The thermodynamic temperature:
Go for an energy-based unit!

The thermodynamic temperature:

Go for an energy based unit!

 

What are the Boltzmann constant ‘k’ and the gas constant ‘R’? Conversion factors! Do away with them by putting them into the thermodynamic temperature in the following way:

 

T’ = ½ R T

 

T’ = consolidated thermodynamic temperature,  J/mol = Bo

R = universal gas constant = kN_A .  S I-unit: 8.314 J/(K mol)  Imperial unit: 4.619 J/ (R mol)

T = conventional thermodynamic temperature, SI-unit: Kelvin, Imperial unit: Rankine

N_A = Avogadro’s number, 6.0224*10^-23

 

Why?

 

Thermodynamic relations will be easier to grasp, superfluous conversion factor will be abandoned, molar heat capacity and entropy will be dimensionless. However: do not introduce the notion into every-day life; Celsius and Fahrenheit are doing well here. The notion is for those occupied with thermodynamics as science.

How come?

 

The General Conference on Weights and Measurement (CGPM) has now assigned an exact value of

 

k = 1.380 6 *10^-23 J/K

 

to the Boltzmann constant. Fisher et al. [Int. J. Thermophysics, 2007] state that “. . .Boltzmann constant is simply the proportionality constant between temperature and thermal energy. . .” Celsius and Fahrenheit did not know about this relation and based their scales on water properties and human functions.

The unit of the thermodynamic temperature defined in this proposed way will be “J/mol” and should be given the name boltzmann, short form Bo, in remembrance of the great Austrian physicist Ludwig Boltzmann who cleared it up:

J/mol = Bo

 

Great efforts have been put into experiments to determine the Boltzmann constant. These experiments have in common that it is not k, but the product kT that is investigated. They have been performed under the assumption that the surrounding temperature (mostly triple point of water, TPW) was exactly known. With the decision of CGPM, the uncertainty of the experiment results is put on the temperature – in agreement with this proposal.

What’s up?

 

I have put the factor ½ in front of the RT-product for the single reason that the Boltzmann temperature then represents the internal, kinetic energy per degree of freedom. The total internal energy will simply be the thermodynamic temperature multiplied by the number of degrees of freedom, which implies that molar heat capacity, c , is a pure number. Looking at a monatomic gas, the three  space directions represent the accessible degrees of freedom, so the molar internal energy of such a gas is 3 times the Boltzmann temperature. Check it out! (Tabulated value of c for helium at constant volume is 3.12*10³ J/(kg K), molar mass is 4.0*10^-3  kg/mol)

 

The gas equation takes the form :  pV = 2T’ indicating the well-known fact that the pressure-volume product represents two degrees of freedom, so the molar heat capacity at constant pressure is 5 for monatomic gases.

Number of accessible degrees of freedom is not an integer for more complicated materials as internal excitations come into effect, the notion is however still valid. Theory says that crystalline bodies ideally have 6 degrees of freedom. This is expressed by the rule of Dulong and Petit which states that the specific heat capacity of crystalline, solid bodies is about 25.9 J/(K mol). Applying the rule in the Boltzmann idiom, this figure will transform into the pure number 6.23.

Boltzmann temperature may be deduced from the gas equation without resorting to the Boltzmann or gas constant: it is known that 0.0224 m³ of any gas at pressure 1.0135 N/m² in thermal equilibrium with freezing water contains 1 mole of that gas. The Boltzmann temperature of freezing water is then calculated to be:

T’ =  ½*1.0135^.10⁵ N/m²*0.0224 m³/mol = 1,135 Bo

Exact value of the Boltzmann triple point temperature of water is obtained from the defining equation by inserting values for R and T_TPW.

What about entropy?

A thermodynamic temperature must possess the property of being the “integrating factor” for heat added. The Boltzmann temperature fulfils this requirement. When heat added (δq_m) is divided by T’, the potential  (entropy) arises:

_m

The unit both of the nominator and denominator on the left hand side is J/ mol , making the molar entropy dimensionless.

When striving to fit the radiation law to observations, Max Planck also considered the definition of entropy that Ludwig Boltzmann had generalized to be valid for all physical systems. Planck writes: “The entropy of a system in a certain state depends only on the probability of that state”. Boltzmann had introduced the logarithmic function, but this function may have any constant in front of it. Max Planck deduced that the constant should be set to the Boltzmann constant k. In the context proposed here, the Boltzmann constant is replaced by 2/N_A the so-called Boltzmann equation reads in this idiom:

In this formulation the equation consists of pure numbers only which reveals the profound relation between entropy and probability and is in line with Shannon’s information theory.

 

So what?

 

Discard the Boltzmann and the gas constant as mere conversion factors; go for the Boltzmann temperature!