Boltzmann Constant in definition of entropy

[Curl]

I don't remember why the Boltzmann constant is the perfect number that let's us convert the log of the multiplicity to a unit that works with other SI units. I understand that this constant was "given" units of J/K but why does it work exactly to convert a rather odd made-up number (log of the multiplicity) to a unit which macroscopically has meaning (one joule of heat per Kelvin). Why doesn't there have to be a factor of say, 2, in there as well?

 

[Zacku]

I don't remember why the Boltzmann constant is the perfect number that let's us convert the log of the multiplicity to a unit that works with other SI units. I understand that this constant was "given" units of J/K but why does it work exactly to convert a rather odd made-up number (log of the multiplicity) to a unit which macroscopically has meaning (one joule of heat per Kelvin). Why doesn't there have to be a factor of say, 2, in there as well?

Hi,

A good and simple explanation according to me comes from statistical mechanics. First, you can define an entropy with an arbitrary multiplicative constant that you will call "A" or whatever. Then you can define a temperature from that as the derivative of the energy with respect to S. You will find that two systems in contact with each other will share the same temperature at equilibrium.

This result is independent from the constant "A" that you put in you definition of statistical entropy. Now, you may be interested in ideal gases and in the microcanonical ensemble can find an equation of state relating the pressure P, the volume V, the number of particles and the temperature. This result does depend on the constant "A" you put in your definition for the entropy and can then be estimated using the ideal gas law which is well known and established since the 19th century.
If you do that correctly, you find that, speacking in term of molecules rather than in moles, "A" equals kB the Boltzmann constant.

 

[vanhees71]

The Boltzmann constant is simply converting the units of thermal energy to temperatures. For a non-relativistic ideal gas, made up by [tex]N[/tex] particles at absolute temperature [tex]T[/tex] the total mean of the kinetic energy is

[tex]U=\frac{f}{2} k T[/tex],

where [tex]f[/tex] is the number of momentum-degrees of freedom, entering the Hamiltonian quadratically, e.g., for a monatomic gas it's 3, for a two-atomic one its 5 and for a more general one its 6.

Of course, at higher temperatures also vibrational degrees of freedom are excited, and this changes the factor again.

In natural units, one usually sets [tex]k=1[/tex] an gives temperatures in energy units (e.g., MeV or GeV in relativistic heavy-ion collisions to give the temperature of the created hot and dense fireball undergoing a phase transition from a deconfined QGP phase to a hadron-gas phase at a temperature of around [tex]T_c=160 \; \mathrm{MeV}[/tex].)

 

[Curl]

Oh right, Temperature was defined from entropy (which was defined to include k) so it works out that way.

Dumb question I guess, sorry.

 

[sardar]

The Boltzmann constant, denoted by k, is a fundamental constant in thermodynamics and statistical mechanics that relates the average kinetic energy of particles in a system to its temperature. It is defined as the ratio of the gas constant to Avogadro's constant, and has a value of approximately 1.38 x 10^-23 J/K.

The significance of the Boltzmann constant in the definition of entropy lies in its ability to convert the logarithm of the multiplicity (the number of microstates corresponding to a macrostate) to a unit that is consistent with other SI units. This is because the logarithm of the multiplicity is a dimensionless quantity, while entropy is a thermodynamic quantity with units of energy per temperature.

The Boltzmann constant is necessary in this conversion because it provides a scale for the logarithm of the multiplicity, allowing it to be expressed in units of energy per temperature. Without this constant, the logarithm of the multiplicity would remain a dimensionless quantity and would not have a physical meaning in terms of energy and temperature.

As for why there is no need for a factor of 2 in the Boltzmann constant, this can be understood from the definition of entropy. Entropy is a measure of the disorder or randomness in a system, and it is directly proportional to the logarithm of the multiplicity. This means that a change in entropy is equal to the change in the logarithm of the multiplicity. Therefore, the value of the Boltzmann constant is already taken into account in the definition of entropy, and there is no need for an additional factor.

In summary, the Boltzmann constant is a crucial component in the definition of entropy as it allows for the conversion of a dimensionless quantity to a physical quantity with units of energy per temperature. Its value is determined by the relationship between the gas constant and Avogadro's constant, and it is necessary for the consistent application of thermodynamic principles.