Question

Organ pipes in unheated churches develop "tin disease," in which white tin is converted to gray tin. Given $$ \begin{array}{l} \text { white Sn: } \Delta H_{\mathrm{f}}^{\circ}=0.00 \mathrm{~kJ} / \mathrm{mol} ; S^{\circ}=51.55 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \\ \text { gray Sn: } \Delta H_{\mathrm{f}}^{\circ}=-2.09 \mathrm{k} \mathrm{J} / \mathrm{mol} ; S^{\circ}=44.14 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K} \end{array} $$ calculate the equilibrium temperature for the transition.

Short answer

Answer: The equilibrium temperature for the transition between white tin and gray tin is approximately 281.9 K.

Step-by-step solution

Calculate the difference in enthalpy and entropy between white and gray tin.

We are given the standard enthalpy of formation values and standard entropy values for both white and gray tin. We need to find the difference in these values. White tin: $$ \Delta H_{\mathrm{f}}^{\circ}(white) = 0.00 \,\mathrm{kJ/mol} $$ $$ S^{\circ}(white) = 51.55 \,\mathrm{J/mol \cdot K} $$ Gray tin: $$ \Delta H_{\mathrm{f}}^{\circ}(gray) = -2.09 \,\mathrm{kJ/mol} $$ $$ S^{\circ}(gray) = 44.14 \,\mathrm{J/mol \cdot K} $$ Now we find the differences, making sure to convert the units so they match: $$ \Delta H = \Delta H_{\mathrm{f}}^{\circ}(gray) - \Delta H_{\mathrm{f}}^{\circ}(white) = (-2.09 - 0.00) \,\mathrm{kJ/mol} = -2.09 \,\mathrm{kJ/mol} $$ $$ \Delta H = -2.09 \times 10^3 \,\mathrm{J/mol} $$ $$ \Delta S = S^{\circ}(gray) - S^{\circ}(white) = (44.14 - 51.55) \,\mathrm{J/mol \cdot K} = -7.41 \,\mathrm{J/mol \cdot K} $$

Calculate the equilibrium temperature using the Gibbs free energy equation.

At equilibrium, \(\Delta G=0\). We can use the Gibbs free energy equation and the calculated values of \(\Delta H\) and \(\Delta S\) to solve for the equilibrium temperature: $$ 0 = \Delta H - T \Delta S $$ $$ T = \frac{\Delta H}{\Delta S} $$ Plugging in the values for \(\Delta H\) and \(\Delta S\): $$ T = \frac{-2.09 \times 10^3 \,\mathrm{J/mol}}{-7.41 \,\mathrm{J/mol \cdot K}} $$ $$ T \approx 281.9\,\mathrm{K} $$ The equilibrium temperature for the transition between white tin and gray tin is approximately 281.9 K.