Wien displacement law -Jayam chemistry learners
State Wien displacement law:
When we casually look into the sky at night, we observe
blue, white, and red colored twinkling stars. Have you ever thought about how
hot the stars are? Or how to determine the temperature of stars or other
celestial bodies? Surprisingly, today we are discussing all this stuff in this
blog post. Then why late? Let us start our learning journey.
Wien displacement law is an empirical generalization to
measure the surface temperature of stars and other terrestrial objects when
their temperature is higher than their surroundings. It enumerates the highest
spectral radiance of a blackbody radiator as a function of its wavelength at a
particular temperature. And it unfolded the movement of the maximum intense
wavelength peak toward shorter wavelengths at higher temperatures. Hence,
Wien's law later became Wien displacement law. The term 'displacement' depicts
the transfer of wavelength peak having maximum emissive power towards higher
energies when the temperature escalates.
What do you mean by Wien displacement law?
It determines the spectral intensity per unit wavelength of emitted thermal electromagnetic radiation at λmax for any temperature.
Moreover, the Wien displacement law interprets an inversely proportional relationship of maximum intense blackbody radiation wavelength (λmax) with its absolute temperature.
As a result, the temperature of the emitted thermal radiation is low at longer wavelengths. And at shorter wavelengths, higher is the temperature of released blackbody radiation.
Besides, the law held good at shorter wavelengths of radiation emissions but failed to give accurate results at their longer wavelengths.
Wien displacement law formula:
It is a temperature-specific mathematical relationship to estimate
the blackbody spectral brightness as a function of its wavelength or frequency.
We know that frequency and wavelength of any radiation are proportionally
related to a constant term called the speed of light in a vacuum. Thus, we have
two different forms of the Wien displacement formula separately for the peak wavelength
and frequency.
- Wien displacement law formula for intense wavelength peak:
The product of the maximum emissive power wavelength and the
body's absolute temperature is constant. And it is equal to a fixed numerical
value called Wien's constant.
λm x T = b
λm denotes the wavelength of thermal radiation with
peak intensity.
T= absolute temperature of the body.
b=Wien's constant. And its value is 2.89 x 10-3 meter
Kelvin
It shows the inversely proportional relationship between the maximum intense wavelength peak and the absolute temperature.
λm = b⁄T
(Or) λm ∝ 1⁄T
If we assume two blackbody curves with the maximum intense wavelength peaks at λm1 and λm2 wavelengths at T1 and T2 temperature conditions. We can rewrite the Wien displacement wavelength formula as below;
λm,1 = b⁄T1-----(1)
λm,2 = b⁄T2-----(2)
Dividing eq (1) with eq (2), we get;
λm,1 ⁄ λm,2 = T2 ⁄ T1
- Wien displacement law formula for intense frequency peak:
The product of Wien's constant and the body's absolute temperature
gives the maximum intense frequency value of released thermal radiation.
νm = b x T
νm = frequency peak with maximum intensity
T= body’s absolute temperature on the Kelvin scale
b = Wien's constant. And its value is 0.058 terahertz/Kelvin
It shows the directly proportional relationship between the
highest intense frequency and the absolute temperature of the radiator.
νm ⁄ T = b
(Or)νm ∝ T
If we assume two blackbody curves with the maximum intense frequency peaks at νm1 and νm2 frequencies at T1 and T2 temperature conditions. We can rewrite the Wien displacement frequency formula as below;
νm,1 ⁄ T1 = b-------(1)
νm,2 ⁄ T2 = b-------(2)
Dividing eq (1) with eq (2), we get;
νm,1⁄νm,2 = T1⁄T2
Theory of Wien displacement law:
Wilhelm Wein's research on blackbody radiation emissions revealed
their spectral energy densities at various temperature conditions. He noticed
that the thermal energy radiated per unit wavelength increases with the
temperature of the blackbody enclosure. He combined this analysis of blackbody
emissions with the Boltzmann distribution of atoms to derive a formula for
blackbody radiant emittance as a function of wavelength and temperature. It is
Wien's radiation law (or) Wien's approximation (or) Wien's law.
