Thermodynamic Concepts of Radiation and Energy Levels in Molecules

Thermodynamic concepts of radiation (a) Black-body radiation - classical theory predicted (wrongly) increasing emission with increasing frequency (v) from an amorphous black surface. Quantum theory introduced by Planck agreed with observations. The blackbody radiation is defined by the Planck function. B(T)   = (2hv³/c²) * 1 / (e^(hv/kT) - 1) B(T) = (2hc/λ⁵) * 1 / (e^(hc/λkT) - 1) which agrees with observation and provided the first evidence for the quantum theory (h = Planck constant, k = Boltzmann constant). Asymptotes:   λ → 0, B → 0; λ → ∞, B → 2kT/c²; Wien wing (UV) λ → ∞, B → 2kT/c²; Rayleigh-Jeans wing (radio) Stefan's Law: J = ∫₀^∞ B(T) dλ = (15c²h/π⁴) * (kT/h)⁴ Stefan's constant (σ) = 5.67 x 10^-8 W/m²K⁴ 12.815: Atmospheric Radiation Prof. Ronald Prinn Wien law: dB(T)/dv = 0, dB(T)/dλ = 0 (defines the maximum in B) cT/v = 0.51 (cm·K) ΔT = 0.29 (cm·K) (peak in flux per unit frequency) (peak in flux per unit wavelength) (b) ideal black body: one which is in equilibrium with the Planck distribution. It absorbs all the radiation incident upon it and emits the Planck distribution for its temperature. Practical Examples: (i) a hole in a blackened sphere (radiation entering the hole is totally absorbed and the hole radiates the Planck distribution). (ii) an infinite crystal   (solid and liquid particles, where dimensions >> wavelength of radiation of interest) - these bodies are very very closely spaced and possess a Boltzmann distribution of energy levels. They absorb very strongly, so their oscillator energies are populated according to Boltzmann's distribution. (iii) a multiple scattering cloud (see later) (c) application to real gases - providing the discrete energy levels in the molecule are populated according to Boltzmann's distribution, we can show these levels will be in m with black body radiation. Consider an enclosed gas at constant temperature (i.e. in local thermodynamic equilibrium (LTE))   (excited) 0 (ground state) Rate of induced emission = N₁b₁₀Iᵥ₀ Rate of induced absorption = N₀b₀₁Iᵥ₀ where b₁₀ = Einstein "b" coefficient (see later) and Nᵢ/N₀ = gᵢe^(-λᵢEᵢ/kT) (gᵢ = 1 for vibrations = 2j+1 for rotations) For vibrations Nᵢ << N₀ (room temp.) so induced absorption far exceeds induced emission. Einstein argued the fact that molecules do not fall apart at low pressures that the statistical Boltzmann distribution for energy should be maintained even at very low pressures where collisions do not occur. He therefore proposed the existence   of spontaneous emission with a rate   ae(sec⁻¹ster⁻¹). Equivalently define the radiative relaxation time τ₀ = 1/4ла. 12.815: Atmospheric Radiation Prof. Ronald Prinn Therefore in LTE (in absence of collisions): Dominant 4πN₁(b₁₀Iᵥ₀ + a₁₀hν₀) - 4πN₀(b₀₁Iᵥ₀) (total rate of emission) (total rate of absorption) i.e. N₁/N₀ = b₁₀Iᵥ₀ / (b₁₀Iᵥ₀ + a₁₀hν₀) = e^(-hν₀/kT) i.e. a₁₀/b₁₀ = hv₀ / (e^(hν₀/kT) - 1) = Bᵥ₀ From time-dependent quantum theory a₁₀ = 2πν₀²/c²b₁₀, so that Iᵥ₀ = Bᵥ₀. Hence total rate of emission given by 4πN₀b₁₀Bᵥ₀ (Kirchhoff's Law) in LTE (as well as non-LTE). Also due to "broadening" of absorption and emission lines (discussed later) we allow for absorption or emission over a range   of frequencies by generalizing v₀ to v. 4πN₀ ∫₀^∞ κᵥBᵥ dv (kᵥ = absorption cross-section) (m²) = 4πfᵥBᵥ dv (fᵥ = absorption coefficient) (m⁻¹) (d) Application to the real atmosphere: the atmosphere is a non-enclosed space and generally has external radiation (e.g. solar, thermal from ground or distant atmosphere). Clearly, Iₑ   ≠ Bₑ (as defined by Tlocal) If 4π ∫₀^∞ fᵥIᵥ dv > 4π ∫₀^∞ fᵥBᵥ dv (absorption) (emission) get radiative heating If 4π ∫₀^∞ fᵥIᵥ dv < 4π ∫₀^∞ fᵥBᵥ dv (absorption) (emission) get radiative cooling 12.815: Atmospheric Radiation Prof. Ronald Prinn