Chemical shifts are a sensitive probe of protanation states of titratable groups, and they can be used to measure the pKas of weak acids and bases.

## General Considerations

1. Shortened NMR tube. pH electrodes that can fit inside NMR tubes tend to be too short for the probe to reach the bottom of the tube and make contact with the sample. The top of an NMR tube (1-2 inches) can be cut off with the help of a glass cutter.

2. Weak acids and bases. Adding an aliquot of strong acid or base (HCl or NaOH) can push the pH of the solution to extremes. A weak acid or base with a pKa in the vicinity of the endpoint pH of the titration avoids this problem.

3. Sample volume. If using a standard NMR tube, make sure that the sample volume is large enough to fill the receiver coil length. You will likely lose some sample from inserting the pH probe into the NMR sample; add volume to your initial sample accordingly. I use about 350ul for a 5mm NMR tube.

4. Capillary to add aliquots. It’s usually easiest to titrate directly in the NMR tube, and a capillary tube can be used to add each aliquot. The capillary tube can be affixed to the end of a micropipettor tip directly, or a short piece of flexible tubing (1-2 inches) can be used to join the capillary to a micropipette tip.

5. Centrifuge thoroughly. If you are adding acid/base aliquots directly to the NMR tube, centrifuge the tube thoroughly (for a minute) before mixing. Titrant that falls into the NMR sample after the pH has been measured can skew results–this is particularly problematic for titrants of high concentration and low volume. Measure the pH before and after the NMR experiment. Swing-bucket centrifuges can break the NMR tube if spun too quickly–spin slowly and for a long time.

6. Vortex. Mix the contexts of the tube with a vortexer on high. This is done by shutting off the vortexer, placing the NMR tube in the head of the vortexer, then turning it on.

## Titration of 13C groups

1. D2O titrations. Backbone Hα groups resonate at the same 1H frequency as 1H2O, and titrations in D2O make the spectroscopy easier. Dissolve your titrant in D2O, lyophilize, then dissolve again in 100% D2O. Rinse your NMR tube with 99% D2O and try thoroughly. Calibrate the pD offset of the pH electrode with a reference buffer solution in H2O and D2O–ex : 50mM Tris pH 7.40 in H2O or D2O, diluted from a 1M stock. Before every measurement, soak the pH electrode in KCl/D2O for an hour, then rinse with D2O.

## Titration of 15N groups

1. HN titrations. Above pH ~4, the rate of chemical exchange for amide protons increases by a factor of 10 with every pH unit. Consequently, HN resonances are broadened by the exchange rate to the point that amide signals can no longer be seen in an 15N-HSQC.

## Fitting the Data

For a titratable group with the following equilibrium :

$\ce{HA <=> A^- +H_3O^+}$

The average chemical shift of the two exchanging species in the fast exchange regime can be written :

$CS = CS_{HA} \cdot \chi_{HA} + CS_{A} \cdot (1 - \chi_{HA})$

The average chemical shift is a weighted sum (average) of the mole fractions of the acid (HA) and conjugate base (A) species, in the fast-exchange regime. The Hendersen-Hasselbalch equation for a buffer solution can also be written :

$pH = pK_a + log \left( \frac{\chi_A}{\chi_HA} \right)$

And solving for the mole-fraction of HA, and inserting into the equation for the average chemical shift yields:

$CS = \frac{CS_A +CS_{HA} \cdot 10^{pK_a - pH}}{1+10^{pK_a-pH}}$