30 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS • k7* \So -- a, (but see the Appendix) (5) where vc is the inoculum volume which reduces si from its initial value to Sc, the critical value below which the preservation criteria are not satisfied. Since v is the total volume of the microbe inoculum (Lang and Rye presumably report their internal concentration data on this basis) the ratio v/V can be converted to the number of microbes per unit volume by dividing by the volume of a single microbe. To use eq 5 to compute capacities, we need values for the microbiological constants k* and Sc in addition to the solubilities, ai. The work reported by Lang and Rye can be used to calculate consistent sets of these constants. These authors found intra- to extracellular concentration ratios of 3.5, 6.4 and 18.5 for methyl, ethyl and propyl parabens over a wide range of external concentrations equitoxic to Escherichia co/i. According to eq 1, this ratio is given by: = k*/G (6) Using the solubilities of these three parabens as given in the trade literature (2.5, 1.6 and 0.5 g/l) the corresponding values of the Ferguson constant, k*, are 8.8, 10.2 and 9.3 g/1. The mean value, 9.4 g/l, was used in eq 5 to compute the capacities shown in Table I for a 50ø70 reduction in the growth constant which occurred at an average external saturation fraction of 0.27 (Sc in eq 5). The cell volume of E. co/i was taken as 5 cubic microns. The initial concentrations chosen for Table I are the respective saturation levels for the first four alkyl parabens. Capacities for other initial concentrations are shown graphically in Figure 3. This figure shows that at an initial concentration of 1.0 g/l, for example, the capacities of methyl and ethyl paraben are about 2.6 and 4.3 ) 10•ø/cm3 the propyl and butyl esters would have capacities of 6.7 and 7.6 ) 10•ø/cm -• on the same basis if they were soluble to this extent (they are not, but we may suppose that these capacities would be realized if the excess over their solubilities were well dispersed in the medium). Figure 4 shows the capacities predicted by eq 5 for Sc = 0.5 which is roughly the mean saturation corresponding to the concentrations at which the growth rate constant is zero in Lang and Rye's Figure 1 (1). Since this corresponds to complete inhibition of growth, it is closer to a practical criterion of good preservation. The slopes of the Table I Capacities of Alkyl Parabens for 50% Reduction of Growth Rate Constant of E. Co/i in Water at 25øC N• X 10-9/ml Initial Conc., g/1 Paraben Soly., g/l 0.2 0.5 1.6 2.5 Methyl 2.5 nil nil 73 144 Ethyl 1.6 nil 5.4 92 163 a Propyl 0.5 5.1 29 115 a 186 a Butyl 0.2 12 35 a 122 a 193 a •Initial concentration greater than solubility.

PARABENS 31 [Vie lOO Et 50 Pr -50 11 ! •l ii I I i I I I I I I I I I I I I I ! I ! 1.o 1.5 concenlralion g/L. t 2.0 2.5 Figure 3. Capacity of parabens to cause 50% reduction in the growth rate constant of E. coli in water at 25øC.