A recurring theme throughout the study of quantum physics has been the omnipresence of
Planck’s constant h. This universal constant of the microscopic world first made its
appearance in 1900 in the study of the radiation of so-called blackbodies.
A blackbody is a substance that absorbs radiation of all wavelengths and radiates in a continuous
spectrum at all wavelengths. It is given the name blackbody because an object that absorbs light at all
wavelengths appears black to the human eye.
By the end of the 19th century, several properties of blackbody radiation had been established. First, the
total intensity I (the average rate of radiation of energy per unit surface area) emitted from a blackbody
was shown to be proportional to the fourth power of its temperature:
This is called the Stefan-Boltzmann law for a blackbody. The constant of proportionality ? is known as
the Stefan-Boltzmann constant and was determined to be ?=5.67×10?8W/(m2?K4). It had also
been discovered that the wavelength at which the radiation intensity was maximum varied inversely with
temperature. This result, known as the Wien displacement law, is written
where ?m is the wavelength with the greatest radiated intensity.
One aspect of blackbody radiation that remained unexplained was the full wavelength dependence of the
intensity of the radiation, I(?). In 1900, largely through trial and error, Max Planck formulated the following
equation that successfully explained the wavelength dependence of the intensity:
where h is Planck’s constant, c is the speed of light in vacuum, and kB is Boltzmann’s constant.
Planck justified his law by claiming that different modes of electromagnetic oscillations within the cavity
could only emit radiation in increments of energy equal to Planck’s constant h multiplied by the
frequency f. At first, Planck did not believe in that idea himself, but the revolutionary concept
ofquantization (or “clumping”) of energy paved the way for the “quantum revolution” in physics.
Consider a blackbody that radiates with an intensity I1 at a room temperature of
intensity I2 will this blackbody radiate when it is at a temperature of 400K?
300K. At what
Express your answer in terms of I1.
At what wavelength ?m would the intensity of blackbody radiation be at a maximum when the blackbody
is at 2900K?
Express your answer in meters to two significant figures.
?m = ?????? m
An astronomer is trying to estimate the surface temperature of a star with a radius of 5.0×108m by
modeling it as an ideal blackbody. The astronomer has measured the intensity of radiation due to the star
at a distance of 2.5×1013m and found it to be equal to 0.055W/m2. Given this information, what is
the temperature of the surface of the star?
Express your answer in kelvins to two significant digits.
T = ????? K
Exciting an Oxygen Molecule An oxygen molecule (O2) vibrates with an energy identical to
that of a single particle of mass m=1.340×10?26kg attached to a spring with a force constant of k =
1215N/m . The energy levels of the system are uniformly spaced, as indicated in the figure( Figure 1) ,
with a separation given by hf.
What is the vibration frequency of this molecule?
Express your answer using four significant figures.
f = ???? Hz
How much energy must be added to the molecule to excite it from one energy level to the next higher
Express your answer using three significant figures.
E = ??????? J
3. A hydrogen atom, initially at rest, emits an ultraviolet photon with a wavelength of ? =
What is the recoil speed of the atom after emitting the photon?
v = ???? m/s
4. A blue-green photon (? = 488nm ) is absorbed by a free hydrogen atom, initially at rest.
What is the recoil speed of the hydrogen atom after absorbing the photon?
v = ????? m/s
5. An X-ray photon with a wavelength of 0.260nm scatters from a free electron at rest. The
scattered photon moves at an angle of 110? relative to its incident direction.
Find the initial momentum of the photon.
Find the final momentum of the photon.