Spreadsheet #17:  Titration of a Weak Acid with a Strong Base (cont.) - Using Gran's Plot to Determine the Endpoint and Ka

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The titration data for this spreadsheet is identical to the prior spreadsheet.  However, you will use a different graphical method to determine the endpoint and pKa.  Recall the dissociation of a weak acid is represented by

Equation 1:           HA  =  H+   +    A-,       Ka = [H+]*[A-] / [HA]

The ratio of  [A-] / [HA] may be expressed as

Equation 2:          [A-] / [HA]  =  ( Vb)  /   ( Veq - Vb )

Where Vb is the volume of base added and Veq is the volume of base added at equivalence (endpoint).  Substituting Eqn 2 into Eqn 1 yields

Equation 3:         Ka  =  [H+] * [ Vb  /  (Veq - Vb) ]

Rearranging yields a linear equation:

Equation 4: Vb*[H+]  =  -Ka*Vb + Ka*Veq

   Linear Eqn:            y          =    m * x  +      b

So, by plotting Vb on the x-axis, and Vb*[H+] on the y-axis, the slope is -Ka and the y-intercept is Ka*Veq.  If we follow the plot down to the x-axis intercept, where y = 0, then

Equation 5:        Ka*Vb   =  Ka*Veq

which simplifies to  Vb = Veq.    This means that the volume of base added at the x-axis intercept is equal to the volume of base at the endpoint of the titration.

This relationship is linear and well-behaved for most of the titration data, however the linearity is lost at the extremes near the beginning and endpoint of the titration.  So any linear fitting with regression, etc., must be done on a selected set of the data, ignoring the first few and the later data points near the endpoint.  However, it is interesting that extrapolation of the regression to the x-axis intercept means that an endpoint for such an acid-base titration can be determined without ever adding enough base to reach the endpoint!

Download the data from the titration.  This time, calculate [H+] from the pH and also determine Vb*[H+] at each point.  Plot these data as described above and then fit data points #20-#50 with a best-fit line that  extends to both axes.  Calculate the volume of base needed for equivalence and also report the value of Ka and pKa.  (Click here to download data:  17pHGran )

Spreadsheet # 17
Titration Curve for Weak Acid - Strong Base
Gran's Plot to Determine Endpoint and pKa
#	Vb	pH	[H]	Vb*[H+]	Linear Regression For Data Points 20 - 70:
1	0.004	2.890	.	.	Slope:	.	.
2	0.048	2.964	.	.	y-Intercept	.	.
3	0.094	3.038	.	.	R^2:	.	.
4	0.142	3.112	.	.	x-Intercept:	.	.
5	0.194	3.186	.	.	Ka =	.	.
6	0.251	3.260	.	.	EndPoint =	.	mL
7	0.314	3.335	.	.
...	...	...	...	...
129	16.211	12.375	.	.
130	17.818	12.449	.	.
131	20.000	12.523	.	.