Buffer Intensity and Buffer Capacity
Buffers and Alkalimetric Titration
A buffer is a mixture of a weak acid and a base. Let’s consider a diprotic acid H_{2}A to which a monoacidic base BOH (with B^{+} = Na^{+}, K^{+}, or NH_{4}^{+}) is added:^{1}
(1)  H_{2}A + n BOH = B_{n}H_{2n }A + n H_{2}O 
Example: For H_{2}CO_{3} and NaOH we obtain – in the special case of n = 0, 1, and 2 – pure solutions of H_{2}CO_{3}, NaHCO_{3} and Na_{2}CO_{3} (see Equivalence Points).
The parameter n acts as a stoichiometric coefficient that embodies the ratio of added amount of strong base, C_{B}, to the total amount of the diprotic acid C_{T}:
(2)  n = C_{B }/C_{T} or C_{B} = n C_{T} 
Both C_{B} and C_{T} are given in molar units (mol/L) while n is a dimensionless number (unitfree).
The entity B_{n}H_{2n }A in reaction (1) does not survive in water; it dissociates into several aqueous species – as indicated in the right picture.
As we will see below, the amount of strong base is just the total alkalinity, Alk = C_{B}. Thus we have:
(3)  n = Alk /C_{T} or Alk = n C_{T} 
The variation of n (or C_{B}) in reaction formula (1) by adding a strong base is called alkalimetric titration. Note: The reason for our focus on diprotic acids is that it comprises carbonic acid H_{2}CO_{3} as the most prominent representative (i.e. A^{2} = CO_{3}^{2}).
Notation. To keep the notation straight we abbreviate the concentration of H^{+} simply by x. Its relationship with pH is given by:^{2}
(4)  x = [H^{+}] = 10^{pH}  ⇔  pH = – lg [H^{+}] = – lg x 
The (molar) concentrations of the acid species [H_{2 }A], [HA^{}] and [A^{2}] are abbreviated as
(5)  [j] = [H_{2j }A^{j}]  with j = 0, 1, 2 
In addition, we introduce the “alkalinity of pure water” by
(6)  w = [OH^{}] – [H^{+}] = K_{w }/x – x 
where K_{w} = 10^{14} is the equilibrium constant of autoprotolysis (selfionization of water). As we’ll see later, w represents the (additive) contribution of selfionization to the alkalinity. For pure water – defined by x = (K_{w})^{1/2} = 10^{7}, i.e. pH = 7, – this contribution vanishes: w = 0.
StartingPoint: Basic Set of Equations
The set of mathematical equations to describe the alkalimetric titration is exactly the same as we already introduced for diprotic acids and their conjugate bases:^{3}
(1.1a)  K_{1}  = {H^{+}} {HA^{}} / {H_{2}A}  (1^{st} diss. step)  
(1.1b)  K_{2}  = {H^{+}} {A^{2}} / {HA^{}}  (2^{nd} diss. step)  
(1.1c)  K_{w}  = {H^{+}} {OH^{}}  (selfionization)  
(1.1d)  C_{T}  = [H_{2}A] + [HA^{}] + [A^{2}]  (mass balance)  
(1.1e)  n C_{T}  = [HA^{}] + 2 [A^{2}] + [OH^{}] – [H^{+}]  (charge balance) 
The only difference to the previous approach is that n is now a smooth variable of any value (rather than merely a fixed integer n = 0, 1, 2).
For the special case of n = 0, the set reduces to the description of the pure diproticacid system, where [H^{+}] (or pH) is determined by the amount of C_{T}, and vice versa. Now, with the new variable n the system gets an additional degree of freedom.
