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Stefan-Boltzmann Law

Last Updated : 04 Feb, 2024
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Stefan-Boltzmann Law relates the power radiated by the black body to its temperature and surface area. In the study of thermodynamics and astrophysics, the Stefan-Boltzmann Law is widely used to better our understanding of the subject. Other than this, Stefan-Boltzmann Law helps scientists understand the behavior of objects that emit radiation, such as stars and planets. Stefan-Boltzmann Law also has some applications in the real world as well such as, in designing solar panels and other energy conversion instruments.

What is Stefan-Boltzmann Law?

The Stefan Boltzmann law states that the total energy emitted per unit surface area of a black body across all wavelengths per unit of time is directly proportional to the fourth power of the black body’s thermodynamic temperature and emissivity. It expresses the power emitted by a black substance as a function of temperature and emissivity. 

Stefan-Boltzmann Law

 

Formula for Stefan-Boltzmann Law

Mathematically, Stefan Boltzmann’s law for the black body is given by 

P/A ∝T4

OR

P = σAT4

Where, 

  • P is the power radiated, 
  • A is the surface area of the black body, 
  • T is the temperature of the body and
  • σ is the Stefan-Boltzmann constant.

Objects which are not black bodies emit less radiation as they can absorb radiation as well. For such objects, Stefan Boltzmann’s law is as follows:

P/A ∝ eT4

OR

P = σeAT4

Where, e is the emissivity of the object and can have value from 0 to 1.

Stefan-Boltzmann Constant

Stefan-Boltzmann Constant is the physical constant that is named after Josef Stefan and Ludwig Boltzmann, an Austrian and Dutch physicist respectively. The Stefan-Boltzmann Constant is denoted by the Greek alphabet σ. Stefan-Boltzmann Constant is the proportionality constant in Stefan-Boltzmann law.

Value of Stefan-Boltzmann Constant

The value of Stefan-Boltzmann contact in SI units is approximately 5.67 ×10-8 Watt per square meter per Kelvin to the fourth power and other units and the values in other systems of the unit are given as follows:

Unit system

Value of σ

Units

SI

5.670367×10-8

W/(m2K4)

CGS

5.6704 x 105

erg/(cm2 s1 K4)

Thermochemistry

11.7 x 108

cal/(cm2 day K4)

U.S. Customary Units

1.714 x 109

BTU/(ft2 hr ˚R4)

The dimension of the Stefan-Boltzmann Constant is [M]1[L]0[T]-3[K]-4, where [M], [L], [T], and [K] are the dimension of mass, length, time, and temperature respectively.

Formula for Stefan-Boltzmann Constant

Other than the Stefan-Boltzmann law, there are some more formulas involving the Stefan-Boltzmann Constant. One such formula can be derived from Planck’s law of radiation i.e., integrating Planck’s Radiation formula, and is given as follows:

\bold{\sigma=\dfrac{2 \pi^5 k_b^4}{15 h^3 c^2}=\dfrac{2 \pi^5 R^4}{15 h^3 c^2 N_A^4}}\quad \left[\text{As, }k_b = \frac{R}{N_A} \right]

Where,

  • R is Universal Gas Constant which is equal to 8.3144598 J per mole per K (J x mol-1 x K-1)
  • NA is Avogadro constant which is equal to 6.02214076 x 1023 mol-1
  • h is the Planck’s constant which is equal to 6.62607015 × 10-34 m2 kg / s,
  • c is the speed of light which is equal to 299,792,458 m / s
  • kb is the Boltzmann Constant which is equal to 1.380649 × 10-23 m2 kg s-2 K-1

Black Body Radiation

Black Body Radiation is the electromagnetic radiation by the Black Body which is an opaque ideal body that absorbs all the radiation and is also a perfect emitter of radiation i.e., an object that can emit radiation of all the wavelengths depending on the temperature of the surface of the body.

Planck’s Law for Black Body Radiation

Planck’s Law or Planck’s Radiation Law states the relationship between temperature and radiation emitted by the black body and is given as follows:

\bold{B(\nu, T) = \left(\dfrac{2h\nu^3}{c^2}\right) \left(\frac{1}{e^{\frac{h\nu}{k_bT}} - 1}\right)}

Where,

  • B(ν, T) is the energy radiated per unit area per unit time form the body,
  • ν is Frequency,
  • kb is Boltzmann Constant,
  • h is Planck’s Constant, and
  • c is speed of light in vacuum.