The basis of Wien's law is in classical physics, where energy
propagates continuously as a wave from one point to the other. So, he assumed a
slow adiabatic expansion for a blackbody pinhole that holds a bundle of
blackbody radiation waves at thermal equilibrium. As a result, the blackbody
cavity does not transfer matter or heat to the surroundings. It leads to zero
net heat flow with the surroundings in the thermal equilibrium state by
following the second law of thermodynamics.
Heating the blackbody enclosure disturbs its thermodynamic
equilibrium state. It correspondingly increases the magnitude of absorbed
thermal electromagnetic energies of the blackbody enclosure. It enhances the
quantity of released blackbody radiant energies following the energy conservation
law. Consequently, the maximum intense wavelength peaks move toward shorter
wavelengths (i.e., higher energy end) upon increasing the temperature. It is
the essence of Wien's displacement law which narrates the inversely
proportional relationship between the λmax peak and the body's absolute
temperature.
The mentioned Wien's law and Wien's displacement law formulas give
precise results at shorter radiation wavelengths or higher frequencies as they
correctly measure the continuous variation of spectral intensity as a function
of temperature.
But they fail to enumerate the spectral intensity changes at
longer radiation wavelengths or lower frequencies.
Wien displacement law graph:
It is a graphical explanation of a
blackbody's emissive power and radiation wavelength at various temperatures. We
know blackbody emissions are temperature specific. And a blackbody graph is an
overall variation of a blackbody's emissive power as a function of wavelength
measured at different temperatures. But the Wien displacement law graph focuses
on spectral intensities of blackbody emissions as a function of wavelength at a
peak wavelength state λmax at a specific temperature T. Here is an infographic
with different scenarios of the Wien displacement law graph.
The first picture of the Wien
displacement law infographic shows the variation of blackbody emissive power
with temperature. It depicts that the height of the λmax peak increases toward
higher temperatures suggesting a rise in their spectral intensities.
If E1, E2, and E3 are emissive power
magnitudes at temperatures T1, T2, and T3. When E1<E2<E3, then
T1<T2<T3 exhibiting their directly varying relationship.
The second picture of the Wien
displacement law infographic shows the variation of λmax peaks with the radiation
wavelengths at different temperature conditions. Here we can notice the
movement of λmax crests toward shorter wavelengths with the temperature rise,
which indicates an escalation of the amount of emitted radiant energies.
If λm1, λm2, and λm3 are the
maximum intense wavelengths of emitted blackbody radiation at T1, T2, and T3.
When T3>T2>T1, then λm3<λm2<λm1. It
points out their inversely changing connection.
The picture is a graph between the
intensity and wavelength of emitted thermal electromagnetic radiations. It
exhibits that the released thermal radiation intensity shows a distorted linear
decline from higher temperatures due to a decrease in their wavelengths, as it
violates Stefan-Boltzmann law.
Next is a graph between the fourth
root of radiation intensity with wavelength. It exhibits a steep linear
decrease in radiation wavelengths for λmax peaks having higher temperatures.
Finally, it accurately depicts the Wien displacement law.
The last graph of this infographic describes the Wien displacement law limitation. The Wien displacement law graph is a uniformly changing curve for radiation emissive power and wavelength at higher temperatures. But the Wien displacement law graph is discontinuous at lower temperatures for longer radiation wavelengths. As a result, the Wien displacement law gives approximate results at longer radiation wavelengths.
Wien's constant:
It is a proportionality constant measuring spectral intensities of
released thermal radiation as a function of wavelength or frequency at a
particular temperature.
It is a physical quantity with different numerical values for the
maximum intense wavelength and frequency peaks formulas. Hence, it is not a
universal constant.
