# Planck’s constant: formulas, values and exercises

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

The name of Planck’s constant comes from the physicist Max Planck, who obtained it by proposing the equation of the radiant energy density of a thermodynamic equilibrium cavity as a function of the radiation frequency.

**Story**

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

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

To solve the problem, Max Planck stated that blackbody atoms behave like harmonic oscillators that absorb and emit energy in an amount proportional to their frequency.

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

The discovery of the Planck constant served as a major contribution to laying the foundations for Quantum Mechanics.

**What is the Planck constant for?**

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

Planck’s constant explains 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 characteristic of the yellow color.

Likewise, Planck’s constant explains why a human whose body temperature is around 37°C emits radiation with infrared wavelengths. This radiation can be detected using an infrared thermal camera.

Another application is the redefinition of fundamental physical units, such as kilogram, ampere, kelvin, and mol, from watt balance experiments. 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 proportionality relationship between the energy of electromagnetic radiation and its frequency. Planck’s formulation assumes that each atom behaves like 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 a multiple amount of energy from hv, the possible energy values being 0, hv, 2hv, 3hv, 4hv…nhv.

Energy quantization allowed Planck to mathematically establish the ratio of radiant energy density of a blackbody as a function of frequency and temperature through the equation.

E (v) = (8Пhv3 / c3). [1 / (ehv / kT-1)]

E (v) = energy density

c = speed of light

k = Boltzman constant

T = temperature

The energy density equation corresponds to experimental results for different temperatures at which a maximum radiant energy appears. As the temperature increases, the frequency at the maximum energy point also increases.

**Planck constant value**

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

The most 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

**Exercises solved in Planck’s constant**

**1- Calculate the energy of a blue light photon**

Blue light is part of the visible light that the human eye is able to perceive. Its length varies between 400 nm and 475 nm, corresponding to higher and lower 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 contains a yellow light beam with a wavelength of 589nm and 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 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 taken 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 its own frequency, can be made to have an arbitrarily chosen energy, adjusting the number of oscillations accordingly.