Salt Concentration

Salt concentration from pulse width (p1) (1) [Adapted from reference 2.]

Where P₉₀ = 90 degree pulse width for sample with salt,

P_{90 no salt}= 90 degree pulse width for aprotic solvent

d = inner diameter of the NMR tube (4.2mm for a 5mm tube)

Signal-to-noise ratio for a sample (2)

Where Rc = Resistance of the RF coil,

Tc = Temperature of the coil,

Ta = Temperature of the preamplifier,

Rs = Resistance of the sample,

Ts = Temperature of the sample,

Sample resistance (3)

Where ω^2= angular frequency,

r_s = sample radius, (2.1mm for a 5mm tube)

σ= conductivity of the sample

Sample conductivity (4)

Where c = concentration of the ionic species,

q = magnitude of charge of the ionic species,

l = mobility of the ionic species

Here the ionic mobility itself can be calculated from Diffusion coefficient by the following eqn: (5)

Where D = diffusion coefficient,

q = magnitude of charge of the ion,

k = Boltzmann’s constant,

T = absolute temperature.

Thus Rs can now be calculated from:

Sensitivity factor (6)

For Ts = 298 K,

If Ta = 15 K

Tc = 27 K

Then the equation (5) can be simplified to: (7)

For low buffer concn, the sample resistance (Rs) is small and becomes negligible relative to 1, hence can be ignored and for such samples, L~ 1 (max sensitivity).

For high salt concn, Rs becomes dominant and the factor 7.45(Rs/Rc) becomes large relative to 1, hence L~ Rs which itself is ~ cql of the ionic species. Thus

L ~ √ l for two buffers with different ionic mobility (8)

If we have the values for D and Rc, we can calculate the sensitivity factor, by using eqn 3,4 and 5 to calculate Rs . The factor L can vary between 0-1, with 1 being highest achievable sensitivity.

Reference:

1. Low-Conductivity Buffers for High-Sensitivity NMR Measurements. Alexander E. Kelly, JACS, 2002.

2. Performance of cryogenic probes as a function of ionic strength and sample tube geometry. Markus W. Voehler, JMR, 2006.

3. Improving NMR sensitivity in room temperature and cooled probes with dipolar ions. Andrew N. Lane, JMR, 2005.