Alkalinity. The righthand side of 1.1e is just the definition of total alkalinity (also known as M alkalinity in carbonate systems),
(1.2)  Alk ≡ ([OH^{}] – [H^{+}]) + [HA^{}] + 2 [A^{2}] 
This justifies our statement, which we originally made in 3:
(1.3)  Alk = C_{B} = n C_{T} 
Ionization Fractions (a_{0}, a_{1}, a_{2})
The basic system of equations consists of 5 equations. Everything concerning the acid itself, however, is contained in three equations: (1.1a), (1.1b) and (1.1c). The information contained in this subsystem is usually provided by the ionization fractions (instead of the three acid species [j]):
(2.1)  a_{j} = [j] / C_{T}  with j = 0, 1, 2 
These normalized “concentrations”, which express the degree of dissociation, are completely independent of the amount of acid C_{T}. Dividing 1.1d by C_{T} simplifies the mass balance equation:
(2.2)  1 = a_{0} + a_{1} + a_{2}  (mass balance) 
From a mathematical point of view, the ionization fractions are nothing else than a clever way to combine 1.1a and (1.1b) using 2.2. They are given by
(2.3a)  a_{0} = [ 1 + K_{1}/x + K_{1}K_{2}/x^{2} ]^{1}  
(2.3b)  a_{1} = [ x/K_{1} + 1 + K_{2}/x ]^{1}  
(2.3c)  a_{2} = [ x^{2}/(K_{1}K_{2}) + x/K_{2} + 1 ]^{1} 
They are solely functions of x (or pH); the only other ingredients are two equilibrium constants, K_{1} and K_{2}. Their pH dependence is shown here. Each a_{j} is imprisoned between 0 and 1, i.e. it neither become negative nor greater than 1. Strictly speaking, the functions will come very close to the boundaries, but will never actually reach the values 0 and 1:
(2.4)  0 < a_{j} < 1  for j = 0, 1, 2 
There are three exceptional points of pH (or x):^{4}
(2.5a)  pH = pK_{1}  x = K_{1}  ⇔  a_{1} = a_{0} ≈ ½  a_{2} ≈ 0  
(2.5b)  pH = ½ (pK_{1}+pK_{2})  x = (K_{1}K_{2})^{1/2}  ⇔  a_{0} = a_{2} ≈ 0  a_{1} ≈ 1  
(2.5c)  pH = pK_{2}  x = K_{2}  ⇔  a_{1} = a_{2} ≈ ½  a_{0} ≈ 0 
In what follows, the three ionization fractions a_{0}, a_{1}, and a_{2} will become the building blocks of almost all relevant quantities. In this respect, don’t confuse the smooth variable n with the index j for three integers j = 0, 1, 2. The latter are dedicated to the three ionization fractions a_{j} and to the three dissolved species, H_{2}A, HA^{}, and A^{2}.
Analytical Solution of the System of Equations
The set of 5 equations in (1.1) can be cast into a single analytical formula:
(3.1a) 
Here, the pH dependence (in form of x = 10^{pH}) is also in a_{1} and a_{2}. To make it evident insert (2.3b) and (2.3c) into 3.1a:
(3.1b) 
Both formulas calculate C_{T} for a given pH (or x) and n. Often, however, one is interested in the “inverse task”: to calculate the pH for a given C_{T} and n. For this purpose, you need to rearrange 2.1b to isolate x. But that’s impossible. The only thing you can achieve is a polynomial of 4^{th} order in x:
(3.2)  x^{4} + {K_{1} + nC_{T}} x^{3} + {K_{1}K_{2} + (n–1)C_{T }K_{1} – K_{w}} x^{2} + K_{1} {(n–2)C_{T }K_{2} – K_{w}} x – K_{1}K_{2}K_{w} = 0 
This quartic equation (quartic!, not quadratic)^{5} predicts the pH for any given pair of C_{T} and n. Replacing nC_{T} with C_{B} in 3.2, you get an alternative form:
(3.3)  x^{4} + {K_{1} + C_{B}} x^{3} + {K_{1}K_{2} + (C_{B} – C_{T}) K_{1} – K_{w}} x^{2} + K_{1} {(C_{B} – 2C_{T}) K_{2} – K_{w}} x – K_{1}K_{2}K_{w} = 0 
Once the pH value (or x) is known, the concentrations of the acid species, H_{2 }A, HA^{}, and A^{2}, can be calculated via the ionization fractions a_{j} as follows