The English alphabet 'b' denotes it.
(a) For λmax calculations:
1. The formula of b:
λm x T = b
Where, b = hc⁄(4.965)k
2. Value of Wien's constant:
The value of b is 2.89 x 10-3 meter Kelvin in the SI
system.
The value of b is 0.289 centimeter Kelvin in the CGS system
3. Units of b:
Its SI unit is meter Kelvin.
Its CGS unit is centimeter Kelvin.
4. Dimensional formula:
Its dimensional formula is [M0L1T0K1]
5. Method to calculate b value:
We have, b = hc⁄(4.965)k
h = Planck’s constant.
And its value is 6.626 x 10-34 joule second
c = Velocity of light.
And its value is 3 x 108 meter per second
k = Boltzmann
constant. And its value is 1.3807 x 10-23 joules per Kelvin
By substituting all
the values of the physical quantities in the above equation, we get;
λm = (6.626 x 10-34) x (3 x 108)/ 4.965 x (1.3807 x 10-23)
λm = 19.878
x 10-26/ 6.855 x 10-23
λm = 2.899
x 10-3 meter Kelvin
(b) For νmax
calculations:
1. The formula of b:
νm = b x T
Where, b = [(2.821)k]⁄h
2. Value of Wien's constant:
The value of b is 0.58 x 1011 Hertz/Kelvin in both the
SI and CGS systems.
3. Units of b:
The unit of b in both the SI and CGS system is Hertz/Kelvin.
4. Dimensional formula:
Its dimensional formula is [M0L0T-1K-1]
5. Method to calculate b value:
We have, b = [(2.821)k]⁄h
b = [(2.821) (1.3807 x 10-23)]/
(6.626 x 10-34)
b = 0.58 x 1011
Wien displacement law calculation:
Wien's law measures the highest intense wavelength of
blackbodies at their absolute temperature conditions with a simple empirical
formula.
With the invention of the mathematical formula, Wien
concluded that the product of the zenith wavelength with the body's temperature
is always constant in thermal equilibrium conditions.
It's possible to quantify the temperature of an object from
the radiation wavelength by Wien's law and vice versa.
At room temperature, for 25 degrees centigrade, the highest
wavelength of intense radiation is 9.6 micrometers.
Similarly, for 30 degrees centigrade, the highest wavelength
of intense radiation is 9.3 micrometers.
Both these examples show that, at room temperature
conditions, hot objects emit energies in the infrared region predominantly.
Below is a table explaining the relationship between λmwith the object's absolute temperature.
| Wavelength in nm | Temperature in Kelvin | Color of radiation |
|---|---|---|
| 350 | 8257 | Ultraviolet light |
| 400 | 7225 | White |
| 450 | 6422 | Blue |
| 550 | 5255 | Yellow |
| 650 | 4446 | Reddish Orange |
| 750 | 3853 | Dull red |
I will show how to calculate the body's temperature using
the Wien displacement formula from the known wavelength data.
When a heated object emits thermal electromagnetic radiation
at 750 nm, let us calculate its temperature in the Kelvin scale.
The peak wavelength of the body with maximum intensity=750
nm
Wien's constant value = 2.89 x 10-3 mK
Now, the Wien displacement law formula is
λm
x T = b
T = 2.89 x
10-3 mK/750 x 10-9 m
T= 3853 K
When the radiation color is yellow with 550 nm wavelength,
its temperature will be 5255 Kelvin, following Wien's formula.
Likewise, if the radiation temperature is 7225 Kelvin, it is
white-colored with 400 nm wavelength.
It contradicts our assumption that red objects are hot and
white bodies are cold. White-colored materials possess the highest temperature
of all the other colors of visible radiations.
Next, a further drop in wavelength to 350 nm moves the radiation to the ultraviolet region with an 8257 Kelvin temperature.
The Wien displacement law graph for the peak wavelength positions of hot material at its absolute temperatures shows a linearly declining curve.
