J. Phys. Chem. B 2008, 112, 15941–15942
15941
COMMENTS Comment on “Osmotic Pressure beyond Concentration Restrictions”
nonstationary diffusion through the system of interest is equal to
Andriy Yaroshchuk tbt ≡
ICREA and Department of Chemical Engineering, Polytechnic UniVersity of Catalonia, 08028 Barcelona, Spain ReceiVed: April 14, 2008; ReVised Manuscript ReceiVed: August 1, 2008 Several months ago, in this journal, A. Grattoni, M. Merlo, and M. Ferrari1 reported on the application of a novel theory to the interpretation of their measurements of osmotic pressure in aqueous solutions of sucrose up to very high concentrations. Generally, a good agreement between the theory and experiment was found. However, the experimental data were systematically somewhat smaller than the theoretical predictions. By multiplying the theoretical predictions by 0.927, the resulting curve overlaid the measured data, with a standard deviation of 2.5% and a correlation factor of R ) 1. This small but consistent deviation of experimental data from the theoretical predictions was not explained. The purpose of this comment is to put forward a simple explanation for this observation. In the experimental setup used in ref 1, a sintered stainless steel porous disk was used to support the membrane and prevent it from bursting. (Reverse osmosis membranes themselves also have a multilayer structure and consist of semipermeable barrier layers and relatively coarse porous supports. However, these “own” membrane supports are much thinner than the porous disk, and their presence can be neglected in the first approximation.) During the experimentation, a tiny sucrose flow into the “solvent” compartment was observed, but no more specific information on the flow magnitude was provided. Nevertheless, this means that there was a solution concentration gradient across the porous disk. Accordingly, the sucrose molality at the interface between the membrane and the disk was not exactly zero, contrary to the assumption of a pure solvent on one side of the membrane made while using eq 161 for the theoretical estimates of osmotic pressure. Osmotic pressure arises due to a solute concentration (or molality) gradient across semipermeable layers alone. If this gradient is actually smaller than the assumed one, the experimental data will deviate downward from the theoretically predicted data, as observed experimentally. In a hypothetical steady state, the diffusion solute flux through the membrane must be equal to its flux through the disk. From this, it follows that the activity at the membrane/disk interface is equal to
ai ) a′ ·
Pm Pd + Pm
(1)
where a′ is the solute activity and Pm and Pd are the diffusion permeabilities of the membrane and porous disk. However, most probably a steady state was not achieved in experiments described in ref 1. Indeed, by using the approach developed in ref 2, it can be shown that the so-called breakthrough time for
tch 1 + Pm /3Pd · 2 1 + Pm /Pd
(2)
γ · L2 Deff
(3)
where
tch ≡
is the classical relaxation time for nonstationary diffusion, γ is the porous disk porosity, L is its thickness, Deff is the effective diffusion coefficient of the solute in the disk including the effects of finite porosity and pore tortuosity. [The problem of nonstationary diffusion in ref 2 was solved by means of Fourier transforms in the approximation of the ideal solution (otherwise, no analytical solution could be obtained). The sucrose solutions used in ref 1 were far from ideal. Nevertheless, from Figure 2 of the commented paper, it is seen that the measured osmotic pressure was always larger than that predicted in the approximation of ideal solution. Accordingly, the driving force for diffusion was larger, and the estimates below can be considered semiquantitative underestimates.] From, eq 2, it is seen that the dependence of breakthrough time on the ratio of diffusion permeabilities of the membrane and porous disk is not very strong. According to ref 3, the disk porosity was ∼45%, and its thickness was 3.1 mm. It is difficult to know precisely the pore tortuosity, but a factor of 2 appears to be a reasonable estimate. The diffusion coefficient of sucrose in aqueous solutions is ∼5 × 10-10 m2/s. By using these values, the characteristic relaxation time can be estimated at ∼4 × 104 s. Accordingly, the breakthrough time could be between 2 and 5 h. The typical measurement time in ref 1 was as short as ∼1 h. Therefore, it is not surprising that only trace amounts of sucrose were detected in the “solvent” compartment; within the typical measurement times, the solute concentration gradients could be essentially confined to the porous disk and did not reach its boundary with the “solvent” compartment. Further, by using the approaches developed in ref 2, it can be shown that at short times (compared with the breakthrough time), the time evolution of solute concentration at the membrane/ disk interface can be modeled as if the disk were semi-infinite. In this limiting case, we can use the results obtained in ref 4 for the so-called in-diffusion mode. For the purpose of present rough estimates, we can also assume that the “solution” compartment was very (formally, infinitely) large. In this case, for the time evolution of the solute concentration at the membrane/disk interface, we obtain this
( ( ) ) ( )
ci(t) t Pm · ) 1 - exp c′ tch Pd
10.1021/jp8031995 CCC: $40.75 2008 American Chemical Society Published on Web 11/14/2008
2
· erfc
t Pm · tch Pd
(4)
15942 J. Phys. Chem. B, Vol. 112, No. 49, 2008
Comments
where erfc is the complementary error function. Above, we have seen that the measurement times were probably as short as one tenth of the characteristic time. Below, we shall see that the ratio Pm/Pd was rather small, too. At sufficiently small values of the argument, eq 4 reduces to this
ci(t) 2Pm · ≈ c′ √πPd
t tch
(5)
According to our interpretation, after ∼1 h, the ratio of sucrose concentration at the membrane/disk interface to that in the “solution” compartment was 0.073. By using this, and the characteristic relaxation time of ∼10 h, from eq 5, we can estimate the ratio of diffusion permeabilities of the membrane and porous disk to obtain
Pm ≈ 0.2 Pd
(6)
Proceeding from the information on the properties of the porous disk,3 we obtain
Pd ≈ 4 × 10-8 m/s
(7)
Pm ≈ 8 × 10-9 m/s
(8)
and
according to eq 6. Within the scope of the so-called solutiondiffusion model, at a transmembrane volume flow of 7 µm/s, this solute permeability corresponds to the solute rejection of 99.9% under reverse osmosis conditions. According to ref 5, the rejection of NaCl at such a permeation flow by the membranes used in this study (AD of GE Osmonics3) is 99.5%. Due to the larger molecular weight, the rejection of sucrose by RO membranes should be higher than that of NaCl, which is in agreement with our estimates. Apparently, the time of each single measurement was not kept the same in ref 1, which could bring about a scattering of the results. Although the standard deviation of experimental data from the maximum likelihood regression line was just 2.5% (compared to the value of osmotic pressure), the deviation between the theory and the experiment was scattered much stronger (∼35%). According to eq 5, this 35% scattering corresponds to ∼50% scattering of the measurement time. In a setup where attempts were made to keep the measurements as short as possible by “presetting” the hydrostatic pressure difference but where the outcome of these attempts depended on poorly controllable details, this estimate of scattering of the measurement times does not appear to be insensible. References and Notes (1) Grattoni, A.; Merlo, M.; Ferrari, M. J. Phys. Chem. B 2007, 111, 11770–11775. (2) Yaroshchuk, A.; Glaus, M. A.; Van Loon, L. R. J. Membr. Sci. 2008, 319, 133–140. (3) Grattoni, A. 2008, personal communication. (4) Yaroshchuk, A.; Van Loon, L. R. J. Contam. Hydrol. 2008, 97, 67– 74. (5) Sterlitech Corporation. http://www.sterlitech.com/products/bench/ flatMembranes/ordering.htm.
JP8031995