(3.4)  [H_{2j }A^{j}] = C_{T} a_{j }(x)  for j = 0, 1, 2 
Exact Relationships between pH, C_{T}, and n (or Alk = nC_{T})
The set of five mathematical equations in 1.1a to (1.1e) contains all information about the “diproticacid plus strong base” system (alkalimetric titration). In particular, the actual equilibrium state (i.e. the concentrations of the acid species H_{2}A, HA^{}, and A^{2}) is controlled by two parameters chosen from the triple (C_{T}, n, pH) or (C_{T}, Alk, pH). Once you know two of them, the third is automatically determined:
(4.1a)  pH (C_{T},n)  = –lg x_{1} with x_{1} = positive root of 3.2  [j] = C_{T} a_{j}  
(4.1b)  pH (C_{T},Alk)  = –lg x_{1} with x_{1} = positive root of 3.3  [j] = C_{T} a_{j} 
(4.2a)  n (C_{T},pH)  = a_{1} + 2a_{2} + w/C_{T}  [j] = C_{T} a_{j}  
(4.2b)  Alk (C_{T},pH)  = C_{T }(a_{1} + 2a_{2}) + w  [j] = C_{T} a_{j}  
(4.3a)  C_{T }(n,pH)  = w/(n – a_{1} – 2a_{2})  [j] = [w/(n – a_{1} – 2a_{2})] a_{j}  
(4.3b)  C_{T }(Alk,pH)  = (Alk – w)/(a_{1} + 2a_{2})  [j] = [(Alk – w)/(a_{1} + 2a_{2})] a_{j} 
with w(x) defined in 6. The abbreviation [j] stands for the molar concentration of the three acid species.
To sum up: All formulas presented here are different encodings of one and the same thing, namely the set of equations (1.1a) to (1.1e).
Cross Plots. It’s quite instructive to exhibit the nonlinearity of these equations graphically. The diagrams below display all possible combinations of one dependent and two independent variables (taken from the triplet C_{T}, n, and pH) for the carbonate system:
The diagrams can be redrawn for the case when the ‘stoichiometric coefficient’ n is replaced by the Alkalinity (Alk = n C_{T}):
Other Examples. Application of 4.2a to a pure CO_{2} system provides the (blue) titration curves in the diagram below. On the other hand, for the special case of n = 0, 1, 2, 4.3a was used to plot the pH dependence of three equivalence points of the carbonate system.
Buffer Capacity
Alkalinity is known to be a buffer capacity (and therefore it is also C_{B}). Thus, the titration curves based on 4.2b display the buffer capacity as a function of pH. In this sense, n(pH) = C_{B}/C_{T} can be understood as the “normalized” or unitless buffer capacity. From the above formulas we get:
(5.1a)  buffer capacity (in mol/L):  C_{B}  =  (a_{1} + 2a_{2}) C_{T} + w  
(5.1b)  normalized buffer capacity:  n  =  (a_{1} + 2a_{2}) + w/C_{T} 
Each individual term in these “titration formulas” is pHdependent: a_{1}(x), a_{2}(x), C_{T}(x), and w(x). In particular, 5.1b can be written as:
(5.2)  n (C_{T},x) = 
Buffer Intensity
The buffer intensity is the 1^{st} derivative of the buffer capacity with respect to pH. Again, we distinguish between two types of buffer intensities:
(6.1a)  buffer intensity (in mol/L):  β_{c}  =  dC_{B} / d pH = β C_{T}  
(6.1b)  normalized buffer intensity:  β_{ }  =  dn / d pH 
The steeper the slope of a titration curve the higher is the buffer intensity β = dn/d(pH), i.e. the higher is the system’s resistance to pH changes (caused by a strong base). Thus, the pH where β reaches its maximum represents the optimal buffer range (bounded by pH_{max }±1) – see example below:
Formulas. The 1^{st} derivative of 5.2 with respect to pH yields (see Appendix):
(6.2a)  β = ln 10 {(a_{2} + a_{0}) – (a_{2} – a_{0})^{2}} + ln 10 (w+2x) /C_{T} 
or, more explicitely,
(6.2b) 
Equation (6.2b), that contains positive summands only, tells us that β is always positive (it never becomes negative or zero).