Here is a linear curve depicting the shift of peak wavelengths towards shorter wavelengths based on the previous table data attached above.
It shows the peak wavelengths of blackbodies move to shorter wavelengths at higher temperature and hold an inversely proportional correlation.
Applications of Wien's displacement law:
1. In astronomy:
As mentioned earlier,
the Wien displacement law helps to measure the surface temperature of stars and
other stellar objects. The spectrometer is the device used to find the spectral
energy distributions of emitted blackbody radiation as a function of
wavelength. With the help of the Wien displacement law formula, we can easily
calculate their surface temperature.
2. In thermal equipment
design:
It plays a principal
role in designing thermal equipment for heating and medical treatment purposes.
For example- Mammals
absorb and radiate thermal energies to a large extent in the far infrared
regions at room temperature conditions of 25 degrees centigrade to 35 degrees
Celsius. So, the temperature of the equipment such as thermal sensors and room
heaters are adjusted based on their requirement following Wien’s displacement
law.
3. Color-temperature
relation:
Wien displacement law
aids in revealing the connection between the temperature and the object's color
in the visible region.
For example- A hot
object emits white-colored radiation on heating at about 7500 Kelvin, but it
seems red when heated at about 3500 Kelvin temperature. It contradicts our
general presumption that red-colored objects are the hottest and white-colored
bodies are cold.
The same principle applies even to star colors, such as white color stars are hotter than blue stars.
Limitations of Wien displacement law:
Wien displacement law fails at longer radiation wavelengths (or) shorter radiation frequencies at lower temperature conditions.
Thus, it calculates approximately the maximum
intense wavelength peaks at lower temperatures. It is due to the obtained
discontinuous Wien's displacement law graphs depicting the non-uniform
variation of radiation spectral intensities with wavelengths. One more reason
for the failure of the Wien displacement law formula is its classical physics
assumption of blackbody radiations as never-ending energy waves.
Apart from its
restrictions, its ease in calculating the peak intense wavelengths of thermal
electromagnetic radiations with their temperature data makes it convenient in
demanding situations.
As William Van Horne's
saying
Nothing is too small to know. Nothing is too big to attempt.
Every invention has
its own merits and demerits. It paved the way for winning the Nobel Prize in
1911 to Wilhelm Wein.
Frequently asked questions and answers of Wien displacement law:
1. What are the
differences between the Wien displacement law and Planck quantum law?
Wien displacement law:
It measures the
maximum intense wavelength peaks of blackbody radiation from the body's
absolute temperature values.
Wilhelm Wien, in 1893,
invented the Wien displacement formula by inserting a fixed physical quantity
called Wien's constant in the equation.
It explained the
movement of λmax peaks toward shorter wavelengths at higher temperatures.
It works well at
shorter radiation wavelengths at higher temperatures and fails at longer
radiation wavelengths at lower temperatures.
Classical physics
assumption of energy forms its basis. As a result, it considered blackbody
radiations as continuously propagating energy waves.
Planck quantum law:
It measures the spectral
intensities of blackbody emissions as a function of wavelength at all
temperature conditions.
Max Planck, in 1900,
discovered Planck's quantum formula by introducing a universal physical
quantity called Planck's constant.
It described the shift
of the whole blackbody spectrum towards shorter wavelengths at higher
temperatures.
It applies to all
radiation wavelengths seamlessly without limits at all temperature conditions.
Quantum physics
assumption of energy forms its basis. Consequently, it considers blackbody
radiations comprised of tiny energy particles called quantum.
2. What is the color of emitted
radiation for an object when heated to 2000 kelvin temperature?
We have;
λm= b/T
λm = 2.89 x 10-3 meter Kelvin/2000
Kelvin = 1445 nanometer
By Wien’s law, an object heated to 2000 kelvin temperature emits λm radiation
at 1445 nm. It lies in the near-infrared region close to the visible zone of
the electromagnetic spectrum. Hence, the object appears in dull red. But it
keeps us warm due to the transfer of more heat energy as infrared radiation to
the surrounding.
