Chapter 14. Boltzmann's Constant

The SI value of the magnitude of Boltzmann's constant, k, is, as follows:

k = 1.4 × 10^-23 J·K^-1 (129

A. Boltzmann's Constant in SE Units

Boltzmann's constant converts from SI to SE units, as follows:

k = (1.4 × 10^-23 J·K^-1) (8.2 × 10^-14 J·E_e^-1)^-1 ×

(5.9 × 10⁹ K·k_e^-1) = 1 E_e·k_e^-1 (130

Therefore, in SE units, the value of Boltzmann's constant is an electron's rest-mass energy divided by its threshold temperature for a value of unity.

B. Boltzmann's Constant in SG Units

The value of Boltzmann's constant in SG units is the same as it is in SE units because both the masson's rest-mass energy and threshold temperature are greater than those of the electron by the same factor θ, which equals 2.0 × 10²¹, a dimensionless number. See Equation 74.

Converting from SE to SG units gives:

k = (1 E_e·k_e^-1) (2.0 × 10²¹ E_e·E_g^-1)^-1 ×

(2.0 × 10²¹ k_e·k_g^-1) = 1 E_g·k_g^-1 (131

This SG value for Boltzmann's constant is equal to the Planck unit of entropy, s_Pl, which we use in calculating Bekenstein and Hawking's equation for the entropy of a Black Hole. See Equation 189.

C. Boltzmann's Constant, Historically

Historically, Boltzmann's constant is defined using SI units, as follows:

k = R N₀^-1 = 1.4 × 10^-23 J·K^-1 (132

where R is the universal gas constant, and N₀ is Avogadro's number.

However, when using SE units, (k = 1); therefore, R and N₀ must equal each other and, ideally, should each be equal to one (and they are).

D. Avogadro's Number

The SI value of Avogadro's number, N₀, is:

N₀ = 6.0 × 10²⁶ atoms·kilomole^-1 or amu·kg^-1 = 1 (133

where the amu (atomic mass unit) equals one-twelfth of the mass of the carbon-12 isotope.

We see that Avogadro's number is a dimensionless conversion factor from one unit of mass to another and its magnitude, therefore, must be one.

E. Universal Gas Constant in SI Units

The SI value of the universal gas constant, R, is:

R = 8.3 × 10³ J·kilomole^-1·K^-1 (134

where the kilomole is the amu-to-kg conversion factor of 6.0 × 10²⁶. Therefore, the value of R in SI units (without the kilomole) is:

R = (8.3 × 10³ J·K^-1) (6.0 × 10²⁶)^-1 =

1.4 × 10^-23 J·K^-1 (135

F. Universal Gas Constant in SE Units

We compute the SE value of the universal gas constant, R, as follows:

R = (1.4 × 10^-23) [(8.2 × 10^-14)^-1 E_e] ×

[(5.9 × 10⁹)^-1 k_e]^-1 = 1 E_e·k_e^-1 (136

G. Boltzmann's Constant, Revisited

Now, when we look at Boltzmann's constant in relation to Avogadro's number and the universal gas constant, we see:

k = R N₀^-1 = (1 E_e·k_e^-1)(1) = 1 E_e·k_e^-1 (137

which, when compared to Equation 130, confirms the SE value of Boltzmann's constant.

Boltzmann's constant, Avogadro's number, and the universal gas constant do not exist, as such.