$buffer-solutions

Buffer solutions: To control pH values in electrolytes.

$buffer-solutions                          optional    !
 buffer-name             character         required    !
 number-of-ions          integer           required    !
 ion-valency             double_array      required    !
 ion-name-1              character         required    !
 ion-name-2              character         required    !
 ion-name-3              character         optional    ! (only necessary for some buffers, e.g. PBS)
 ion-name-4              character         optional    ! (only necessary for some buffers, e.g. PBS)
 pKa                     double_array      required    ! pKa at 25° C ( = 298.15 K)
 dpKa_dT                 double_array      required    ! d pKa / d T
 z_acid                  double_array      required    ! charge on the conjugate acid species
$end_buffer-solutions                      optional    !

pK_a at 25° C ( = 298.15 K)

d pK_a / d T

Syntax

buffer-name = ACETATE    ! Acetate (sodium acetate + acetic acid) (CH3COO-Na+ + CH3COOH)
            = HEPES      ! HEPES   (C8H18N2O4S)
            = TRIS       ! TRIS    (C4H11NO3)
            = MOPS       ! MOPS    (C7H15NO4S)
            = PBS        ! PBS     (phosphate buffer)

Name of buffer

Acetate (sodium acetate + acetic acid): CH₃COO^-Na⁺ + CH₃COOH
HEPES: C₈H₁₈N₂O₄S
TRIS: C₄H₁₁NO₃
MOPS: C₇H₁₅NO₄S
PBS (phosphate buffer): …

number-of-ions = 2   ! 2 for ACETATE, HEPES, TRIS and MOPS
               = 4   ! 4 for PBS

Number of ions that are contained in the buffer

ion-valency    = -1.0  +1.0              ! for ACETATE:  (CH3COO)-     Na+
               = +1.0  -1.0              ! for TRIS:     TrisH+        (OH)-
               = -1.0  +1.0              ! for MOPS:     (C7H14NO4S)-  Na+
               = -1.0  -2.0  -3.0  1.0   ! for PBS:      (H2PO4)-      (HPO4)2-    (PO4)3-    Na+

Valency of the ions that are contained in the buffer. The number of expected entries corresponds to number-of-ions.

ACETATE: (CH₃COO)^- Na⁺

TRIS: TrisH⁺ (OH)^-

MOPS: (C₇H₁₄NO₄S)^- Na⁺

PBS: (H₂PO₄)^- (HPO₄)^2- (PO₄)^3- Na⁺

Name of the ions that are contained in the buffer:

ion-name-1     = CH3COO^-  ! name of ion no. 1 (for ACETATE)
ion-name-2     = Na^+      ! name of ion no. 2 (for ACETATE)

CH₃COOH <==> (CH₃COO)^- + H⁺

ion-name-1     = TrisH^+   ! name of ion no. 1 (for TRIS)
ion-name-2     = OH^-      ! name of ion no. 2 (for TRIS)

TrisH⁺ <==> Tris + H⁺ ( “NH₃⁺” <==> “NH₂” + H⁺ )

Tris + H₂O <==> TrisH⁺ + (OH)^- ( “NH₂” + H₂O <==> “NH₃⁺” + (OH)^- )

                           !                                  ("SO3H" <==> "(SO3)-"      + H+)
ion-name-1     = Mops^-    ! name of ion no. 1 (for MOPS)   C7H15NO4S <==> (C7H14NO4S)- + H+
ion-name-2     = Na^+      ! name of ion no. 2 (for MOPS)

(“SO₃H” <==> “(SO₃)^-” + H⁺)

C₇H₁₅NO₄S <==> (C₇H₁₄NO₄S)^- +H⁺

ion-name-1     = H2PO4^-   ! name of ion no. 1 (for PBS)     NaH2PO4 <==> (H2PO4)- +   Na+
ion-name-2     = HPO4^2-   ! name of ion no. 2 (for PBS)     Na2HPO4 <==> (HPO4)2- + 2 Na+
ion-name-3     = PO4^3-    ! name of ion no. 3 (for PBS)    (HPO4)2- <==> (PO4)3-  +   H+
ion-name-4     = Na^+      ! name of ion no. 4 (for PBS)

NaH₂PO₄ <==> (H₂PO₄)^- + Na⁺

Na₂HPO₄ <==> (HPO₄)^2-+ 2 Na⁺

(HPO₄)^2- <==> (PO₄)^3-+ H⁺

pKa = 4.76                     ! for ACETATE: pKa                        at 25° C ( = 298.15 K)
    = 7.66                     ! for HEPES:   pKa                        at 25° C ( = 298.15 K)
    = 8.06                     ! for TRIS:    pKa                        at 25° C ( = 298.15 K)
    = 7.31                     ! for MOPS:    pKa                        at 25° C ( = 298.15 K)
    = 2.15     7.21    12.33   ! for PBS:     pKa,1    pKa,2    pKa,3    at 25° C ( = 298.15 K)

pK_a value(s) of buffer reactions. One can calculate the pH value from the pK_a value and the concentrations of acid and base (Henderson-Hasselbalch equation):

pH = pK_a+ log₁₀( [base] / [acid] )

