Boltzmann constant

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Boltzmann constant (symbol: \(k_B\) or \(k\)) is a physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It is named after the Austrian physicist Ludwig Boltzmann, who made significant contributions to the field of statistical mechanics during the 19th century. The Boltzmann constant plays a crucial role in the development of statistical mechanics, a branch of physics that deals with the behavior of systems composed of a large number of particles.

Definition

The Boltzmann constant is defined as the ratio of the gas constant (\(R\)) to the Avogadro constant (\(N_A\)): \[k = \frac{R}{N_A}\]

This relationship provides a bridge between the macroscopic and microscopic worlds, connecting macroscopic properties such as temperature and pressure to the microscopic behavior of atoms and molecules.

Value

As of the latest standards, the value of the Boltzmann constant is exactly \(1.380649 \times 10^{-23}\) joules per kelvin (J/K), following the redefinition of the SI base units in 2019, where the kelvin is now defined in terms of the Boltzmann constant.

Applications

The Boltzmann constant appears in several fundamental equations in physics and chemistry, including:

• The Boltzmann distribution, which predicts the distribution of energy among particles in a gas.
• The Stefan-Boltzmann law, which relates the total energy radiated per unit surface area of a black body to the fourth power of its temperature.
• The Maxwell-Boltzmann distribution, which describes the distribution of speeds among particles in a gas.
• The Boltzmann equation, which provides a statistical description of the dynamical state of a gas.

Significance

The Boltzmann constant is a key factor in the entropy formula \(S = k \ln \Omega\), where \(S\) is the entropy and \(\Omega\) is the number of microstates corresponding to a given macrostate. This formula, inscribed on Boltzmann's tombstone, highlights the fundamental connection between entropy and the number of ways a system can be arranged, emphasizing the statistical nature of the second law of thermodynamics.

See also

• Thermodynamics
• Kinetic theory of gases
• Planck constant
• Quantum mechanics