Quantification of osmosis (ideal systems): Van’t Hoff equation

5.15 Quantification of osmosis (ideal systems): Van’t Hoff equation

To quantify osmosis we start from the equilibrium condition

\begin{equation*} \mu_A^*(p)=\mu_A(solution,p+\pi) =\mu_A^*(p+\pi) + R\,T\, \ln a_A \label{eq:osmosis_eq_2} \end{equation*}

(5.47)

To calculate $\mu_A^*(p+\pi)$ we use the pressure dependence of $G_{A,m}$

\begin{equation*} \begin{split} \left(\frac{\partial \mu_A^*}{\partial p} \right)_T &=\;\;\left(\frac{\partial G_{A,m}}{\partial p} \right)_T = V_{A,m}^* \quad \Rightarrow \quad \int_p^{p+\pi} d\mu_A^* = \int_p^{p+\pi} V_{A,m}^* dp\\ \mu_A^*(p+\pi) &=\;\; \mu_A^*(p) + V_{A,m}^* \int_p^{p+\pi}dp = \mu_A^*(p) + V_{A,m}^*\;\pi \quad \mbox{assuming $V_{A,m}^*$ independent of $p$}\\ \end{split} \label{eq:osmosis_mu_p} \end{equation*}

(5.48)

Including this result into the equilibrium condition of Eq. (5.47) we find

\begin{equation*} \begin{split} \mu_A^*(p) &=\;\;\mu_A(solution,p+\pi) + R\,T\, \ln a_A = \mu_A^*(p) + V_{A,m}^*\;\pi + R\,T\, \ln a_A \\ \Rightarrow - V_{A,m}^*\;\pi &=\;\; R\,T\, \ln a_A\\ \end{split} \label{eq:osmosis_eq_3} \end{equation*}

(5.49)

i.e. the activity of the solvent can be determined from measurements of $\pi$.
Using the approximations for Raoult’s standard state applied to $R\,T\, \ln a_A$ we get

\begin{equation*} \begin{split} R\,T\, \ln a_A &\approx\;\; R\,T\, \ln x_A = R\,T\, \ln (1-x_B) \approx - R\,T\, x_B = - R\,T\, \frac{n_B}{n_A+n_B} \approx - R\,T\, \frac{n_B}{n_A}\\ \Rightarrow - \pi \;V_{A,m}^* &= -\pi \frac{V_{A}^*}{n_A} = \;\; R\,T\, \ln a_A \approx - R\,T\, \frac{n_B}{n_A} \quad \Rightarrow \quad \pi V_{A}^* = n_B \,R \,T\\ \end{split} \label{eq:osmosis_eq_4} \end{equation*}

(5.50)

Figure 5.14: Representation of measured data according to Eq. (5.52).

Rewriting Eq. (5.50) we get the van’t Hoff equation

\begin{equation*} \pi = \frac{n_B}{V_{A}^*} \,R \,T = \frac{m_B}{M_B\,V_{A}^*} \,R \,T = \frac{\,R \,T}{M_B} \frac{m_B}{V_{A}^*}= \frac{\,R \,T}{M_B} c_B \label{eq:osmosis_eq_5} \end{equation*}

(5.51)

here $c_B$ is a modified concentration in [g/cm$^3$].
Measuring the osmotic pressure $\pi$ as a hydrostatic pressure $\rho_A\, g\, h$ ($\rho_A$: density of the solvent) we get a generalization of Eq. (5.51) by a virial approach

\begin{equation*} \begin{split} \pi = \rho_A \,g\, h &=\;\; \frac{\,R \,T}{M_B} c_B \left(1 + \frac{B\, c_B}{M_B} + \cdots \right) \\ \Rightarrow \quad \frac{h}{c_B} & =\;\;\frac{R\,T}{\rho_A\, g\, M_B} + \left( \frac{R\,T\,B }{\rho_A\, g\, M_B^2} \right)\, c_B \, + \cdots \end{split} \label{eq:osmosis_eq_6} \end{equation*}

(5.52)

Plotting of $h/c_B$ against $c_B$ typically gives a straight line as shown in Fig. 5.14 . The extrapolation of this line to $c_B = 0$ allows to calculate $R\,T\,/\rho_A\,g\,M_B$ and thus finally $M_B$. The $B$ parameter contains additional information about the intermolecular interactions.