chemistry

Derivation of Planks Black body spectrum equation

Planks Black body spectrum
Planks Black body spectrum equation is given by,

Planks Spectral energy density, µ(ʋ,T)

\dpi{120} \fn_jvn \large \dpi{120} \fn_jvn U\left ( \lambda,T \ \right )= \frac{8\pi h\nu ^{2}}{c^{3}}. \frac{1}{e^{\frac{h\nu }{KT}-1}}

The factor µ(ʋ,T) is called  the Black body spectrum

♦DERIVATION

Consider a box of side ‘l’

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The resonating modes of the electronegative radiation inside the box have the frequency, [docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p2-2.odt” comments=”true”]

The energy of the proton is given by planks hypothesis as[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p3.odt” comments=”true”]

For a three dimensional box the energy expression will be[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p4-1.odt” comments=”true”]

                                                  

The total energy, U, in a 1-D box is[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p5-2.odt” comments=”true”]

Therefor in a 3-D box the energy U will be[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/p6-1.odt” comments=”true”]

Assuming the ni s to be continuous variables the summations can be replaced by integrals (fermi approximation)

Thus the total energy U is given by[docxpresso file=”https://learnwithstudy.com/wp-content/uploads/2018/04/plnk-last.odt” comments=”true”]

U(ʋ,T) is called the Black body spectrum and the equation is called Planks Black body spectrum equation

  

In terns of λ

In terms of wavelength λ the expression is

   \dpi{120} \fn_jvn \dpi{120} \large \frac{U}{l^{3}}= \int_{0}^{\infty } U\left ( \lambda ,T \right )d\lambda

\dpi{120} \large (d\nu = \frac{c}{\lambda ^{2}}d\lambda)

where  \dpi{120} \fn_jvn \large \dpi{120} \fn_jvn U\left ( \lambda,T \ \right )= \frac{8\pi h\nu ^{3}}{\lambda ^{3}}. \frac{1}{e^{\frac{hc}{\lambda KT}-1}}

This spectral density function has units of energy per unit wavelength per unit volume.

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Total energy

 Evaluation of the integral to obtains the total energy inside the box yields

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