Example: pH = pK_a+ log₁₀( [Cl^-] / [HCl] )

HCl + H₂O <==> Cl^- + H₃O⁺

dpKa_dT = -0.0002                   ! for ACETATE:  d pKa   / d T
        = -0.014                    ! for HEPES:    d pKa   / d T
        = -0.028                    ! for TRIS:     d pKa   / d T
        = -0.011                    ! for MOPS:     d pKa   / d T
        =  0.0044  -0.0028  -0.026  ! for PBS:      d pKa,1 / d T     d pKa,2 / d T     d pKa,3 / d T

d pK_a / d T value(s) of pK_a value(s), i.e. temperature dependence of pK_a value(s) (temperature coefficient, i.e. the rate of change of pK_a with temperature). If the buffer solution warms up, the pK_a value drops. For further details, see $buffer-solutions.

z_acid =  0.0                    ! for ACETATE:   0 = CH3COOH
       =  0.0                    ! for MOPS:      0 = C7H15NO4S
       = +1.0                    ! for TRIS:     +1 = TrisH+
       =  0.0    -1.0    -2.0    ! for PBS:       0 = H3PO4, -1 = (H2PO4)- , -2 = (HPO4)^2-

Charge on the conjugate acid species.

Note: ‘conjugate acid’ + H₂0<==> ‘conjugate base’ + H₃0⁺

This value enters the equation for the ionic strength dependence of the pK_a value (modified pK_a value: pK_a’).

forACETATE:: 0 = CH₃COOH
forMOPS:: 0 = C₇H₁₅NO₄S
forTRIS:: +1 = TrisH⁺
forPBS:: 0 = H₃PO₄, -1 = (H₂PO₄)^- , -2 = (HPO₄)^2-

Examples: Acetate, MOPS and PBS (phosphate buffer)

The phosphate buffer is special (and thus more complicated) because it consists of three pK_a values (and it thus has four different ions).

$buffer-solutions

 buffer-name    = ACETATE                    ! Acetate (sodium acetate + acetic acid) (CH3COO-Na+ + CH3COOH)
 number-of-ions = 2                          ! CH3COOH  <==> (CH3COO)- + H+
 ion-valency    = -1.0 +1.0                  !
 ion-name-1     = CH3COO^-                   ! (CH3COO)-
 ion-name-2     = Na^+                       ! Na+
 pKa            = 4.76                       ! pKa at 25° C ( = 298.15 K)
 dpKa_dT        = -0.0002                    ! d pKa   / d T
 z_acid         = 0.0                        ! charge on the conjugate acid species (0 = CH3COOH)

(CH₃COO^-Na⁺ + CH₃COOH)

CH₃COOH<==> (CH₃COO)^- + H⁺

buffer-name    = MOPS                       ! MOPS  (C7H15NO4S)  +  NaOH
number-of-ions = 2                          !
ion-valency    = -1.0  +1.0                 ! (C7H14NO4S)-  Na+
ion-name-1     = Mops^-                     !  C7H15NO4S <==> (C7H14NO4S)- + H+
ion-name-2     = Na^+                       !
pKa            = 7.31                       ! pKa at 25° C ( = 298.15 K)
                                            ! (Note: This pKa is thermodynamic value. The working pKa' is 7.20.)
dpKa_dT        = -0.011                     ! d pKa   / d T
z_acid         = 0.0                        ! charge on the conjugate acid species (0 = C7H15NO4S)

(C₇H₁₅NO₄S) + NaOH

(C₇H₁₄NO₄S)^- Na⁺

C₇H₁₅NO₄S <==> (C₇H₁₄NO₄S)^- + H⁺

(0 = C₇H₁₅NO₄S)

 buffer-name    = PBS                        ! PBS (phosphate buffer)
 number-of-ions = 4                          !
 ion-valency    = -1.0  -2.0  -3.0  1.0      ! (H2PO4)-     (HPO4)2-     (PO4)3-     Na+
 ion-name-1     = H2PO4^-                    ! NaH2PO4 <==> (H2PO4)- +   Na+
 ion-name-2     = HPO4^2-                    ! Na2HPO4 <==> (HPO4)2- + 2 Na+
 ion-name-3     = PO4^3-                     !
 ion-name-4     = Na^+                       !
 pKa            = 2.15     7.21    12.33     ! pKa,1    pKa,2    pKa,3    at 25° C ( = 298.15 K)
 dpKa_dT        = 0.0044  -0.0028  -0.026    ! d pKa   / d T
 z_acid         = 0.0     -1.0     -2.0      ! charge on the conjugate acid species (0 = H3PO4, -1 = (H2PO4)- , -2 = (HPO4)2-)

