Every object whose temperature is above absolute zero emits thermal radiation. Some bodies are more efficient radiators than others. You will remember that a perfect emitter of thermal radiation is called a blackbody. Although perfect emit- ters do not exist, cavities—voids in opaque materials that have small openings to the outside—are a close approximation. The energy emitted by a cavity is called blackbody radiation. The rate at which energy is emitted follows Stefan’s law,
S 5 sT 4 , (28.1) where S is the rate of radiated energy emitted per square meter, s is the Stefan-Boltzmann constant, and T is the cavity’s absolute temperature.
lower T 28.2 Wien’s Displacement Law Relative Intensity A blackbody cavity emits a continuous spec- trum of radiation that includes radiation at λmax all frequencies. At higher temperatures the (a) Wavelength cavity emits more of its radiation at higher frequencies with shorter wavelengths. Wilhelm Wien (1864–1928) was a Figure 28-2 shows the relative intensity of German physicist who won the energy emitted at each wavelength. Figure Nobel Prize in Physics in 1911. Phys 28-02-G (a) is for a lower absolute temperature than figure (b). The symbol lmax represents the higher T wavelengths at which the most energy is According to Wien’s displacement
Relative Intensity emitted. law, the wavelength of the most in- In some ways the predictions of classical tense radiant energy emitted by a blackbody radiator is inversely pro- physics were correct. For instance, Wien’s portional to the radiator’s Kelvin λmax displacement law says that the most intense temperature. (b) Wavelength radiant energy emitted by a cavity progresses to shorter wavelengths as the cavity’s tem- 28-2 Energy emitted by a blackbody vs. perature rises: Sir James H. Jeans (1877–1946) wavelengths emitted at (a) low temperature; 23 was an English astronomer, physi- (b) high temperature lmaxT 5 2.8978 3 10 m?K (28.2) cist, and mathematician. Since frequencyPhys 28-03-Gincreases as wavelength decreases, Wien’s displacement law pre- dicts the experimental finding that Figure 28-2 shows. The constant k in Equation 28.3 is 28.3 The Rayleigh-Jeans Function called Boltzmann’s constant, where k 5 1.381 3 10223 J/K. You were Another formula based on classical phys- first introduced to this constant in Rayleigh-Jeans ics, developed by Lord Rayleigh and James Chapter 14 in connection with the classical prediction Jeans, was much less successful. The func- universal gas constant (R). Ludwig tion below is their attempt to find the rela- Boltzmann, who developed the sta- ) tionship between frequency and the energy
tistical basis for thermodynamics in , E ( f experimental emitted at that frequency from a blackbody observations the nineteenth century, showed that radiator at a given temperature. If E( f ) is the probability of a molecule being Energy the emitted energy as a function of fre- in any of two states was a function of its energy, which was proportional quency, the Rayleigh-Jeans function is to its absolute temperature. The pro- 8pf 2kT portionality constant in that function Frequency, f E( f ) 5 3 . (28.3) is Boltzmann’s constant. Boltzmann’s c 28-3 At higher frequencies, the Rayleigh-Jeans constant is considered to be one of The symbol k is Boltzmann’s constant, and function does not agree with the observed energies the four fundamental constants of c is the speed of light. All other quantities physics, along with G, c, and h. and frequencies emitted from a blackbody at a given temperature. in Equation 28.3 are constant at a given ART28-03G-A Fig.28-2 642 Chapter 28