\[ \mu_{w}\left(l,x_{w},p+\pi\right)=\mu_{w}^{0}\left(l,p+\pi\right)+\nu RT\ln a_{w}
\]
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.
