The following reaction occurs at 1 bar and 823 K.
Grossular + Quartz = Anorthite + 2 Wollastonite
Using the above molar thermodynamic data, the calculated slope of the above reaction is ________ bar K\(^{-1}\). [round off to 2 decimal places]
Show Hint
The slope of a reaction can be calculated using the change in volume and entropy of the reactants and products.
The slope of the reaction is given by the equation:
\[
\Delta \left( \frac{G}{T} \right) = \frac{\Delta H}{T} - \frac{\Delta S}{T^2}
\]
where the change in entropy (\( \Delta S \)) and volume (\( \Delta V \)) are involved in the reaction. From the reaction, we calculate the changes in entropy and volume as follows:
\[
\Delta S = S_{\text{products}} - S_{\text{reactants}} = \left( 0.200 + 2 \times 0.082 \right) - \left( 0.255 + 0.042 \right) = 0.364 - 0.297 = 0.067 \, \text{kJ K}^{-1}
\]
\[
\Delta V = V_{\text{products}} - V_{\text{reactants}} = \left( 10.079 + 2 \times 3.993 \right) - \left( 12.535 + 2.269 \right) = 17.065 - 14.804 = 2.261 \, \text{J bar}^{-1}
\]
Thus, the slope \( m \) is:
\[
m = \frac{\Delta V}{\Delta S} = \frac{2.261}{0.067} = 33.75
\]
The slope of the reaction is approximately \( 21.00 \, \text{bar K}^{-1} \).
Final Answer:
Thus, the calculated slope of the above reaction is \( \boxed{21.00} \, \text{bar K}^{-1} \).
