Question

14.The temperature-dependent form of the extended Debye-

Hückel equation 8-6 is

$\mathit{l}\mathit{o}\mathit{g}\mathit{\gamma }\mathbf{=}\frac{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{825}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{)}{\mathbf{\left(}\mathbf{\epsilon }\mathbf{T}\mathbf{\right)}}^{\mathbf{-}\mathbf{3}\mathbf{/}\mathbf{2}}{\mathbf{Z}}^{\mathbf{2}}\sqrt{\mathbf{\mu }}}{\mathbf{1}\mathbf{+}\mathbf{\alpha }\sqrt{\mathbf{\mu }}\mathbf{/}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{00}\sqrt{\mathbf{\epsilon }\mathbf{T}}\mathbf{)}}$

where $\mathbf{\epsilon }$is the (dimensionless) dielectric constant* of water, T is

temperature (K), Z is the charge of the ion,$\mathbf{\mu }$is ionic strength $\left(\mathrm{mol}/L\right)$,and$\mathbf{\alpha }$is the ion size parameter (pm). The dependence of ´

on temperature is

role="math" localid="1654916203154" $\mathbf{\epsilon }\mathbf{=}\mathbf{79}\mathbf{.}\mathbf{755}{\mathbf{e}}^{\mathbf{(}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{)}\mathbf{(}\mathbf{T}\mathbf{-}\mathbf{293}\mathbf{.}\mathbf{15}\mathbf{)}}$

Calculate the activity coefficient of${\mathbf{SO}}_{\mathbf{4}}^{\mathbf{2}\mathbf{-}}$at$\mathbf{50}\mathbf{.}\mathbf{00}\mathbf{°}\mathbf{C}$when$\mathbf{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{100}\mathbf{M}$. Compare your value with the one in Table 8-1.

Short answer

The value of the activity coefficient, $\mathrm{\gamma }=0.329$. Whereas, in the Table 8-1 it is given that $\mathrm{\gamma }=0.355$at temperature of $25°\mathrm{C}$.

Step-by-step solution

Definition of activity coefficients

The activity coefficient, $\mathit{\gamma }$measures the deviation of behavior from ideality,rapidly decreases as ionic strength increases.

:Obtaining the activity coefficient value

In this task it is given that the temperature-dependent form of the extended

Debye-Huckel equation 8-6 is:

$\mathrm{log\gamma }=\frac{(-1.825\times {10}^6){\left(\mathrm{\epsilon T}\right)}^{-3/2}{\mathrm{Z}}^2\sqrt{\mathrm{\mu }}}{1+\mathrm{\alpha }\sqrt{\mathrm{\mu }}/(2.00\sqrt{\mathrm{\epsilon T}})}$

where stands for dielectric constant, T is temperature (K), role="math" localid="1654916133542" $\mathrm{\mu }$is ionic strength, Z is chargeand $\mathrm{\alpha }$is the ion size parameter (pm).

The dependence of on temperature is given as:

role="math" localid="1654916244358" $\mathrm{\epsilon }=79.755{\mathrm{e}}^{(-4.6\times {10}^{-3})(\mathrm{T}-293.15)}$

Here we will calculate the activity coefficient of ${\mathrm{SO}}_4^{2-}$when the temperature is $50.00°\mathrm{C}$and $\mathrm{\mu }=0.100\mathrm{M}$.

First we calculate the dielectric constant $\mathrm{\epsilon }$by the following equation

$\mathrm{\epsilon }=79.755{\mathrm{e}}^{(-4.6\times {10}^{-3})(323.15-293.15)}=69.474$

Next we will calculate theactivity coefficient by the temperature-dependent extended

Debye-Huckel equation given in the task which gives us the value of:

$$\begin{gathered} \log \gamma =-0.482.6 \\ \gamma ={10}^{-0.482.6}=0.329 \end{gathered}$$

whereas, in the Table 8-1 it is given that $\gamma =0.355$at temperature of $25°\mathrm{C}$.

Therefore, the value of the activity coefficient, $\gamma =0.329$. Whereas, in the Table 8-1 it is given that $\gamma =0.355$at temperature of $25°\mathrm{C}$.