Planck's Radiation Law PDF

# PLANCK'S LAW OF RADIATION ## 4.6 Quantum Nature of Energy It is given by $E = nhv = n\epsilon$ where _n_ is a positive integer 1, 2, 3, ... It means that the energy of the atomic oscillator is quantized and integer _n_ is known as **quantum number**. ## 4.7 Planck's law of Radiation **Stateme...

# PLANCK'S LAW OF RADIATION ## 4.6 Quantum Nature of Energy It is given by $E = nhv = n\epsilon$ where _n_ is a positive integer 1, 2, 3, ... It means that the energy of the atomic oscillator is quantized and integer _n_ is known as **quantum number**. ## 4.7 Planck's law of Radiation **Statement** The energy density of heat radiation emitted from a black body at temperature _T_ in the wavelength range from λ to λ + dλ is given by: $\newline$ $E_{\lambda} d\lambda = \frac{8\pi hc}{λ^5 (e^{hv/kT} - 1)} dλ$ $\newline$ Here, * _h_ = Planck's constant * _c_ = Speed of the light * _v_ = Frequency of radiation * _k_ = Boltzmann's constant * _T_ = Temperature of the blackbody ## 4.8 Derivation of Planck's Law Where: * $E$ = Total energy due to all the oscillators * $N$ = Total number of oscillators * $N_0$ = Number of atomic oscillators in ground state According to Maxwell's energy distribution law, the number of oscillators with energy $E_n$ is given by: $\newline$ $N = N_0 e^{-E_n/kT}$ $\newline$ where * $T$ = Absolute temperature of the black body * $k$ = Boltzmann's constant. If _N_ is total number of oscillators and $N_0$, $N_1$, $N_2$, ... are the number of oscillators with energies $E_0$, $E_1$, $E_2$,... then $\newline$ $N = N_0 + N_1 + N_2 +...$ From the equation (2), we have $N = N_0 e^{-E_0/kT} + N_0e^{-E_1/kT} + N_0e^{-E_2/kT} +....$ (3) From Planck's quantum theory, $e$ can take only a quantum of values _hv_. Therefore, the possible values of $e$ are 0, 1_hv_, 2_hv_, 3_hv_, ... etc. (see fig. 4.2) ie., $\newline$ $E_n = nhv$, $n = 0$, 1, 2... $e_0 = 0$, $e_1 = hv$, $e_2 = 2hv$ ... Substituting these values in equation (3), we have $\newline$ $N = N_0e^0 + N_0e^{-hv/kT} + N_0e^{-2hv/kT} + ....$ $N = N_0 + N_0e^{-hv/kT} + N_0e^{-2hv/kT} + ...$ (4) $[e^o = 1]$ ## 4.9 Planck's Formula for Energy Density **Energy Diagram for Planck's Oscillator:** | Energy | Quantum Number | Number of oscillators | |---|---|---| | $e_5 = 5hv$ | n = 5 | $N_5$ | | $e_4 = 4hv$ | n = 4 | $N_4$ | | $e_3 = 3hv$ | n = 3 | $N_3$ | | $e_2 = 2hv$ | n = 2 | $N_2$ | | $e_1 = hv$ | n = 1 | $N_1$ | | $e_0= 0$ | n = 0 | $N_0$ | **Total Energy** $E = 0 \times N_0 + hvN_0e^{-hv/kT} + 2hvN_0e^{-2hv/kT} + ... $ $E = hvN_0e^{-hv/kT} + 2hvN_0e^{-2hv/kT} + ...$ ...(8) put $x = e^{-hv/kT}$, we have: $\newline$ $E = hvN_0x + 2hvN_0x^2 + ...$ $E = hvN_0 [x + 2x^2 +...]$\ $E = hvN_0x [1 + 2x + ...]$ $E = hvN_0x(1-x)^{-1}$ ...(9) **Number of Oscillators**: $N = N_0 + N_0x + N_0x^2 + N_0x^3 +....$ $N = N_0 [1 + x + x^2 + ...]$ ...(5) $\frac{1}{(1-x)^2} = 1 + 2x + 3x^2 +...$ by using binomial series. Substituting equations (6) and (10) in equation (1), we get $\newline$ $ E = \frac{hvN_0x}{(1-x)} = \frac{hvN_0x}{N_0(1-x)^2}$ $ E = \frac{hvN_0x}{(1-x)^2}$ $ E = \frac{hv}{1-x} \times \frac{N_0x}{(1-x)} $ $N= \frac{N_0}{(1-x)} $ $N = \frac{N_0}{(1-x)}[1= x + x^2 +....] = \frac{N_0}{(1-x)^2}$(6) by using binomial series $\newline$ Substituting for $E_0$, $E_1$, $E2$, ..., and $N_0$, $N_1$, $N2$, ... in equation (7), we have: $\newline$ $E = 0 \times N_0 + hv \times N_1 + 2hv \times N_2 + ...$ $E = hvN_1 + 2hvN_2 + ...$ ...(7) $E = \frac{hv x}{(1-x)}$ $ E = \frac{hv}{1-x} \times \frac{x}{(1-x)} $ ## 4.10 Planck's Radiation Law in Terms of Wavelength On substituting $x = e^{-hv/kT}$, we have $\newline$ $ E = \frac{hv}{1-e^{-hv/kT}} = \frac{hv}{e^{hv/kT}-1}$ ...(12) ## 4.11 Deduction of Wien's Displacement Law from Planck's Law **Number of oscillators per unit volume in the wavelength range λ and λ + dλ is given by:** $\newline$ $N = \frac{8\pi dλ}{λ^4}$ ...(13) **The energy density of radiation between wavelengths λ and λ + dλ is given by:** $\newline$ $E_{\lambda} dλ = \frac{average energy per oscillator}{ \newline number of oscillators per unit volume}$ $\newline$ $E_{\lambda} dλ = \frac{hv}{e^{hv/kT}-1} \times \frac{8\pi dλ}{λ^4}$ ...(14) **Therefore, when λ is very small, ν is very large, hence $\frac{hv}{kT}$ >> 1 and $e^\frac{hv}{kT}$ is large when compared to 1. ** $\newline$ Thus, '1' is neglected in the denominator of equation (16) $\newline$ i.e., $e^{hv/kT} -1 ≈ e^{hv/kT}$. **Hence the equation (16) reduces to:** $\newline$ $E_{\lambda} dλ = \frac{8\pi hc}{λ^5 (e^{hv/kT})} dλ$ ...(15) $E_{\lambda} dλ = \frac{8\pi hc}{λ^5 (e^{hc/λkT})} dλ$ ...(16) $Eλ = \ \frac {8πhc}{λ^5e^{hc/λkT}} $ $Eλ = \frac{8πhc}{λ^5}e^{-hc/λkT} $ ...(18) This equation (18) represents **Wien's displacement law.**