k_B × F = e_c × R Four Related Constants.

We examine the following 4 constants and define each in terms of the other three, showing a clear link between different areas of physics. 

Calculated Constants vs. Known Values:

Boltzmann constant (k_B)= (e_c * R) / F

  Calculated: 1.380649000021918e-23 J/K

  Actual:     1.380649000000000e-23 J/K

  Difference: 2.192e-34

Faraday constant (F)=(e_c * R) / k_B

  Calculated: 9.648533212153174e+04 C/mol

  Actual:     9.648533212000001e+04 C/mol

  Difference: 1.532e-06

Elementary charge (e_c)= (k_B * F) / R

  Calculated: 1.602176633974565e-19 C

  Actual:     1.602176634000000e-19 C

  Difference: 2.544e-30

Gas constant (R)=(k_B * F) / e_c

  Calculated: 8.314462617868006e+00 J/(mol⋅K) 

  Actual:     8.314462618000000e+00 J/(mol⋅K) 

  Difference: 1.320e-10

Let's perform a dimensional analysis on the derived formulas to ensure they are consistent and to gain further insights into the relationships between the constants. We will use standard SI units.

1. Defining the Units:

Let's first define the SI units for each constant:

• k_B (Boltzmann constant): J/K (Joules per Kelvin) = kg⋅m²⋅s⁻²⋅K⁻¹

• F (Faraday constant): C/mol (Coulombs per mole) = A⋅s⋅mol⁻¹ (Ampere-seconds per mole)

• e_c (Elementary charge): C (Coulombs) = A⋅s (Ampere-seconds)

• R (Ideal Gas constant): J/(mol⋅K) (Joules per mole per Kelvin) = kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹ (This is also the value for the Universal Gas Constant)

2. Dimensional Analysis of the Formulas:

Now let's analyze the units of each derived formula:

• Formula for k_B: 

  • Units of (e_c × R): (A⋅s) * (kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (e_c × R) / F: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (A⋅s⋅mol⁻¹) = kg⋅m²⋅s⁻²⋅K⁻¹

  • Result: The units of the calculated value for k_B are kg⋅m²⋅s⁻²⋅K⁻¹, which matches the units of the Boltzmann constant (J/K).

• Formula for F: 

  • Units of (e_c × R): (A⋅s) * (kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (e_c × R) / k_B: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (kg⋅m²⋅s⁻²⋅K⁻¹) = A⋅s⋅mol⁻¹

  • Result: The units of the calculated value for F are A⋅s⋅mol⁻¹, which match the units of the Faraday constant (C/mol).

• Formula for e_c: 

  • Units of (k_B × F): (kg⋅m²⋅s⁻²⋅K⁻¹) * (A⋅s⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (k_B × F) / R: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹) = A⋅s

  • Result: The units of the calculated value for e_c are A⋅s, which match the units of the elementary charge (C).

• Formula for R: 

  • Units of (k_B × F): (kg⋅m²⋅s⁻²⋅K⁻¹) * (A⋅s⋅mol⁻¹) = kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A

  • Units of (k_B × F) / e_c: (kg⋅m²⋅s⁻¹⋅K⁻¹⋅mol⁻¹⋅A) / (A⋅s) = kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹

  • Result: The units of the calculated value for R are kg⋅m²⋅s⁻²⋅K⁻¹⋅mol⁻¹, which match the units of the ideal gas constant (J/(mol⋅K)).

3. Summary and Interpretation:

• Dimensional Consistency: The dimensional analysis confirms that all four derived formulas are dimensionally consistent. The units on both sides of each equation match, which reinforces the validity of the relationships.

• Units are Fundamental: The results also indicate the fundamental connections between the units we have used. They confirm that all of the base units used to derive each constant are consistent with the other constants, when the derived formulas are used.
  
  
  

  import numpy as np

  from scipy import constants

  
  

  # Define the known constants

  k_B = 1.380649e-23 # Boltzmann constant (J/K)

  F   = 96485.33212    # Faraday constant (C/mol)

  e_c = 1.602176634e-19 # Elementary charge (C)

  R   = 8.314462618     # Gas constant J/(mol K)

  
  

  # Define a function to calculate each constant

  def calculate_constants(k_B, F, e_c, R):

      """

      Calculates each constant in terms of the other three.

      """

      k_B_calc = (e_c * R) / F

      F_calc = (e_c * R) / k_B

      e_c_calc = (k_B * F) / R

      R_calc = (k_B * F) / e_c

      return k_B_calc, F_calc, e_c_calc, R_calc

  
  

  
  

  # Calculate the constants

  k_B_calc, F_calc, e_c_calc, R_calc = calculate_constants(k_B, F, e_c, R)

  
  

  # Print the results and check against known values

  print("Calculated Constants vs. Known Values:\n")

  
  

  print(f"Boltzmann constant (k_B)= (e_c * R) / F")

  print(f"  Calculated: {k_B_calc:.15e} J/K")

  print(f"  Actual:     {k_B:.15e} J/K")

  print(f"  Difference: {np.abs(k_B_calc-k_B):.3e}\n")

  
  

  print(f"Faraday constant (F)=(e_c * R) / k_B")

  print(f"  Calculated: {F_calc:.15e} C/mol")

  print(f"  Actual:     {F:.15e} C/mol")

  print(f"  Difference: {np.abs(F_calc - F):.3e}\n")

  
  

  print(f"Elementary charge (e_c)= (k_B * F) / R")

  print(f"  Calculated: {e_c_calc:.15e} C")

  print(f"  Actual:     {e_c:.15e} C")

  print(f"  Difference: {np.abs(e_c_calc - e_c):.3e}\n")

  
  

  print(f"Gas constant (R)=(k_B * F) / e_c")

  print(f"  Calculated: {R_calc:.15e} J/(mol⋅K)")

  print(f"  Actual:     {R:.15e} J/(mol⋅K)")

  print(f"  Difference: {np.abs(R_calc-R):.3e}\n")

  
  
  Solving for the formulas
  
  

  it was not recursive, it actually reached a definition

  k_B = R / N_A
  F = N_A * e_c
  R = N_A * k_B
  e_c = 1.602176634 × 10^-19 C

  All but one has N_A in it.

  So

  k_B_calc = (e_c * R) / F
  k_B = ( 1.602176634 × 10^-19 * N_A * k_B) / N_A * 1.602176634 × 10^-19 C
  k_B = k_B

  F_calc = (e_c * R) / k_B
  F_calc = (1.602176634 × 10^-19 * N_A * k_B) /R / N_A
  F_calc = (1.602176634 × 10^-19 * N_A * k_B) /N_A * k_B / N_A
  F_calc = (1.602176634 × 10^-19 * N_A ) At this step I had to note that this is the definition for F
  F_calc = F

  e_c_calc = (k_B * F) / R
  e_c_calc = (R / N_A* N_A * e_c) / N_A * k_B
  e_c_calc = (R * e_c) / N_A * k_B
  e_c_calc = (N_A * k_B * e_c) / N_A * k_B
  e_c_calc = 1.602176634 × 10^-19

  R_calc = (k_B * F) / e_c
  R_calc = (R / N_A* N_A * e_c) / 1.602176634 × 10^-19
  R_calc = (R / N_A* N_A * 1.602176634 × 10^-19) / 1.602176634 × 10^-19
  R_calc = R