Buffer solutions: To control pH values in electrolytes.

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

Syntax

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

!number of ions that are contained in the buffer number-of-ions = 2 ! 2 for ACETATE,HEPES,TRISandMOPS = 4 ! 4 for PBS

                                                 !valency of the ions that are contained in the buffer ion-valency    = -1d0 +1d0                     !forACETATE: (CH₃COO)^-Na⁺
                = +1d0 -1d0                     !forTRIS:        TrisH⁺(OH)^-
                = -1d0 +1d0                     !forMOPS:        (C₇H₁₄NO₄S)^- Na⁺
                = -1d0 -2d0 -3d0 1d0          !for PBS:          (H₂PO₄)^-(HPO₄)^2-(PO₄)^3-Na⁺
                                                 !(the number of expected entries corresponds to number-of-ions)

                                                 !name of the ions that are contained in the buffer                                                 !                                                  !CH₃COOH <==> (CH₃COO)^-+H⁺ ion-name-1     = CH3COO^-                       !name of ion no. 1 (for ACETATE) ion-name-2     = Na^+                           !name of ion no. 2 (for ACETATE)
                                                 !                                                  !TrisH⁺     <==> Tris   +H⁺           ( "NH₃⁺"      <==>"NH₂" + H⁺)                                                 !Tris + H₂0 <==> TrisH⁺+(OH)^-       ( "NH₂" + H₂0 <==> "NH₃⁺"+(OH)^-) ion-name-1     = TrisH^+                        !name of ion no. 1 (for TRIS) ion-name-2     = OH^-                           !name of ion no. 2 (for TRIS)
                                                 !                          ("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)
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-+ 2Na⁺ 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)

!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 + H₂0<==>Cl^- + H₃0⁺ pKa = 4.76d0 !for ACETATE: pK_a at 25° C ( = 298.15 K) = 7.66d0 !for HEPES: pK_a at 25° C ( = 298.15 K) = 8.06d0 !for TRIS: pK_a at 25° C ( = 298.15 K) = 7.31d0 !for MOPS: pK_a at 25° C ( = 298.15 K) = 2.15d0 7.21d0 12.33d0 !for PBS: pK_a,1 pK_a,2 pK_a,3 at 25° C ( = 298.15 K)

!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$electrolyte. dpKa_dT = -0.0002d0 !for ACETATE: d pK_a/ d T = -0.014d0 !for HEPES: d pK_a/ d T = -0.028d0 !for TRIS: d pK_a/ d T = -0.011d0 !for MOPS: d pK_a/ d T = 0.0044d0 -0.0028d0 -0.026d0 !for PBS: d pK_a,1/ d T d pK_a,2/ d T d pK_a,3/ d T

!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' ). z_acid = 0d0 !forACETATE: 0 = CH₃COOH = 0d0 !forMOPS: 0 = C₇H₁₅NO₄S = +1d0 !forTRIS: +1 = TrisH⁺ = 0d0 -1d0 -2d0 !for PBS: 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 = -1d0 +1d0 ! ion-name-1 = CH3COO^- !(CH₃COO)^- ion-name-2 = Na^+ !Na⁺ pKa = 4.76d0 !pK_a at 25° C ( = 298.15 K) dpKa_dT = -0.0002d0 !d pK_a/ d T z_acid = 0d0 !charge on the conjugate acid species (0 = CH₃COOH) ! ! buffer-name = MOPS !MOPS (C₇H₁₅NO₄S) + NaOH number-of-ions = 2 ! ion-valency = -1d0 +1d0 !(C₇H₁₄NO₄S)^- Na⁺ ion-name-1 = Mops^- ! C₇H₁₅NO₄S <==> (C₇H₁₄NO₄S)^-+H⁺ ion-name-2 = Na^+ ! pKa = 7.31d0 !pK_a at 25° C ( = 298.15 K)
!(Note: This pK_a is thermodynamic value. The working pK_a' is 7.20.) dpKa_dT = -0.011d0 !d pK_a/ d T z_acid = 0d0 !charge on the conjugate acid species (0 = C₇H₁₅NO₄S) ! ! buffer-name = PBS !PBS (phosphate buffer) number-of-ions = 4 ! ion-valency = -1d0 -2d0 -3d0 1d0 !(H₂PO₄)^- (HPO₄)^2- (PO₄)^3- Na⁺ ion-name-1 = H2PO4^- !NaH₂PO₄ <==> (H₂PO₄)^-+ Na⁺ ion-name-2 = HPO4^2- !Na₂HPO₄ <==> (HPO₄)^2-+ 2Na⁺ ion-name-3 = PO4^3- ! ion-name-4 = Na^+ ! pKa = 2.15d0 7.21d0 12.33d0 !pK_a,1 pK_a,2 pK_a,3 at 25° C ( = 298.15 K) dpKa_dT = 0.0044d0 -0.0028d0 -0.026d0 !d pK_a/ d T z_acid = 0d0 -1d0 -2d0 !charge on the conjugate acid species (0 = H₃PO₄, -1 = (H₂PO₄)^- , -2 = (HPO₄)^2-) ! $end_buffer-solutions ! !------------------------------------------------!

For more details on buffers, please have a look at the excellent book of
R.J. Beynon, J.S. Easterby, "Buffer solutions: The basics", Oxford University Press (1996).

Rob Beynon also provides a web interface "A recipe calculator for thermodynamically correct buffers for pH control" at:http://www.liv.ac.uk/buffers/

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 nextnano³ 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_pH1D.dat

If you want to obtain the nextnano³ input file (Buffer_Acetate.in) 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 for HEPES andTRIS)
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 nextnano³ input file (Buffer_PBS.in) 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.