Download Blackbody Radiation Problems

Blackbody/Planck Problem Solving
Modern Physics
Mr. Youker |
1. The Sun has a peak luminous intensity at a wavelength of approximately 500 nm.
Estimate the temperature of the Sun’s surface.
2. Suppose a star has a surface temperature of 32,500 K. What color would this star
appear to be?
3. What is the peak wavelength of electromagnetic radiation emitted by a healthy
human being? Within what portion of the electro-magnetic spectrum does this
fall?
4. A metal rod is inserted into a fire. Its temperature goes from room temperature
(20°C) to that of the fire (700°C). By what factor does the electro-magnetic
radiation output increase? What color will the rod appear when in the fire?
5. The above problems illustrate some basic calculations of luminous “black-bodies”.
What was the problem with our physical understanding of the production of light
by these items in 1900? What colorful name was used to describe this issue?
6. What hypothesis did Max Planck introduce in an attempt to explain the production
of light by “black-bodies”?
7. Sketch a typical “black-body” emission spectrum and explain why it has the shape
that it does.
8. As a macroscopic example of Planck’s Theory of Quantization consider a 500 gram
mass vibrating up and down on a spring.
The equation of oscillation frequency for this system is: f = 1/2 √(k/m)
a. What is the frequency of the 500 gram mass on the spring (k = 50 N/m)?
b. What is the size of one energy “quantum” for this system?
The equation for the energy of oscillation for a mass on a spring is: E = ½ kx2
c. How many quanta of energy does the 500 gram mass/spring system
possess if it vibrates up and down with a displacement (x) of 5 cm?
9. An H-Cl molecule vibrates with a natural frequency of 8.1 x1014 Hz. What is the
difference in energy (in J and eV) between possible values of the oscillation energy?
10. Shortly after Planck’s hypothesis was put forth, Einstein proposed that not only
was the vibratory energy of objects quantized, but the light energy emitted by
them was also quantized. He called the quanta of light energy “photons”. Their
energy followed Planck’s formula:  = hf
a. What is the energy carried by one photon of infrared light ( = 900 nm)?
b. What is the energy carried by one photon of ultraviolet light ( = 300 nm)?
Blackbody/Planck Problem Solving
Modern Physics
Mr. Youker |
1. The Sun has a peak luminous intensity at a wavelength of approximately 500 nm.
Estimate the temperature of the Sun’s surface.
peak = 0.0029/T
500 x10-9 = 0.0029/T
T = 5800 K
2. Suppose a star has a surface temperature of 32,500 K. What color would this star
appear to be?
peak = 0.0029/T
peak = 0.0029/32,500
peak = 8.9 x10-8 m
peak = 89 x10-9 m
peak = 89 nm

Peak emission is in the UV-range.
The star would likely appear white or blue.
3. What is the peak wavelength of electromagnetic radiation emitted by a healthy
human being? Within what portion of the electro-magnetic spectrum does this
fall?
peak = 0.0029/T
peak = 0.0029/310
peak = 9.4 x10-6 m
peak = 9400 x10-9 m
peak = 9400 nm

Peak emission is in the IR-range.
4. A metal rod is inserted into a fire. Its temperature goes from room temperature
(20°C) to that of the fire (700°C). By what factor does the electro-magnetic
radiation output increase? What color will the rod appear when in the fire?
ΔE/A.Δt = (5.7 x10-8)T4
ΔE/A.Δt = (5.7 x10-8)T4
ΔE/A.Δt = (5.7 x10-8)(293)4
ΔE/A.Δt = (5.7 x10-8)(973)4
ΔE/A.Δt = 420 J/m2s
ΔE/A.Δt = 51,088 J/m2s
The radiation output increases by a factor of 122.
peak = 0.0029/T
peak = 0.0029/973
peak = 2.98 x10-6 m
peak = 2980 nm

Peak emission is in the IR-range.
It might be glowing a dim red.
5. The above problems illustrate some basic calculations of luminous “black-bodies”.
What was the problem with our physical understanding of the production of light
by these items in 1900? What colorful name was used to describe this issue?
The physics theories present in 1900 (Maxwellian electromagnetic theory,
Newtonian mechanics, the wave model of light, etc.) were inadequate to
produce an explanation of blackbody radiation. The most successful result
was known as the Rayleigh-Jeans Law… which fit the blackbody curve well
at long wavelengths, however it predicted a prodigious amount of UV
radiation to be produced. This was known as the “ultra-violet catastrophe”.
6. What hypothesis did Max Planck introduce in an attempt to explain the production
of light by “black-bodies”?
Planck hypothesized that a “quantum restriction” could be placed on the
energy of vibrating systems. For a black-body to produce UV light waves
the frequency of oscillations within the molecular matter would be
constrained by the fact that higher frequencies corresponded to higher
energies:  = hf.
7. Sketch a typical “black-body” emission spectrum and explain why it has the shape
that it does.
This curve has the total energy of the hot
object spread out over a range of
wavelengths. The long wavelengths
(infrared and red) require little energy to
produce… the short wavelengths
(ultraviolet and blue) require more
energy to produce.
8. As a macroscopic example of Planck’s Theory of Quantization consider a 500 gram
mass vibrating up and down on a spring.
The equation of oscillation frequency for this system is: f = 1/2 √(k/m)
a. What is the frequency of the 500 gram mass on the spring (k = 50 N/m)?
b. What is the size of one energy “quantum” for this system?
The equation for the energy of oscillation for a mass on a spring is: E = ½ kx2
c. How many quanta of energy does the 500 gram mass/spring system
possess if it vibrates up and down with a displacement (x) of 5 cm?
f = 1/2 √(k/m)
 = hf
Enet = n
f = 1/2 √(50/0.500) = (6.63 x10-34)(1.6)
f = 1.6 Hz
 = 1 x10-33 J
½ kx2 = n
(0.5)(50)(0.05)2 = n
0.0625 = n 1 x10-33
n = 6.25 x1031 quanta
9. An H-Cl molecule vibrates with a natural frequency of 8.1 x1014 Hz. What is the
difference in energy (in J and eV) between possible values of the oscillation energy?
 = hf
 = (6.63 x10-34)( 8.1 x1014)
 = 5.4 x10-19 J
 = 3.36 eV
10. Shortly after Planck’s hypothesis was put forth, Einstein proposed that not only
was the vibratory energy of objects quantized, but the light energy emitted by
them was also quantized. He called the quanta of light energy “photons”. Their
energy followed Planck’s formula:  = hf
a. What is the energy carried by one photon of infrared light ( = 900 nm)?
b. What is the energy carried by one photon of ultraviolet light ( = 300 nm)?
 = hf
 = h(c/)
 = (6.63 x10-34)(3.0 x108)/(900 x10-9)
 = 2.2 x10-19 J
 = 1.38 eV
 = hf
 = h(c/)
 = (6.63 x10-34)(3.0 x108)/(300 x10-9)
 = 6.6 x10-19 J
 = 4.14 eV