What is the origin of Boltzmann's constant?

[Galileo]

I know that measurements have established the following
empirical laws for the ideal gas, contained in some closed volume:

-Keeping pressure and temperature constant, the volume is proportional to the number of moles.
-The volume varies inversely with pressure.
-The pressure is proportional to the absolute temperature.

These three relations can be put in an equation, called the ideal-gas equation:
$$pV=nRT$$

where the constant of proportionality is the gas constant.
As far as I have understood, R is an experimentally measured quantity:

$$R=8.314510(70) J/mol\cdot K$$
when we want to work with the number of particles instead of moles (which we often do) we define:
$$k=\frac{R}{N_A}$$
where $N_A$ is Avogadro's number and k is called the Boltzmann constant.
The gas equation then becomes:
$$pV=NkT$$
with N the number of particles.

These laws can be 'derived' or 'proven' from statistical mechanics.
When applying statistical considerations to the ideal gas and derive the Maxwell-Boltzmann distribution we end up with two constants.
One has to be found by normalization to give the right number of particles and the other one is found by comparing with the above gas equation and they find the Boltzmann constant (times temperature).

So am I correct that the Boltmann constant is (essentially) an experimental value? It occurs to me this constant should be derivable by statistical methods as well.

 

[pervect]

So am I correct that the Boltmann constant is (essentially) an experimental value? It occurs to me this constant should be derivable by statistical methods as well.

You might want to take a look at the Wikipedia article on Boltzmann's constant


http://en.wikipedia.org/wiki/Boltzmann%27s_constant

In principle, the Boltzmann constant is a derived physical constant, as its value is determined by other physical constants. However, calculating the Boltzmann constant from first principles is far too complex to be done with current knowledge.

So why is it so hard to compute the value of Boltzmann's constant? Boltmznn's constant relates the thermodynamic temperature scale to energy. Let's look at the SI definition of the thermodynamic temperature scale

kelvin The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

We know by the equipartition theorem that every degree of freedom of water at the triple point has an energy of
$$\frac {1}{2} K 273.16 degrees$$

But computing this value of energy accurately from first principles is not currently within our capabilities.

 

[krab]

Boltzmann's constant relates energy and temperature, and so is dependent upon our chosen temperature scale. In a sense, it is an historical artefact. The common scale is one in which the boiling and freezing points of water at atmospheric pressure are separated by 100 units. If, on the other hand, we choose a temperature scale that is basically the same as our energy units, we would not have a Boltzmann's constant, i.e. k=1.

 

[Rogerio]

Boltzmann's constant relates energy and temperature, and so is dependent upon our chosen temperature scale. In a sense, it is an historical artefact. The common scale is one in which the boiling and freezing points of water at atmospheric pressure are separated by 100 units. If, on the other hand, we choose a temperature scale that is basically the same as our energy units, we would not have a Boltzmann's constant, i.e. k=1.

By this point of view, it seems any particular dimensional constant is an artifact, since it could be unitary.

Boltzmann's constant (symbol: k) is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the temperature of that gas. It plays a crucial role in statistical mechanics, thermodynamics, and the understanding of the behavior of matter at the microscopic level. The constant is named after the Austrian physicist Ludwig Boltzmann, who made significant contributions to the field of statistical mechanics.

The origin of Boltzmann's constant can be traced back to Boltzmann's work in the late 19th century, where he sought to provide a theoretical foundation for the principles of thermodynamics. His work involved the statistical interpretation of entropy and the development of the kinetic theory of gases. In particular, Boltzmann introduced the concept of entropy as a measure of the number of microstates associated with a particular macrostate of a system.

Boltzmann's constant emerges when he related the entropy (S) of a system to the number of microstates (W) corresponding to that macrostate and the fundamental constants of nature:

S = k * ln(W),

where:

• S is the entropy,
• k is Boltzmann's constant,
• ln represents the natural logarithm,
• W is the number of microstates.

This equation, now known as the Boltzmann entropy formula, is a fundamental concept in statistical mechanics. Boltzmann's constant was introduced to link the macroscopic properties of a system (entropy) to the microscopic behavior of its constituent particles. It allows us to bridge the gap between the statistical behavior of individual particles and the macroscopic properties of matter.

Boltzmann's work had a profound impact on our understanding of the behavior of gases, the concept of temperature, and the laws of thermodynamics. His constant, k, serves as a critical bridge between the microscopic world of particles and the macroscopic world of thermodynamics, and it remains a fundamental constant in the field of physics and chemistry. The value of Boltzmann's constant is approximately 1.380649 × 10^(-23) joules per kelvin (J/K).