Borax, sodium tetraborate decahydrate, is an important mineral found in dry lakebeds in California. It is used to make soap and glass, and used as a preservative. You can use the values of $K_{\mathrm{sp}}$ of borax at different temperatures to determine $\Delta H^{\circ}, \Delta S^{\circ}$, and $\Delta G^{\circ}$ for the dissolution of borax:
(Borax)
(Borate)
The relationship:
$$
\ln \left(K_{\mathrm{sp}}\right)=\frac{-\Delta H^\rho}{R T}+\frac{\Delta S^\rho}{R}
$$
has the form of a linear equation $y=m x+b$, where $y$ is the $\ln K_{\text {sp }}$ and $x$ is $1 / T$. The slope is equal to $\left(\Delta H^{\circ} / R\right)$, and the $y$ intercept is $\Delta S^{\circ} / R$, where $R$ is the gas constant, $8.314 \mathrm{~J} / \mathrm{Kmol}$. If you measure $K_{\mathrm{sp}}$ at several different temperatures, you can plot the $\ln K$ versus $1 / T$ ( $T$ in kelvin) as shown here.
(FIGURE CAN'T COPY)
Knowing the values of $\Delta H^{\mathrm{a}}$ and $\Delta S^{\circ}$ at a specific temperature allows the calculation of the change in Gibbs free energy for the reaction: $\Delta G^a=\Delta H^a-T \Delta S^{\circ}$.
The following table lists $K_{\mathrm{sp}}$ values for the dissolution of borax at several temperatures $\left({ }^{\circ} \mathrm{C}\right)$.
(TABLE CAN'T COPY)
a. Plot a graph of $\ln K_{\text {sp }}$ versus $1 / T$ ( $T$ in kelvin) and find the best-fitting line.
b. Determine $\Delta H^{\circ}$. Is this process endothermic or exothermic?
c. Determine $\Delta S^a$.
d. Determine $\Delta G^{\circ}$ at $298 \mathrm{~K}$.
c. Sketch a graph of $\ln K$ versus $1 / T$ for an exothermic process.