Derivation of Stefan-Boltzmann Constant

The formula for the Stefan-Boltzmann constant can be derived from the integral of Plank’s Radiation Law, as integrating Planck’s Radiation Formula gives a similar relation as the Stefan-Boltzmann law. From where we can compare the Stefan-Boltzmann constant in both formulas and formulate the formula for Stefan-Boltzmann Constant. The complete derivation for the same is given as follows:

\frac{d\left(\frac{P}{A}\right)}{d \lambda}=\frac{2 \pi h c^2}{\lambda^5\left(e^{\left.\frac{h \nu}{ k_b T}-1\right.}\right)}

\frac{d\left(\frac{P}{A}\right)}{d \lambda}=\frac{2 \pi h c^2}{\lambda^5\left(e^{\left.\frac{h c}{\lambda k_b T}-1\right.}\right)}

On integrating both sides with suitable limits,

\frac{P}{A}=\int_0^{\infty}\left[\frac{2 \pi h c^2}{\lambda^5\left(e^{\frac{h c}{\lambda k_b T}}-1\right)}\right] d \lambda

\frac{P}{A}=\frac{1}{\lambda^5}\int_0^{\infty}\left[\frac{d \lambda}{\left(e^{\frac{h c}{\lambda k_b T}}-1\right)}\right]  . . .(1)

Now,  let x=\frac{h c}{\lambda k_b T}  . . .(2)

Therefore,  d x=-\frac{h c}{\lambda^2 k_b T} d \lambda

\Rightarrow d \lambda=-\frac{\lambda^2 k T}{h c} d x

Putting all the results in equation (1)

\begin{aligned} & \Rightarrow \frac{P}{A}=\frac{2 \pi}{\lambda^5}\left(\frac{x \lambda k_b T}{c}\right)\left(\frac{x \lambda k_b T}{h}\right)^2 \int_0^{\infty}\left[\frac{\left(-\frac{\lambda^2 k_b T}{h c}\right) d x}{e^x-1}\right] \\ &\left[\text{From equation (2), } h=\frac{x \lambda k T}{c} \text{ and } c=\frac{x \lambda k T}{h} \right] \\ & \Rightarrow \frac{P}{A}=2 \pi\left(\frac{x^3 \lambda^5 k_b^4 T^4}{h^3 c^2 \lambda^5}\right) \int_0^{\infty}\left[\frac{d x}{e^x-1}\right] \\ & \Rightarrow \frac{P}{A}=\frac{2 \pi(k_b)^4}{h^3 c^2} \int_0^{\infty}\left[\frac{x^3}{e^x-1}\right] d x \end{aligned}

As, we know the improper integral,\int_0^{\infty}\left[\frac{x^3}{e^x-1}\right] d x=\frac{\pi^4}{15}

Therefore, \frac{P}{A}=\frac{2 \pi(k_b T)^4}{h^3 c^2} \frac{\pi^4}{15}

\Rightarrow \frac{P}{A}=\left(\frac{2 k^4 \pi^5}{15 h^3 c^2}\right) T^4

According to Stefan-Boltzmann law, energy radiated per unit area is directly proportional to the fourth power of temperature and the Stefan-Boltzmann constant, is the proportionality constant here.

Thus, \sigma = \frac{2 k_b^4 \pi^5}{15 h^3 c^2} = \frac{2 \pi^5 R^4}{15 h^3 c^2N_A^4} \quad \left[\text{As, }k_b = \frac{R}{N_A} \right]

Calculation of Stefan-Boltzmann Constant

Using the formula we derived we can calculate the value of the Stefan-Boltzmann constant i.e., 

\bold{\sigma = \frac{2 \pi^5 R^4}{15 h^3 c^2N_A^4}}

Substituting, all the values of other constants i.e., 

  • R = 8.3144598 J per mole per K (J x mol-1 x K-1)
  • NA = 6.02214076 x 1023 mol-1
  • h = 6.62607015 × 10-34 m2 kg / s,
  • c = 299,792,458 m / s

\begin{aligned} & \Rightarrow \sigma=\frac{2 \times(3.14)^5 \times(8.3144598)^4}{15 \times\left(6.62607015 \times 10^{-34}\right)^3 \times(2.99 .792 .458)^2 \times\left(6.02214076 \times 10^{23}\right)^4} \\ & \Rightarrow \sigma=\frac{2 \times 6305.25 \times(8.32)^4}{15 \times\left(6.63 \times 10^{-34}\right)^3 \times(299.792 \times 458)^2 \times\left(6.22 \times 10^2\right)^4} \\ & \Rightarrow \sigma=\frac{2 \times 305.25 \times 4791.74}{15 \times 291.43 \times 10^{-102} \times 8.99 \times 10^{16} \times 1315 \cdot 11 \times 10^{92}} \\ & \Rightarrow \sigma=\frac{2925357.27}{51682949.11 \times 10^6}=0.056601 \times 10^{-6} \\ \end{aligned}