3. Why did Wien’s law
called Wien's displacement law?
Answer:
Wien's law explains
the inversely proportional relationship of peak wavelength having maximum
intensity with an absolute temperature of the body. At lower temperatures,
blackbody emissions consist of longer wavelength radiations. But, increasing
the temperature of the blackbody moves the wavelength crest of the blackbody curve
to shorter wavelengths. In this way, Wien's law interprets the displacement in
the peak wavelength of a blackbody at various temperature conditions, hence,
commonly known as Wien's displacement law.
Numerical problems on Wien displacement law:
1. What is
the temperature of an astronomical object present in a star galaxy with a
wavelength of about 1.42 x 10-5 cm?
Answer:
The wavelength of the
astronomical object = 1.42 x 10-5 cm
According to Wien displacement
law, we have;
λm X T = b
Value of Wien’s
constant in the CGS system = 0.289 cm Kelvin
By substituting all
the values in the Wien displacement formula, we have;
T = 0.289/1.42 x 10-5
T = 20352 Kelvin
2. What is
the temperature of blackbody radiation if its maximum intense wavelength is at
4500 A0? Provided that Boltzmann constant = 1.38 x 10-23
joule/Kelvin, Planck’s constant = 6.626 x 10-34 joule seconds, speed
of light in vacuum = 3 x 108 meter/second.
Answer:
The maximum
intense wavelength of blackbody radiation = 4500 A0 = 4500 x 10-10
meters
According
to Wien displacement law, we have;
λm X T =
hc/4.965 k
By substituting the
value of physical quantities as provided in the question, we have;
T = (6.626 x 10-34)
x (3 x 108)/4.965 x (1.38 x 10-23) x (4500 x 10-10)
T = 6447 Kelvin
3. If the temperature
of a heated object is 2324 degrees centigrade, then the intensity is maximum at
12000 A0. If the wavelength of the emitted radiation falls to 4800 A0,
then find the temperature of the radiation?
Answer:
The temperature of the
emitted thermal radiation = 23240 C = 2324 + 273 = 2597 Kelvin
Wavelength of
radiation with maximum intensity = 12000 A0 = 12000 x 10-10
meters
The new radiation
wavelength = 4800 A0 = 4800 x 10-10 meters
According to Wien displacement
formula, we have;
λm1 x T1 = λm2 x T2
T2 = λm1 x T1
/ λm2
T2 = (12000
x 10-10) x 2597 / (4800 x 10-10)
T2 = 6492.5
Kelvin
4. The multiplicative
product of λm and the body’s temperature is one-fourth of a
constant quantity ‘u’ having a value of 1.156 cm Kelvin. The body’s temperature
is 18000 Kelvin, then find λm.
Answer:
According to the
question, we have
λm x T = u/4
Temperature of the
body = 18000 Kelvin
U value = 1.156 cm
Kelvin
By substituting their
values in the above formula, we have;
λm = 1.156 / 4 x 18000
λm = 1.6 x 10-5 cm
5. If blackbody
radiation has the energy of 100 joules at 7225 Kelvin, calculate the number of
photons present in the radiation.
Answer:
The temperature of the
emitted blackbody radiation = 7225 Kelvin
Value of Wien’s
constant = 289 x 10-5 meter Kelvin
According to Wien
displacement law, we have;
λm x T = b
λm = 289 x 10-5 / 7225
λm = 400 x 10-9 meter
Energy of blackbody radiation = 100 joules
The formula to calculate the number of photons is
n = λE / hc
n = (400 x 10-9) x 100 / (6.626 x 10-34) x (3 x 108)
n = 2.012 x 1020
Check your understanding:
Question: Calculate the energy of thermal radiation emitted from a wood
fire at 4041 Kelvin temperature.