Borax (sodium tetraboratedecahydrate), a mineral found in dry lakebeds in California, is used as a preservative and in the manufacturing of soap and glass. By detervining the $K_{\text { sp }}$ of borax at different temperatures, we can determing the $K_{\text { sp }}$ of $\Delta G^{\circ}$ for the dissolution of borax:
$$\begin{array}{c}{\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{5}(\mathrm{OH})_{4} \cdot 8 \mathrm{H}_{2} \mathrm{O}(s) \longrightarrow 2 \mathrm{Na}^{+}(a q)+\mathrm{B}_{4} \mathrm{O}_{5}(\mathrm{OH})_{4}^{2-}(a q)+8 \mathrm{H}_{2} \mathrm{O}(l)} \\ {(\mathrm{Borax}) \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text (Tetraborate)}\end{array}$$
The relationship:
$$\ln \left(K_{\mathrm{sp}}\right)=\frac{-\Delta H^{\circ}}{R T}+\frac{\Delta S^{\circ}}{R}$$
has the form of a linear equation $y=m x+b,$ where $y$ is the $\ln K_{\text { sp }}$ and $x$ is 1$/ T(T$ in Kelvin). The slope is equal to $-\Delta H / R,$ and the $y$ intercept is $\Delta S^{\circ} / R,$ where $R$ is the gas constant, 8.314 $\mathrm{J} /$ $\mathrm{K}$ mol. Determining $K_{\mathrm{sp}}$ at several different temperatures allows us to plot a graph of lnK versus 1$/ T$ as shown in Figure a
Knowing the values of $\Delta H^{\circ}$ and $\Delta S^{\circ}$ at a specific tempera-
ture allows the calculation of the change in Gibbs free energy
for the reaction $\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ}$
$$\begin{array}{|l|l|}\hline Temperature \left(^{\circ} \mathrm{C}\right) & {K_{\mathrm{sp}}} \\ \hline 40.0 & {0.041} \\ \hline 45.0 & {0.083} \\ \hline 50.0 & {0.264} \\ \hline 55.0 & {0.486} \\ \hline 60.0 & {0.552} \\ \hline\end{array}$$
Use the information provided in the figure and table to do the following:
\begin{equation}\begin{array}{l}{\text { a. Plot a graph of } \ln K_{\text { sp versus } 1 / T(T \text { in Kelvin })} \text { . }} \\ {\text { b. Determine } \Delta H^{\circ} . \text { Is this process endothermic or exothermic? }} \\ {\text { c. Determine } \Delta S^{\circ} .} \\ {\text { d. Determine } \Delta G^{\circ} \text { at } 298 \mathrm{K} \text { . }} \\ {\text { e. Sketch a graph of lnK versus 1/T for an exothermic process. }}\end{array}\end{equation}