G_aG_b = F_aF_b · N_bN_a · M_aM_b

where G_a and G_b are the parts by weight of the components a and b, in this example water and alcohol respectively, and F_a and F_b the areas of the signals of H₂O and CH₂. N_a and N_b are the numbers of nuclei that cause the signals, in this case 2 and 2 respectively, and M_a and M_b are the molecular masses of the two components, 18 and 46 respectively. Using the measured integrals (1152, 202.6) the water/alcohol ratio by weight is calculated to be 2.029/1; in other words the vodka contains 33.0% alcohol by weight or, taking into account the density of the actual ethanol–water mixture, 39.6% by volume. This is in good agreement with the alcohol content given on the label (Bison Brand Vodka 40° by volume). Because the OH signal contains both water and ethanol OH protons, one has to subtract the intensity corresponding to one proton from the value of the integral of water before calculating this ratio.

7. Comments

The integrals of NMR signals are relative measures of the numbers of resonating nuclei. If one component is present at much lower concentration than another, the percentage error in measurement of the quantity of this minor component could be quite high (see Exp. 3.12). For quantitative determinations, deconvolution and curve-fitting methods have been proposed in the literature, especially for cases where peaks are not fully resolved. Curve-fitting procedures are often included in the commercially available software packages.

8. Own Observations