chemistry

# Derivation of Planks Black body spectrum equation

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

Planks Spectral energy density, µ(ʋ,T)

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

#### ♦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

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

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

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

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