Local Extrema. The local maxima (and minima) of the buffer intensity are at the following points:
x = K_{1}  maximum  ⇒  optimal buffer range 
x = (K_{1}K_{2})^{1/2}  minimum  
x = K_{2}  maximum  ⇒  optimal buffer range (small C_{T} excluded) 
Please note that this is not strictly exact, but it’s a very good approximation. Mathematically, local extrema of β are found where the firstorder derivative of β is zero.
Example: The buffer intensity of a pure CO_{2} system is displayed as green curve in the diagram below.
Buffer Capacity. The buffer capacity is just the integral of the buffer intensity:
(6.3)  buffer capacity = 
Derivative of the Buffer Intensity
The derivative of the buffer intensity is given by:
(7.1) 
with the abbreviation
(7.2)  f_{ }(x) = (x/K_{1} – K_{2}/x) (x/K_{1} + K_{2}/x – 1 + 8 K_{2}/K_{1}) 
Example: The derivative of the buffer intensity of a pure CO_{2} system is displayed as red curve in the diagram below. The zeros of dβ/d(pH) are marked by blue circles.
Example Calculation for the Carbonate System
Given is a pure CO_{2} system with C_{T} = 100 mM, 10 mM, and 1 mM H_{2}CO_{3}.^{6} Three quantities are plotted as function of pH:
• n(pH)  (normalized) amount of base added^{7}  (as blue curve) 
• β = dn/dpH  (normalized) buffer intensity  (as green curve) 
• dβ/dpH  derivative of the buffer intensity  (as red curve) 
All three quantities as per definition are unitless. The small blue dots mark the zeros of dβ/d(pH).
Since the (blue) titration curve is an everincreasing function, its derivation, i.e. the buffer intensity β, is always positive (see green curve). This is in full accord with Le Châtelier’s principle that every solution resists pH changes.
Appendix – Derivatives with Respect to pH
The first and second derivatives of x = [H^{+}] with respect to pH are
(A1)  dx/d(pH) = (–ln 10) x  and  d^{2}x/d(pH)^{2} = (–ln 10)^{2} x 
This result can be used to differentiate any given function f(x) with respect to pH (by application of the chain rule):
(A2) 
Hence, for w(x) = K_{w}/x – x, introduced in 6, we get the amazing result:
(A3a)  dw/d(pH)  = ln 10 (K_{w}/x + x)  = ln 10 (w + 2x)  
(A3b)  d^{2}w/d(pH)^{2}  = (ln 10)^{2} (w + 2x – 2x)  = (ln 10)^{2} w 
All derivatives of higher degree repeat this pattern:
(A3c) 
Ionization Fractions. The three ionization fractions, introduced in 2.3a to (2.3c), obey the following relations (where the sum runs from i = 0 to 2):
(A4a)  Y_{0} ≡ Σ_{i} a_{i}  = a_{0} + a_{1} + a_{2}  = 1  
(A4b)  Y_{1} ≡ Σ_{i} i a_{i}  = 0⋅a_{0} + 1⋅a_{1} + 2⋅a_{2}  = a_{1} + 2 a_{2}  
(A4c)  Y_{k} ≡ Σ_{i} i^{k} a_{i}  = 0⋅a_{0} + 1⋅a_{1} + 2^{k}⋅a_{2}  = a_{1} + 2^{k }a_{2}  (for all k > 0) 
Quantities so defined are also called kth moments. Here in our case, Y_{0} represents the mass balance, Y_{1} enters the alkalinity (as its main part), and Y_{2} together with Y_{3} will be used for the buffer intensity and its derivative. All Y_{k} (excluding Y_{0}) are positive numbers, living in the range 0 < Y_{k} < 2^{k}.