$end_buffer-solutions

(H₂PO₄)^- (HPO₄)^2- (PO₄)^3- Na⁺

NaH₂PO₄ <==> (H₂PO₄)^- + Na⁺

Na₂HPO₄ <==> (HPO₄)^2- + 2 Na⁺

(0 = H₃PO₄, -1 = (H₂PO₄)^- , -2 = (HPO₄)^2-)

For more details on buffers, please have a look at the excellent book of [Beynon1996]. Rob Beynon also provides a web interface A recipe calculator for thermodynamically correct buffers for pH control at: https://phbuffers.org

Example: Acetate buffer (sodium acetate + acetic acid)

The following figure shows the pH value as a function of the concentration of the (CH3COO)^- ions. Note that this number is equal to the concentration of Na⁺ ions (see equations above). The concentration of the acetate buffer is 0.1 M, i.e. it consists of

0.1 M CH₃COOH (acetic acid) and

0.1 M CH₃COO^-Na⁺ (sodium acetate).

The temperature was set to 298.15 K ( = 25° C). The pK_a value at 25° C is 4.76. Note that the pK_a value depends on temperature and ionic strength. Thus for each pH value a different value of pK_a had been calculated self-consistently=>pK_a’. The calculations were performed by looping over the pH values. Note that the code adds the appropriate concentrations of Cl^- ions (originating from the acid HCl) and Na⁺ ions (originating from the base NaOH) automatically.

CH₃COO^- + H₃O⁺ + Cl^- <==> CH₃COOH + H₂O + Cl^-

CH₃COOH + Na⁺ + OH^- <==> CH₃COO^- + H₂O + Na⁺

The acetate buffer has the best buffering range (i.e. pH = ±1) at around pK_a = 4.76, i.e. it buffers nicely between pH = 3.76 and pH = 5.76.

Below the pH value of 3, the concentration of (CH₃COO)^- is negligible and only CH₃COOH exists.

Above the pH value of 8, the concentration of CH₃COOH is negligible and only (CH₃COO)^- exists.

pH value vs. [CH3COO^-]

The relation between the pH value, the pK_a’ value and the concentration of CH₃COO^- ions is governed by the Henderson-Hasselbalch equation.

pH = pK_a’ + log ( [CH₃COO^-] / [CH₃COOH] )

The concentration of all ions (not only the buffer ions), the ionic strength, as well as the pK_a’ value(s) as a function of pH are contained in this file: BufferIonConc_vs_pH.dat If you want to obtain the input file (Buffer_Acetate.nn3) that calculates the above figure, please submit a support ticket.

Details on the calculations

ACETATE(sodium acetate + acetic acid)

[CH₃COOH] = [ACETATE] / ( 1 + 10^pH-pKa )

[(CH₃COO)^-] = [CH₃COOH] * 10^pH-pKa (Henderson-Hasselbalch equation)

[Na⁺] = [(CH₃COO)^-]
MOPS

[C₇H₁₅NO₄S] = [MOPS] / ( 1 + 10^pH-pKa )

[(C₇H₁₄NO₄S)^-] = [C₇H₁₅NO₄S] * 10^pH-pKa (Henderson-Hasselbalch equation)

[Na⁺] = [(C₇H₁₄NO₄S)^-]

(analogous forHEPESandTRIS)
PBS (phosphate buffer)

[H₃PO₄] = [PBS] / ( 1 + 10^pH-pKa,1 * ( 1 + 10^pH-pKa,2 * ( 1 + 10^pH-pKa,3 ) ) )

[(H₂PO₄)^-] = [H₃PO₄] * 10^pH-pKa,1 (Henderson-Hasselbalch equation)

[(HPO₄)^2-] = [(H₂PO₄)^-] * 10^pH-pKa,2 (Henderson-Hasselbalch equation)

[(PO₄)^3-] = [(HPO₄)^2-] * 10^pH-pKa,3 (Henderson-Hasselbalch equation)

[Na⁺] = (1 * [(H₂PO₄)^-] + 2 * [(HPO₄)^2-] + 3 * [(PO₄)^3-]

The following figure shows the pH value as a functions of the Na⁺ concentration for a PBS buffer. Note that there are three buffer ranges.

In this figure, we plot the concentration of the PBS buffer ions, and the resulting ionic strength as a function of pH.

If you want to obtain the input file Buffer_PBS.nn3 that calculates the above figure, please submit a support ticket.

The entries for $buffer-solutions that are specified in the database can be overwritten in the input file. For details, have a look at the input file keyword $buffer-solutions.

$buffer-solutions

Syntax

pH value vs. [CH3COO-]

pH value vs. [CH3COO^-]