PLANCK'S CONSTANT: FORMULAS, VALUES ​​AND EXERCISES - PHYSICAL - 2024

The Planck constant is a fundamental constant of quantum physics that relates the radiation energy absorbed or emitted by atoms with frequency. Planck's constant is expressed with the letter ho with the reduced expression ћ = h / 2П

The name of Planck's constant is due to the physicist Max Planck, who obtained it by proposing the equation of radiant energy density of a cavity in thermodynamic equilibrium as a function of the radiation frequency.

History

In 1900 Max Planck intuitively proposed an expression to explain blackbody radiation. A black body is an idealistic conception that is defined as a cavity that absorbs the same amount of energy that the atoms in the walls emit.

The black body is in thermodynamic equilibrium with the walls and its radiant energy density remains constant. The experiments on black body radiation showed inconsistencies with the theoretical model based on the laws of classical physics.

To solve the problem, Max Planck stated that the atoms of the black body behave as harmonic oscillators that absorb and emit energy in a quantity proportional to their frequency.

Max Planck assumed that atoms vibrate with energy values ​​that are multiples of a minimum energy hv. He obtained a mathematical expression for the energy density of a radiant body as a function of frequency and temperature. In this expression the Planck constant h appears, whose value adjusted very well to the experimental results.

The discovery of Planck's constant served as a great contribution to lay the foundations of Quantum Mechanics.

Radiation energy intensity of a black body. from Wikimedia Commons

What is Planck's constant for?

The importance of Planck's constant is that it defines the divisibility of the quantum world in many ways. This constant appears in all equations that describe quantum phenomena such as Heisenberg's uncertainty principle, de Broglie wavelength, electron energy levels, and Schrodinger's equation.

Planck's constant allows us to explain why objects in the universe emit color with their own internal energy. For example, the yellow color of the sun is due to the fact that its surface with temperatures around 5600 ° C emits more photons with wavelengths typical of yellow.

Likewise, Planck's constant allows us to explain why human beings whose body temperature is around 37 ° C emits radiation with infrared wavelengths. This radiation can be detected by means of an infrared thermal camera.

Another application is the redefinition of fundamental physical units such as the kilogram, ampere, kelvin and mole, from experiments with the watt balance. The watt balance is an instrument that compares electrical and mechanical energy using quantum effects to relate Planck's constant to mass (1).

Formulas

Planck's constant establishes the proportional relationship between the energy of electromagnetic radiation and its frequency. Planck's formulation assumes that each atom behaves as a harmonic oscillator whose radiant energy is

E = hv

E = energy absorbed or emitted in each electromagnetic interaction process

h = Planck's constant

v = radiation frequency

The constant h is the same for all oscillations and the energy is quantized. This means that the oscillator increases or decreases an amount of energy that is a multiple of hv, the possible energy values ​​being 0, hv, 2hv, 3hv, 4hv… nhv.

The quantization of energy allowed Planck to establish mathematically the relationship of the radiant energy density of a black body as a function of frequency and temperature through the equation.

E (v) = (8Пhv3 / c3).

E (v) = energy density

c = speed of light

k = Boltzman constant

T = temperature

The energy density equation agrees with the experimental results for different temperatures at which a maximum of radiant energy appears. As the temperature increases the frequency at the maximum energy point also increases.

Planck's constant value

In 1900 Max Planck adjusted the experimental data to his energy radiation law and obtained the following value for the constant h = 6.6262 × 10 -34 Js

The best adjusted value of Planck's constant obtained in 2014 by CODATA (2) is h = 6.626070040 (81) × 10 -34 Js

In 1998 Williams et al. (3) obtained the following value for Planck's constant

h = 6.626 068 91 (58) × 10 -34 Js

The most recent measurements that have been made of Planck's constant have been in experiments with the watt balance which measures the current necessary to support a mass.

Watt balance. Wikimedia Commons

Solved exercises on Planck's constant

1- Calculate the energy of a photon of blue light

Blue light is part of the visible light that the human eye is capable of perceiving. Its length oscillates between 400 nm and 475 nm corresponding to greater and lesser energy intensity. The one with the longest wavelength is chosen to perform the exercise

λ = 475nm = 4.75 × 10 -7m

The frequency v = c / λ

v = (3 × 10 8m / s) / (4.75 × 10 -7m) = 6.31 × 10 14s-1

E = hv

E = (6,626 × 10 -34 Js). 6.31 × 10 14s-1

E = 4,181 × 10 -19J

2-How many photons does a beam of yellow light contain that has a wavelength of 589nm and an energy of 180KJ

E = hv = hc / λ

h = 6.626 × 10 -34 Js

c = 3 × 10 8m / s

λ = 589nm = 5.89 × 10 -7m

E = (6.626 × 10 -34 Js). (3 × 10 8m / s) / (5.89 × 10 -7m)

E photon = 3.375 × 10 -19 J

The energy obtained is for a photon of light. It is known that the energy is quantized and that its possible values ​​will depend on the number of photons emitted by the light beam.

The number of photons is obtained from

n = (180 KJ). (1 / 3,375 × 10 -19 J). (1000J / 1KJ) =

n = 4.8 × 10 -23 photons

This result implies that a beam of light, with a natural frequency, can be made to have an arbitrarily chosen energy by adjusting the number of oscillations appropriately.

References

1. Watt balance experiments for the determination of the Planck constant and the redefinition of the kilogram. Stock, M. 1, 2013, Metrologia, Vol. 50, pp. R1-R16.
2. CODATA recommended values ​​of the fundamental physical constants: 2014. Mohr, PJ, Newell, DB and Tay, B N. 3, 2014, Rev. Mod. Phys, Vol. 88, pp. 1-73.
3. Accurate Measurement of the Planck Constant. Williams, ER, Steiner, David B., RL and David, B. 12, 1998, Physical Review Letter, Vol. 81, pp. 2404-2407.
4. Alonso, M and Finn, E. Physics. Mexico: Addison Wesley Longman, 1999. Vol. III.
5. History and progress on accurate measurements of the Planck constant. Steiner, R. 1, 2013, Reports on Progress in Physics, Vol. 76, pp. 1-46.
6. Condon, EU and Odabasi, E H. Atomic Structure. New York: Cambridge University Press, 1980.
7. Wichmann, E H. Quantum Physics. California, USA: Mc Graw Hill, 1971, Vol. IV.