What is the origin of Boltzmann's constant?

**Galileo** · Aug14-04, 08:55 AM

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:

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

$LaTeX Code: R=8.314510(70) J/mol\\cdot K$
when we want to work with the number of particles instead of moles (which we often do) we define:
$LaTeX Code: k=\\frac{R}{N_A}$
where

is Avogadro's number and k is called the Boltzmann constant.
The gas equation then becomes:

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** · Aug14-04, 10:50 AM

Originally Posted by Galileo

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
$LaTeX Code: \\frac {1}{2} K 273.16 degrees$

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

**krab** · Aug14-04, 11:33 AM

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** · Aug16-04, 06:37 PM

Originally Posted by 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.

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