Water Transport

Abstract

The passage of water through artificial or biological membranes is, considering its apparent simplicity, a process that has received and is receiving a great deal of theoretical attention and yet remains poorly understood. A basic example that is of great historic importance is that of a membrane permeable to water but impermeable to a solute. The facts are familiar: Water passes spontaneously across the membrane from the less to the more concentrated solution. This flow was, in the 19th century, attributed to a so-called osmotic pressure that was taken to be equal to the magnitude of the external pressure, which, when applied to the more concentrated solution, just prevents the flow of water. A quantitative theoretical relationship between the equilibrium osmotic pressure and the composition of the solutions on either side of the membrane is easily derived. The chemical potential of water is written:

$${{\mu }_{w}}=\mu _{w}^{0}+{{\bar{V}}_{w}}P+RT\ln {{x}_{w}}$$

(16.1)

where the symbols have the meanings they had in (11.5) and we have assumed ideality. At equilibrium the chemical potential is the same on both sides of the membrane:

$$\mu _{w}^{0}+{{\bar{V}}_{w}}P\left( 1 \right)+RT\ln {{x}_{w}}\left( 1 \right)=\mu _{w}^{0}+{{\bar{V}}_{w}}P\left( 2 \right)+RT\ln {{x}_{w}}\left( 2 \right)$$

(16.2)

where 1 and 2 label the two solutions. Rearranging we find:

$$P(2) - P(1) = \Delta \pi = \frac{{RT}}{{\mathop {{V_w}}\limits^ - }}\ln ({x_w}(1)/{x_w}(2))$$

(16.3)

In dilute solution well-worn approximations, accessible in most undergraduate textbooks of physical chemistry, yield the equation originally due to van’t Hoff:

$$\mu _{w}^{0}+{{\bar{V}}_{w}}P\left( 1 \right)+RT\ln {{x}_{w}}\left( 1 \right)=\mu _{w}^{0}+{{\bar{V}}_{w}}P\left( 2 \right)+RT\ln {{x}_{w}}\left( 2\right)$$

(16.2)

where 1 and 2 label the two solutions. Rearranging we find:

$$\Delta \pi = RT({c_s}(2) - {c_s}(1)) = RT\Delta {c_s}$$

(16.4)

where c^s is the concentration of solute.