The first derivative of the ionization fractions is given by
(A5)  da_{i }/d(pH) = ln 10 (i – a_{1} – 2a_{2}) a_{i} = ln 10 (i – Y_{1}) a_{i}  for i = 0, 1, 2 
Applying it to the two sums in A4a and (A4b) yields
(A6a)  (d/d pH) Σ_{i} a_{i}  = ln 10 Σ_{i} (i – Y_{1}) a_{i}  = ln 10 (Y_{1} – Y_{1}) = 0  
(A6b)  (d/d pH) Σ_{i} i a_{i}  = ln 10 Σ_{i} i (i – Y_{1}) a_{i}  = ln 10 (Y_{2} – Y_{1}^{2}) 
The first relation, which delivers zero, is obvious because it represents the derivation of a constant, namely d^{ }1/d(pH) = 0. In general we have
(A7)  d Y_{k }/d pH = ln 10 Σ_{i} i^{k} (i – Y_{1}) a_{i} = ln 10 (Y_{k+1} – Y_{1}Y_{k}) 
The second derivative of Y_{1} is given by
(A8a)  =  
= (ln 10)^{2} (Y_{3} – Y_{1}Y_{2} – 2Y_{1} (Y_{2} – Y_{1}^{2}))  
= (ln 10)^{2} (Y_{3} – 3Y_{1}Y_{2} + 2Y_{1}^{3}) 
and the third derivative of Y_{1} is
(A8b)  = 
= (ln 10)^{3} {(Y_{4}–Y_{1}Y_{3}) – 3Y_{2} (Y_{2}–Y_{1}^{2}) – 3Y_{1} (Y_{3}–Y_{1}Y_{2}) + 6Y_{1}^{2 }(Y_{2}–Y_{1}^{2})} 
Buffer Intensity & Co. The above equations can now be employed to derive the buffer intensity and its derivative introduced in 6.2 and 7.1. In particular, we have:
(A9a)  n  =  (Y_{1} + w/C_{T})  
(A9b)  β  = d /d pH  (Y_{1} + w/C_{T})  
(A9c)  dβ/d pH  = d^{2}/d (pH)^{2}  (Y_{1} + w/C_{T}) 
which yields
(A10a)  n  = (Y_{1} + w/C_{T})  
(A10b)  β  = (ln 10) {(Y_{2} – Y_{1}^{2}) + (w+2x) /C_{T}}  
(A10c)  dβ/d pH  = (ln 10)^{2} {(Y_{3} – 3Y_{1}Y_{2} + 2Y_{1}^{3}) + w/C_{T}} 
If desired, the Y’s can be replaced by the a’s. For example, the central piece of A10b is given by:
(A11)  Y_{2} – Y_{1}^{2} = (a_{1} + 4a_{2}) – (a_{1} + 2a_{2})^{2} = (a_{2} + a_{0}) – (a_{2} – a_{0})^{2} 
Here, in the last relation A4a was applied.
Remarks & Footnotes

This topic is also discussed in the PowerPoint presentation. A detailed mathematical description is provided as pdf. ↩

To be correct, pH is based on the activity of H^{+}: pH = –log {H^{+}}. ↩

Here and in the following we assume that all activities (expressed by curly braces) are replaced by molar concentrations (expressed by square brackets). This is legitimate for dilute systems or by switching to conditional equilibrium constants. ↩

The equilibrium constant K can also be expressed by pK = –log K. ↩

There are algebraic formulas for quadratic, cubic and quartic equations (4th order polynomial), but the effort for the last two is enormous. (By the way: 5th order and higher polynomials are algebraically already not solvable any more.) ↩

The two equilibrium constants of the carbonic acid are: K_{1} = 10^{6.35} and K_{2} = 10^{10.33} (see also here). ↩

The formula allows also negative values of n. This mimics the withdrawal of the strong base from the solution or the addition of a monoprotic acid (e.g. HCl). ↩