Applications of Stefan-Boltzmann Law

In the real world, there are many applications of Stefan-Boltzmann Law. Some of those applications are as follows:

  • Stefan-Boltzmann Law helps scientists in the calculation of the luminosity of celestial bodies such as stars, planets, and galaxies
  • As we know, the amount of heat absorbed by the atmosphere due to the presence of greenhouse gases, causes the temperature of our earth to rise. By using Stefan-Boltzmann Law, to calculate the amount of energy absorbed by the atmosphere scientists can predict the effect of this rise in temperature.
  • By comparing the surface temperatures for various materials, engineers can make more power-efficient designs that do not need active cooling to work.

Also, Read

Sample Problems on Stefan-Boltzmann Law

Problem 1. Calculate the radiation power for a black body of 100 sq. m at a temperature of 200 K.

Solution:

We have, A = 100 and T = 200

Using the formula for Stefan-Boltzmann Law,

P = σ A T4

⇒ P = 5.670373 × 10−8 × 100 × (200)4

⇒ P = 9073 W

Problem 2. Calculate the radiation power per unit area for a black body at a temperature of 100 K.

Solution:

Given: T = 100

Using the formula for Stefan-Boltzmann Law,

P/A = σ T4

⇒ P/A = 5.670373 × 10−8 × (100)4

⇒ P/A = 5.67 W m-2

Problem 3. Calculate the area of a black body with a radiation power of 300 W for a black body at a temperature of 250 K.

Solution:

Given: P = 300, and T = 250

Using the formula we have,

P/A = σ T4

⇒ A = 300/(5.670373 × 10−8 × (250)4)

⇒ A = 1.3544 m2

Problem 4. Calculate the area of a black body with a surface temperature of 350 K and radiation power of 400 W.

Solution:

Given, P = 400 and T = 350

Using the formula we have,

P/A = σ T4

⇒ A = 400/(5.670373 × 10−8 × (350)4)

⇒ A = 0.4701 m2

Problem 5. Calculate the temperature required to generate radiation power of 1000 W for a black body of area 250 sq. m.

Solution:

Given: P = 1000 and A = 250

Using the formula for Stefan-Boltzmann Law,

P/A = σ T4

⇒ T4 = P/σA

⇒ T4 = 1000/(5.670373 × 10−8 × 250)

⇒ T4 = 70555338.077

⇒ T = 91.65 K

FAQs for Stefan-Boltzmann Law

Q1: What is the Stefan-Boltzmann Law?

Answer:

The Stefan-Boltzmann Law states that the total amount of energy emitted per unit time and per unit area of a blackbody is proportional to the fourth power of the temperature the black body.

Q2: What is the formula for the Stefan-Boltzmann Law?

Answer:

The mathematical formula for Stefan-Boltzmann Law is given by

P = σAT4

Where, 

  • P is the power radiated, 
  • A is the surface area of the black body, 
  • T is the temperature of the body and
  • σ is the Stefan-Boltzmann constant.

Q3: Does the Stefan-Boltzmann Law apply to Non-Blackbody Objects?

Answer:

The Stefan-Boltzmann Law applies to all real objects but as real objects do not absorb the complete radation (emissivity <1), so radiation emitted by non-blackbody objects may not be proportional to the fourth power of their temperature.

Q4: What is the value of the Stefan-Boltzmann constant?

Answer:

The value of the Stefan-Boltzmann constant in SI unit system is approximately 5.67 x 10-8 watts per square meter per Kelvin to the fourth power (W/(m2K4)).

Q5: How is the Stefan-Boltzmann constant derived?

Answer:

The one way, Stefan-Boltzmann constant is derived from the improper integration of Planck’s Radiation Formula.

Q6: What is a Black Body?

Answer:

A black body is an ideal object which can absorb all the radiation that falls on it, emits that absorb radiation at all wavelengths. 

Q7: Can the Stefan-Boltzmann Law be used to calculate the temperature of a blackbody?

Answer:

Yes, the Stefan-Boltzmann Law can be used to calculate the temperature of a blackbody if the total power emitted and its surface area are known. We can use the following formula for the same:

T = (P/σA)1/4



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