Chapter 5, Problem 5F.3P

Interpretation Introduction

Interpretation: Whether the given mean activity coefficients for aqueous solution of $\text{NaCl}$ at $\text{25°}\text{C}$ support the Debye-Huckel limiting law has to be confirmed.  Whether a better fit is obtained with the Davies equation has to be confirmed.

Concept introduction: The ionic strength is introduced in the Debye-Huckel limiting law which relates the activity coefficient as a function of the ionic strength of the solution.  It is dependent on the concentration of all the ions present in the solution.  It is a dimensionless quantity.  When the ionic strength of the solution is too high for the limiting law to be applicable, Davies equation is used which is an extended form of the Debye-Huckel limiting law.

Answer to Problem 5F.3P

The given mean activity coefficients for aqueous solution of $\text{NaCl}$ at $\text{25°}\text{C}$ support the Debye-Huckel limiting law.  A better fir is obtained with the Davies equation.

Explanation of Solution

The ionic strength $\left(I\right)$ of a solution is given by the equation,

    $I=\frac{1}{2}{\displaystyle \sum _i{z}_i{}^2\left(b_i/b^{\text{ο}}\right)}$                                                                                        (1)

The ionic strength of the aqueous solution of $\text{NaCl}$ is given as,

    $I=\frac{1}{2}\left(\frac{z_{{\text{Na}}^{\text{+}}}^2{b}_{{\text{Na}}^{\text{+}}}}{b^{\text{ο}}}+\frac{z_{{\text{Cl}}^{\text{+}}}^2{b}_{{\text{Cl}}^{\text{+}}}}{b^{\text{ο}}}\right)$                                                                            (2)

Where,

• $z_{{\text{Na}}^{\text{+}}}$ is the charge on the ${\text{Na}}^{\text{+}}$ ion.
• $z_{{\text{Cl}}^{\text{-}}}$ is the charge on the ${\text{Cl}}^{\text{-}}$ ion.
• $b_{{\text{Na}}^{\text{+}}}$ is the ,molality of the ${\text{Na}}^{\text{+}}$ ion.
• $b_{{\text{Cl}}^{\text{+}}}$ is the ,molality of the ${\text{Cl}}^{\text{-}}$ ion.
• $b^{\text{ο}}$ is $\text{1}\text{mol}{\text{kg}}^{\text{-1}}$.

The dissociation of $\text{NaCl}$ is represented by the equation.

    $\text{NaCl}\to {\text{Na}}^{+}+{\text{Cl}}^{-}$

The molality of ${\text{Na}}^{\text{+}}$ ion is equal to the molality of ${\text{Cl}}^{\text{-}}$ ion

Let molality of ${\text{Na}}^{\text{+}}$ ion and ${\text{Cl}}^{\text{-}}$ ion be $b$.

The charge on the cation, ${\text{Na}}^{\text{+}}$ $\left(z_{{\text{Na}}^{\text{+}}}\right)$ is $+1$.

The charge on the anion, ${\text{Cl}}^{\text{-}}$ $\left(z_{{\text{Cl}}^{-}}\right)$ is $-1$.

Substitute the values of $z_{{\text{Na}}^{\text{+}}}$, $z_{{\text{Cl}}^{\text{-}}}$, $b_{{\text{Na}}^{\text{+}}}$ and $b_{{\text{Cl}}^{\text{+}}}$ in equation (2).

    $\begin{array}{c}I=\frac{1}{2}\left(\frac{{\left(1\right)}^{2}b}{b^{\text{ο}}}+\frac{{\left(-1\right)}^{2}b}{b^{\text{ο}}}\right)\\ =\frac{1}{2}\left(\frac{2b}{b^{\text{ο}}}\right)\\ =\frac{b}{b^{\text{ο}}}\end{array}$

The Debye-Huckel limiting law is given as,

    $\log {\gamma }_{\pm }=-A\left|z_{{\text{Na}}^{\text{+}}}{z}_{{\text{Cl}}^{\text{-}}}\right|I^{1/2}$                                                                                (3)

Where

• ${\gamma }_{\pm }$ is the activity coefficient.
• $A$ is $0.509$ for an aqueous solution at $25°\text{C}$.
• $I$ is the dimensionless ionic strength.
• $z_{{\text{Na}}^{\text{+}}}$ is the charge on the ${\text{Na}}^{\text{+}}$ ion.
• $z_{{\text{Cl}}^{\text{-}}}$ is the charge on the ${\text{Cl}}^{\text{-}}$ ion.

