\mu_{w}\left(l,x_{w},p+\pi\right)=\mu_{w}^{0}\left(l,p+\pi\right)+\nu RT\ln a_{w}
where, \mu_w is the chemical potential of solvent, which is a function of solvent, l, mole fraction of solvent x_w, external pressure, p, and additional osmotic pressure exerted by solutes, \pi (whereas, for the pure solvent, x_w goes away); \nu is the dissociation coefficient (a solute dissociates into \nu ions); R is the gas constant; T is temperature in consistent with R; a_w is water activity.
The addition to the pressure is expressed through the expression for the energy of expansion:
\mu_{w}^{o}(l,p+\pi)=\mu_{w}^{0}(l,p)+\int_{p}^{p+\pi}V\mathrm{d}p
where, V is the molar volume of solvent.
To balance the chemical potentials of solvent for two solutions with and without solute separated by semipermeable membrane, \mu_{w}\left(l,x_{w},p+\pi\right) and \mu_{w}\left(l,p\right),
-\nu RT\ln a_{w}=\int_{p}^{p+\pi}V\mathrm{d}p
Thus, osmotic pressure is expressed as opposed to pure solvent (water), in which water activity is 1,
\pi=-\nu RT/V\ln a_{w}
Conventionally, the gradient of osmotic pressure is approximated using the concentration or the mass fraction of solute (salt). The derivation is shown as follows.
A definition of water activity is,
a_{w}=l_{w}x_{w}=l_w(1-x_s)
where, l_w is activity coefficient; and x_w is the mole fraction of water in aqueous fraction (x_s=1-x_w is the mole fraction of solute). By Raoult's law, l_w is usually approximated as unity in dilute solution.
Using Taylor series, \ln l_w(1-x_s) \approx -x_s.
Thus, the osmotic pressure becomes (C is the molar concentration of solute),
\pi=-\nu RT/V\ln a_{w} = \nu RT/V x_s = \nu RT C
Also, the relation between the gradients of osmotic pressure and chemical potential is,
\nabla\pi=-\frac{1}{V}\cdot\nabla\mu_{w}
The relation between the gradient of osmotic pressure and the concentration or the mass fraction of solute (salt) becomes,
\nabla\pi\approx\nu RT\nabla C=\nu RT\frac{\rho}{M_{s}}\nabla X
where, \rho is the solution density; M_s is molar mass of solute; and X is the mass fraction of solute.
where, \mu_w is the chemical potential of solvent, which is a function of solvent, l, mole fraction of solvent x_w, external pressure, p, and additional osmotic pressure exerted by solutes, \pi (whereas, for the pure solvent, x_w goes away); \nu is the dissociation coefficient (a solute dissociates into \nu ions); R is the gas constant; T is temperature in consistent with R; a_w is water activity.
The addition to the pressure is expressed through the expression for the energy of expansion:
\mu_{w}^{o}(l,p+\pi)=\mu_{w}^{0}(l,p)+\int_{p}^{p+\pi}V\mathrm{d}p
where, V is the molar volume of solvent.
To balance the chemical potentials of solvent for two solutions with and without solute separated by semipermeable membrane, \mu_{w}\left(l,x_{w},p+\pi\right) and \mu_{w}\left(l,p\right),
-\nu RT\ln a_{w}=\int_{p}^{p+\pi}V\mathrm{d}p
Thus, osmotic pressure is expressed as opposed to pure solvent (water), in which water activity is 1,
\pi=-\nu RT/V\ln a_{w}
Conventionally, the gradient of osmotic pressure is approximated using the concentration or the mass fraction of solute (salt). The derivation is shown as follows.
A definition of water activity is,
a_{w}=l_{w}x_{w}=l_w(1-x_s)
where, l_w is activity coefficient; and x_w is the mole fraction of water in aqueous fraction (x_s=1-x_w is the mole fraction of solute). By Raoult's law, l_w is usually approximated as unity in dilute solution.
Using Taylor series, \ln l_w(1-x_s) \approx -x_s.
Thus, the osmotic pressure becomes (C is the molar concentration of solute),
\pi=-\nu RT/V\ln a_{w} = \nu RT/V x_s = \nu RT C
Also, the relation between the gradients of osmotic pressure and chemical potential is,
\nabla\pi=-\frac{1}{V}\cdot\nabla\mu_{w}
The relation between the gradient of osmotic pressure and the concentration or the mass fraction of solute (salt) becomes,
\nabla\pi\approx\nu RT\nabla C=\nu RT\frac{\rho}{M_{s}}\nabla X
where, \rho is the solution density; M_s is molar mass of solute; and X is the mass fraction of solute.