Chapter 5, Problem 5.10IA

Interpretation Introduction

Interpretation: The osmotic pressure of a real solution is given by $\Pi V=-RT\ln a_{\text{A}}$ has to be shown.  At low concentrations the expression takes the form of $\Pi V=\varphi RT\left[\text{B}\right]$ has to be shown.

Concept introduction: Osmotic pressure is the pressure that is applied to stop the flow of solvent molecules from a region of high concentration to a region of low concentration

across a semi permeable membrane.

Answer to Problem 5.10IA

The osmotic pressure of a real solution is given by $\Pi V=-RT\ln a_{\text{A}}$.  At low concentrations the expression takes the form of $\Pi V=\varphi RT\left[\text{B}\right]$.

Explanation of Solution

Cell I consists of pure solvent while cell II consists of concentrated solution and osmosis occurs from cell I to cell II.

At equilibrium the fugacity of pure solvent in cell I $\left(f_{\text{solvent}}^{\text{I}}\right)$ is equal to the fugacity of the solvent at cell II $\left(f_{\text{solvent}}^{\text{II}}\right)$

    $f_{\text{solvent}}^{\text{I}}=f_{\text{solvent}}^{\text{II}}$                                                                                                (1)

The fugacity of the solvent in the cell I is given by the expression,

    $f_{{\text{solvent,P}}^{\text{II}}}^{\text{II}}=x_{\text{solvent}}^{\text{II}}{\gamma }_{\text{solvent}}^{\text{II}}{f}_{{\text{solvent,T,P}}^{\text{II}}}$                                                                      (2)

Where,

• $x_{\text{solvent}}^{\text{II}}$ is the mole fraction of the solvent in cell II.
• ${\gamma }_{\text{solvent}}^{\text{II}}$ is the activity coefficient of the solvent.
• $f_{\text{solvent}}$ is the fugacity of the pure solvent at system temperature and pressure.
• ${\text{P}}^{\text{II}}$ is the pressure on cell II.
• ${\text{P}}^{\text{I}}$ is the osmotic pressure on cell I.

The fugacity of solvent at pressure ${\text{P}}^{\text{II}}$ is related to the fugacity at pressure ${\text{P}}^{\text{I}}$ by,

    $f_{\text{solvent}\left(T,P^{\text{I}}\right)}=f_{\text{solvent}\left(T,P^{\text{II}}\right)}{e}^{\frac{V_{\text{solvent}}\left(P_{\text{II}}-P_{\text{I}}\right)}{RT}}$                                                                  (3)

Substitute equation (2) in (3)

    $f_{\text{solvent}\left(T,P^{\text{I}}\right)}=x_{\text{solvent}}^{\text{II}}{\gamma }_{\text{solvent}}^{\text{II}}{f}_{{\text{solvent,T,P}}^{\text{II}}}{e}^{\frac{V_{\text{solvent}}\left(P_{\text{II}}-P_{\text{I}}\right)}{RT}}$

Substitute equation (1) in above.

    $\begin{array}{l}{f}_{\text{solvent}\left(T,P^{\text{II}}\right)}=x_{\text{solvent}}^{\text{II}}{\gamma }_{\text{solvent}}^{\text{II}}{f}_{{\text{solvent,T,P}}^{\text{II}}}{e}^{\frac{V_{\text{solvent}}\left(P_{\text{II}}-P_{\text{I}}\right)}{RT}}\\ 1=x_{\text{solvent}}^{\text{II}}{\gamma }_{\text{solvent}}^{\text{II}}{e}^{\frac{V_{\text{solvent}}\left(P_{\text{II}}-P_{\text{I}}\right)}{RT}}\end{array}$                                                  (4)

The osmotic pressure is equal to the difference of pressure in both cells.

    $\Pi =P^{\text{II}}-P^{\text{I}}$

Substitute the value of $P^{\text{II}}-P^{\text{I}}$ in equation (4).

