geometry at the same temperature. At a given temperature a perfect blackbody has the maximum emissive power, so it is convenient to compare the emissive power of a given surface with that of a perfect blackbody at the same temperature.

Definition: the coefficient of emission or emissivity $e$ of a given surface is the ratio of the emissive power $R$ of the surface to the emissive power $R_{B}$ of a perfect black surface at the same temperature.

$$e = \dfrac{R}{R_{B}}$$

• For a perfect blackbody $e = 1$; for a perfect reflector $e = 0$.

• For an ordinary body $0 < e < 1$, depending on the nature of the surface; for example the emissivity of copper is 0.3.

• Emissivity is larger for rough surfaces and smaller for smooth and polished surfaces, and it also varies to some extent with temperature and with the wavelength of the radiation.

• Emissivity is a ratio of two emissive powers, so it is dimensionless and has no unit.

Standard results

$$R = \dfrac{Q}{At} \quad \left[\text{W m}^{-2},\ \left[L^{0}M^{1}T^{-3}\right]\right], \qquad e = \dfrac{R}{R_{B}} \quad \left[\text{dimensionless}\right]$$

• Prevost: every body above 0 K both emits and absorbs; thermal equilibrium means equal rates, not zero rates.

• Heat radiated depends on $T$, on the nature of the body and its surface, on $A$ and on $t$.