# What is the origin of Boltzmann's constant?

1. Aug 14, 2004

### 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.

Last edited: Aug 14, 2004
2. Aug 14, 2004

### pervect

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

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

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

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.

3. Aug 14, 2004

### 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.

4. Aug 16, 2004

### Rogerio

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

Last edited: Aug 16, 2004