Glossary

16.3 Black-body radiation and Planck distribution

3 min read•july 23, 2024

radiation revolutionized physics, revealing classical theory's limitations. It showed that energy is quantized, not continuous, leading to the birth of quantum mechanics and challenging our understanding of light and matter.

Planck's distribution law accurately describes radiation from ideal absorbers and emitters. It explains the spectrum of stars, predicts total energy output, and laid the foundation for quantum concepts like and wave-particle duality.

Black-Body Radiation and Planck Distribution

Concept of black-body radiation

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  • Electromagnetic radiation emitted by an idealized physical body that absorbs all incident radiation and emits radiation at all wavelengths
    • Perfect absorber and emitter of radiation (black body)
    • Radiation emitted depends only on temperature, not composition or shape (ideal radiator)
  • Led to the development of quantum mechanics by revealing limitations of classical physics
    • Observed spectrum of black-body radiation could not be explained by classical physics (ultraviolet catastrophe)
    • Introduced the concept of quantized energy levels (photons)

Classical physics failure for radiation

  • based on classical physics predicted intensity of black-body radiation should increase infinitely with increasing frequency (shorter wavelengths)
    • Prediction known as the "ultraviolet catastrophe" inconsistent with experimental observations (infinite energy)
    • Failed to explain observed spectrum of black-body radiation, particularly at high frequencies (X-rays, gamma rays)
  • Predicted an infinite total energy emitted by a black body, which is physically impossible
    • Violated conservation of energy principle (energy cannot be created or destroyed)
    • Highlighted need for a new theory to explain black-body radiation (quantum mechanics)

Derivation of Planck distribution law

  • proposed energy of electromagnetic radiation is quantized, with each quantum having energy [E = hν](https://www.fiveableKeyTerm:e_=_hν)
    • hh is Planck's constant (6.626×10−346.626 \times 10^{-34} J⋅s)
    • νν is frequency of radiation (Hz)
  • Using quantization of energy, Planck derived distribution law for black-body radiation:
    • Bν(T)=2hν3c21ehν/kT−1B_ν(T) = \frac{2hν^3}{c^2} \frac{1}{e^{hν/kT} - 1}
      • Bν(T)B_ν(T) is spectral radiance (power per unit area per unit solid angle per unit frequency) at frequency νν and temperature TT (W⋅sr−1^{−1}⋅m−2^{−2}⋅Hz−1^{−1})
      • cc is speed of light (3×1083 \times 10^8 m/s)
      • kk is Boltzmann's constant (1.380649×10−231.380649 \times 10^{-23} J/K)

Applications of Planck distribution

  • Calculate spectral radiance Bν(T)B_ν(T) at given frequency νν and temperature TT using equation
    • Determines power emitted per unit area, solid angle, and frequency at specific wavelength (color) and temperature
  • Calculate total radiated power per unit area (intensity) of black body by integrating Planck distribution over all frequencies:
    • I(T)=∫0∞Bν(T)dν=2π5k415c2h3T4=σT4I(T) = \int_0^∞ B_ν(T) dν = \frac{2π^5k^4}{15c^2h^3}T^4 = σT^4
      • σσ is Stefan- (5.670374419×10−85.670374419 \times 10^{-8} W⋅m−2^{-2}⋅K−4^{-4})
      • Relates total energy emitted by black body to its temperature (hotter objects emit more energy)
  • Explains color of stars based on their surface temperatures (Hertzsprung-Russell diagram)
    • Cool stars appear red (Betelgeuse), hot stars appear blue (Rigel)

Implications for quantum mechanics

  • Success of Planck distribution in describing black-body radiation indicated energy is quantized
    • Challenged foundations of classical physics based on continuous energy (wave theory of light)
    • Led to development of quantum mechanics describing behavior of matter and energy at atomic and subatomic scales (particles)
  • Introduced concept of energy quanta (photons) laying groundwork for wave-particle duality
    • Einstein extended concept to explain photoelectric effect, further supporting quantum nature of light and matter (photons ejecting electrons)
    • Paved way for other quantum phenomena like Compton scattering and Bohr model of the atom (discrete energy levels)

Key Terms to Review (18)