Substitute the values of $I$, $z_{{\text{Na}}^{\text{+}}}$ and $z_{{\text{Cl}}^{\text{-}}}$ in equation (3).

    $\begin{array}{c}\log {\gamma }_{\pm }=-A\left|z_{{\text{Na}}^{\text{+}}}{z}_{{\text{Cl}}^{\text{-}}}\right|I^{1/2}\\ =-A\left|\left(+1\right)\times \left(-1\right)\right|\sqrt{\frac{b}{b^{\text{ο}}}}\\ =-A\sqrt{\frac{b}{b^{\text{ο}}}}\end{array}$

The data given is,

$b/\left(\text{mmol}{\text{kg}}^{\text{-1}}\right)$	1.0	2.0	5.0	10.0	20.0
${\gamma }_{\pm }$	0.9649	0.9519	0.9275	0.9024	0.8712

The value of $\log {\gamma }_{\pm }$  and $\sqrt{b}$ calculated from the above data is as follows.

${\gamma }_{\pm }$	$b/\left(\text{mmol}{\text{kg}}^{\text{-1}}\right)$	$\log {\gamma }_{\pm }$	$\sqrt{b}$
0.9649	1	-0.01552	1
0.9519	2	-0.02141	1.414214
0.9275	5	-0.03269	2.236068
0.9024	10	-0.0446	3.162278
0.8712	20	-0.05988	4.472136

The graph between $\log {\gamma }_{\pm }$ taken on the y axis and $\sqrt{b}$ taken on the x axis is shown as,

Therefore, the given mean activity coefficients for aqueous solution of $\text{NaCl}$ at $\text{25°}\text{C}$ support the Debye-Huckel limiting law $\left(\log {\gamma }_{\pm }=-A\sqrt{\frac{b}{b^{\text{ο}}}}\right)$.

The Davies equation is given as,

    $\log {\gamma }_{\pm }=-\frac{A\left|z_{+}{z}_{-}\right|I^{1/2}}{1+B{I}^{1/2}}+CI$                                                                            (4)

Where

• ${\gamma }_{\pm }$ is the activity coefficient.
• $A$ is $0.509$ for an aqueous solution at $25°\text{C}$.
• $I$ is the dimensionless ionic strength.
• $z_{{\text{Na}}^{\text{+}}}$ is the charge on the ${\text{Na}}^{\text{+}}$ ion.
• $z_{{\text{Cl}}^{\text{-}}}$ is the charge on the ${\text{Cl}}^{\text{-}}$ ion.
• $B$ is a dimensionless constant.
• $C$ is a dimensionless constant.

Substitute the values of $I$, $z_{{\text{Na}}^{\text{+}}}$ and $z_{{\text{Cl}}^{\text{-}}}$ in equation (4).

    $\begin{array}{c}\log {\gamma }_{\pm }=-\frac{A\left|\left(+1\right)\times \left(-1\right)\right|{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}}{1+B{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}}+C{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}\\ =\frac{A{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}}{1+B{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}}+C{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}\end{array}$

For $B{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}>>1$, the above equation can be written as,

  $\begin{array}{c}\log {\gamma }_{\pm }=C\left(\frac{b}{b^{\text{ο}}}\right)-\frac{A{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}}{B{\left(\sqrt{\frac{b}{b^{\text{ο}}}}\right)}^{1/2}}\\ =C\left(\frac{b}{b^{\text{ο}}}\right)-\frac{A}{B}\end{array}$

The graph between $\log {\gamma }_{\pm }$ taken on the y axis and $b$ taken on the x axis is shown as,

In the above graph, the slope $\left(C\right)$ is $-0.0022$ and the intercept on the y axis $\left(\frac{A}{B}\right)$ is $0.0177$.

Therefore, an improved fir is obtained for the Davies equation, $\log {\gamma }_{\pm }=C\left(\frac{b}{b^{\text{ο}}}\right)-\frac{A}{B}$.