    $\begin{array}{l}1=x_{\text{solvent}}^{\text{II}}{\gamma }_{\text{solvent}}^{\text{II}}{e}^{\frac{V_{\text{solvent}}\Pi }{RT}}\\ \Pi =-\frac{RT}{V_{\text{solvent}}}\ln x_{\text{solvent}}^{\text{II}}{\gamma }_{\text{solvent}}^{\text{II}}\end{array}$                                                                            (5)

If the solvent is represented by a and solute by b, then equation (5) can be written as,

    $\Pi =-\frac{RT}{V_{\text{solvent}}}\ln x_{\text{a}}{\gamma }_{\text{a}}$                                                                                       (6)

The activity of a solvent $\left(a_{\text{a}}\right)$ is given by the expression,

    $a_{\text{a}}=x_{\text{a}}{\gamma }_{\text{a}}$

Substitute the value of $x_{\text{a}}{\gamma }_{\text{a}}$ in equation (6).

    $\Pi =-\frac{RT}{V_{\text{solvent}}}\ln a_{\text{a}}$                                                                                            (7)

Multiply and divide equation (7) by $x_{\text{a}}{x}_{\text{b}}$.

    $\begin{array}{c}\Pi =-\frac{x_{\text{a}}{x}_{\text{b}}}{x_{\text{a}}{x}_{\text{b}}}\times \frac{RT}{V_{\text{solvent}}}\ln a_{\text{a}}\\ =-\frac{x_{\text{a}}}{x_{\text{b}}}\ln a_{\text{a}}\frac{RT}{V_{\text{solvent}}}\times \frac{x_{\text{b}}}{x_{\text{a}}}\\ \end{array}$                                                                              (8)

Where,

• $x_{\text{b}}$ is the mole fraction of the solute.

Let,

    $-\frac{x_{\text{a}}}{x_{\text{b}}}\ln a_{\text{a}}=\varphi$

Where,

• $\phi$ is the osmotic pressure coefficient.

Substitute the value of $-\frac{x_{\text{a}}}{x_{\text{b}}}\ln a_{\text{a}}$ in equation (8).

    $\Pi V_{\text{solvent}}=\phi RT\frac{x_{\text{b}}}{x_{\text{a}}}$                                                                                         (9)

The value of mole fraction of solute $x_{\text{a}}$ in dilute solutions is equal to 1.

    $x_{\text{a}}\approx 1$

Substitute the value of $x_{\text{a}}$ in equation (9).

    $\Pi V_{\text{solvent}}=\phi RT{x}_{\text{b}}$                                                                                         (10)

The mole fraction of the solute, $x_{\text{b}}$ is given by the formula,

    $x_{\text{b}}=\frac{n_{\text{b}}}{n_{\text{a}}+n_{\text{b}}}$

Where,

• $n_{\text{a}}$ is the number of moles of solvent.
• $n_{\text{b}}$ is the number of moles if the solute.

The number of moles of solute in a dilute solution is very less.

Hence,

    $x_{\text{b}}=\frac{n_{\text{b}}}{n_{\text{a}}}$

Substitute the value of $x_{\text{b}}$ in equation (10).

    $\begin{array}{l}\Pi V_{\text{solvent}}=\phi RT\frac{n_{\text{b}}}{n_{\text{a}}}\\ \Pi V_{\text{solvent}}{n}_{\text{a}}=\phi RT{n}_{\text{b}}\end{array}$                                                                                      (11)

In equation (11), $V{n}_{\text{a}}$ is the total volume.

Hence, equation (11) can be written as,

    $\Pi V=\phi RT{n}_{\text{b}}$                                                                                         (12)

The number of moles of solute, $n_{\text{b}}$ is

    $n_{\text{b}}=\frac{\text{Weight}\text{of}\text{solute}}{\text{Molar}\text{mass}\text{of}\text{solute}}=\left[\text{B}\right]$

Hence, equation (12) can be written as,

    $\Pi V=\phi RT\left[\text{B}\right]$