Physical Chemistry , 1st ed.

where is the total power flux(in units of J/m^2 s, or W/m^2 ) and the constants

have their usual meaning. The groups of constants in parentheses illustrate

that the total power flux is proportional to the fourth power of the absolute

temperature. That is, Planck’s equation produces the Stefan-Boltzmann law

(equation 9.18) and predicts the correct value, in terms of fundamental con-

stants, of the Stefan-Boltzmann constant . This was another prediction of

Planck’s derivation that was supported by observation.

Collectively, these correspondences suggested that Planck’s derivation could

not be ignored, and that the assumptions made by Planck in deriving equa-

tions 9.22 and 9.24 should not be discounted. However, many scientists (in-

cluding Planck himself, initially) suspected that Planck’s equations were more

of a mathematical curiosity and did not have any physical importance.

Planck’s quantum theory was a mere mathematical curiosity for only five

years. In 1905, the 26-year-old German physicist Albert Einstein (Figure 9.16)

published a paper about the photoelectric effect. In this paper, Einstein applied

Planck’s quantized-energy assumption not to the electrical oscillators in mat-

ter but to light itself. Thus, a quantum of light was assumed to be the energy

that light has, and the amount of that energy is proportional to its frequency:

Elighth

Einstein made several assumptions about the photoelectric effect:

1. Light is absorbed by electrons in a metal, and the energy of the light in-
   creases the energy of the electron.


2. An electron is bound to a metal sample with some characteristic energy.
   When light is absorbed by the electron, this binding energy must be
   overcome before the electron can be ejected from the metal. The charac-
   teristic binding energy is termed the work functionof the metal and is la-
   beled .


3. If any energy is left over energy after overcoming the work function,
   the excess energy will be converted to kinetic energy, or energy of
   motion.
   Kinetic energy has the formula ^12 mv^2. By assuming that each electron ab-
   sorbs the energy of one quantum of light, Einstein deduced the relationship

h^12 mv^2 (9.25)

where the energy of the light,h , is converted into overcoming the work func-

tion and into kinetic energy. Needless to say, if the energy of the light is less

than the work function, no electrons will be ejected because kinetic energy

cannot be less than zero. The work function therefore represents a threshold

energy for the photoelectric effect. Because the intensity of light is not part of

the equation, changing the intensity of light does not change the speed of the

ejected electrons. However, increasing the light intensity means more photons,

so one would expect a greater number of electrons to be ejected. However, if

the frequency of the light on the sample were increased, the kinetic energy of

the ejected electrons would increase (meaning that their velocity would in-

crease), since the work function is a constant for a particular metal. If one

plotted the kinetic energy of the ejected electrons versus the frequency of light

used, one should get a straight line as indicated by Figure 9.17. Using available

data (Einstein was not an experimentalist!), Einstein showed that this inter-

pretation indeed fit the facts as they were known regarding the photoelectric

effect.

9.8 Quantum Theory 259

Figure 9.16 Albert Einstein (1879–1955).

Einstein’s work had an enormous role in the de-

velopment of modern science. His 1921 Nobel

Prize was awarded for his work on the photoelec-

tric effect and the application of Planck’s law to

the nature of light itself. (His work on relativity

was still being evaluated by experimentalists.)

Figure 9.17 A simple diagram of the kinetic

energy of an ejected electron (directly related to

its speed) versus the frequency of light shined on

a metal sample. Below some threshold frequency

of light, no electrons are emitted. This threshold

frequency,, is called the work function of the

metal. The higher the frequency of light, the more

kinetic energy the emitted electron has, so the

faster it moves. Einstein related the frequency of

the light to the kinetic energy of the ejected elec-

trons using Planck’s ideas about quantized ener-

gies, and in doing so provided an independent

physical basis for Planck’s radiation law as well as

the concept of the quantization of light energy.

Frequency of light shined on metal

(No electron

Kinetic energy ofemitted electronemitted) 

© CORBIS-Bettmann