Absorbance: Absorbance is a measure of the amount of light absorbed by a substance when light passes through it. It quantifies how much incident light is absorbed by a material and is defined mathematically as the logarithm of the ratio of incident light intensity to transmitted light intensity. This concept is critical in understanding black-body radiation and the Planck distribution, as it helps to characterize how different materials interact with electromagnetic radiation at various wavelengths.
Black-body: A black-body is an idealized physical object that perfectly absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It also emits radiation in a characteristic spectrum that depends solely on its temperature, described by Planck's law. This concept is crucial in understanding thermal radiation and the distribution of energy across different wavelengths.
Boltzmann Constant: The Boltzmann constant is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of that gas. It plays a crucial role in statistical mechanics, linking macroscopic properties like temperature to microscopic behaviors of particles. This constant is vital in various fields such as kinetic theory, quantum mechanics, and thermodynamics, emphasizing the connection between energy, temperature, and probability distributions.
E = hν: The equation e = hν represents the relationship between the energy of a photon (e) and its frequency (ν), where h is Planck's constant. This fundamental equation connects quantum mechanics and electromagnetic radiation, highlighting how energy is quantized and proportional to the frequency of radiation emitted or absorbed by matter. This relationship is key to understanding black-body radiation and the Planck distribution, which describe how objects emit thermal radiation based on their temperature.
Emissivity: Emissivity is a measure of an object's ability to emit thermal radiation compared to that of a perfect black body. It is expressed as a ratio, ranging from 0 to 1, where a value of 1 indicates perfect emission, while lower values signify less efficient emission. Emissivity is critical in understanding how heat transfer occurs through radiation and plays a significant role in the study of black-body radiation and the Planck distribution.
I(ν) = (2hν^3/c^2)(1/(e^(hν/kt)-1)): This equation represents the spectral radiance of a black body as described by Planck's law, which quantifies the amount of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. The term connects temperature, frequency, and energy in the context of black-body radiation, showcasing how the energy distribution among different frequencies leads to the observed spectrum.
Infrared spectroscopy: Infrared spectroscopy is an analytical technique used to identify and study the molecular composition and structure of substances by measuring their absorption of infrared radiation. This technique relies on the interaction between infrared light and molecular vibrations, providing insight into functional groups and chemical bonds present in the sample, making it essential in various scientific fields.
Ludwig Boltzmann: Ludwig Boltzmann was an Austrian physicist who made significant contributions to the field of statistical mechanics and thermodynamics, particularly known for his formulation of the statistical interpretation of entropy. His work established a crucial link between microscopic particle behavior and macroscopic physical properties, providing a deeper understanding of the second law of thermodynamics and the nature of entropy.
Max Planck: Max Planck was a German physicist who is best known for his role in developing quantum theory, which fundamentally changed our understanding of atomic and subatomic processes. He introduced the concept of quantization of energy, proposing that energy is emitted or absorbed in discrete units called quanta, which laid the groundwork for modern quantum mechanics and thermodynamics.
Photons: Photons are elementary particles that represent the quantum of light and all other forms of electromagnetic radiation. They are massless and travel at the speed of light in a vacuum, carrying energy proportional to their frequency. This relationship is key to understanding phenomena like black-body radiation and the Planck distribution, which describe how objects emit and absorb electromagnetic radiation.
Planck constant: The Planck constant is a fundamental physical constant denoted by 'h', with a value of approximately 6.62607015 × 10^-34 Js. It plays a critical role in quantum mechanics, defining the relationship between the energy of a photon and its frequency, and is essential for understanding phenomena like black-body radiation and the quantization of energy levels.
Planck distribution: The Planck distribution describes the spectral energy density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. This function is critical in understanding black-body radiation, as it provides a mathematical model that explains how energy is distributed across different wavelengths, revealing the quantized nature of light and energy.
Quantum theory: Quantum theory is a fundamental theory in physics that describes the behavior of matter and energy on very small scales, such as atoms and subatomic particles. It introduces the concept that energy is quantized, meaning it exists in discrete units called quanta, which has profound implications for understanding phenomena like black-body radiation and the distribution of energy across different frequencies.
Rayleigh-Jeans Law: The Rayleigh-Jeans Law describes the spectral radiance of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature, predicting that the radiance increases with frequency and diverges at high frequencies. This law successfully explains the behavior of black-body radiation at long wavelengths but fails at short wavelengths, leading to the ultraviolet catastrophe, which paved the way for the development of quantum mechanics and Planck's distribution.
Stefan-Boltzmann Law: The Stefan-Boltzmann Law states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its absolute temperature. This principle highlights how much more energy an object emits as its temperature increases and is fundamental in understanding black-body radiation and thermal radiation in various fields, including astrophysics and cosmology.
Temperature measurement: Temperature measurement refers to the process of determining the thermal state or intensity of heat within a system. It is essential for understanding the behavior of matter, particularly in relation to energy transfer, phase changes, and thermodynamic processes. This concept connects deeply with black-body radiation and the Planck distribution, as they describe how objects emit and absorb radiation based on their temperature, which can be quantified through various measuring instruments.
Thermal radiation: Thermal radiation is the emission of electromagnetic waves from the surface of an object due to its temperature. All objects with a temperature above absolute zero emit thermal radiation, which carries energy away from the body and is key in understanding energy transfer processes in thermodynamics. This concept is closely linked to the behavior of particles in a canonical ensemble and the characteristics of black-body radiation, as described by Planck's distribution.
Wien's Displacement Law: Wien's Displacement Law states that the wavelength at which the emission of a black body spectrum is maximized is inversely proportional to its temperature. This means that as the temperature of a black body increases, the peak wavelength of its emitted radiation shifts to shorter wavelengths, indicating a shift from infrared towards visible light. This law is essential in understanding how objects emit radiation based on their thermal energy.
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