Question

Solid tin exists in two forms: white and gray. For the transformation

$\mathbf{Sn}\mathbf{(}\mathit{s}\mathbf{,}\mathbf{white}\mathbf{)}\mathbf{\to }\mathbf{Sn}\mathbf{(}\mathit{s}\mathbf{,}\mathbf{gray}\mathbf{)}$

The enthalpy change is -2.1kJand the entropy change is -7.4JK^-1.

(a) Calculate the Gibbs free energy change for the conversion of1.00MOLwhite tin to gray tin at -30⁰C.

(b) Calculate the Gibbs free energy change for the conversion of 2.50MOLwhite tin to gray tin at -30⁰C.

(c) Will white tin convert spontaneously to gray tin at -30⁰C?

(d) At what temperature are white and gray tin in equilibrium at a pressure

e of1 atm?

Short answer

(a) The Gibbs free energy for one mole,$\mathit{△}G=-301.8J$

(b) The Gibbs free energy for$\mathit{△}G=-754.5J$

(c) The reaction is spontaneous.

(d) The temperature at which white and gray tin will be at equilibrium, $T=-283.78K$.

Step-by-step solution

Given data 

The value of standard entropy and enthalpy is given below:

$$\begin{gathered} △H=-2.1J \\ △S=-7.4J{k}^{-1} \end{gathered}$$.

Concept of the Gibbs free energy 

When a system achieves chemical reaction at constant pressure and temperature without being driven by an applied electrolytic voltage, the Gibbs energy is minimized. Its derivative with relevance to the reaction coordinate of the system vanishes at the equilibrium point.

(a) Calculation of Gibbs free energy  

The change in Gibbs energy is calculated by:

$△G=△H-T△S$

Where, is Gibbs free energy, $△H$enthalpy.

Put the value of the given data in the above equation.

data-custom-editor="chemistry" $\begin{array}{c}\mathit{△}G=-2.1\times {10}^3-243\times (-7.4)\\ \mathit{△}G=-301.8J\end{array}$

(b) Calculation of Gibbs free energy per mol 

The Gibb's free energy per mol can be determined by the formula:

$\Delta G=n\times \Delta G_m$

Where $△G$is Gibbs free energy and n is the number of moles

Put the value of given data in the above equation

$\begin{array}{c}\Delta G=2.5\times (-301.8)\\ \Delta G=-754.5J\end{array}$

(c) explanation of spontaneity

The change in Gibbs's free energy is the primary indicator of whether a reaction is spontaneous or not. The response is spontaneous if the change is negative.

From the Gibbs free energy calculation change:

$\Delta G=-301.8\mathrm{J}$

It can conclude that the white tin will spontaneously turn to grey tin.

(d) Calculation of Gibbs free energy at equilibrium 

The calculation of Gibbs free energy is shown below:

$\Delta G=\Delta H_{fus}-T\Delta S_{fus}$

Put $\Delta H=-2.1kJ$and $\Delta S=-7.4J{K}^{-1}$in the above equation.

It is known that, at equilibrium$\Delta G°$

$\begin{array}{c}0=\Delta H_{fus}-T\Delta S_{fus}\\ \Delta H_{fus}=T\Delta S_{fus}\\ T=\frac{\Delta H_{fus}}{\Delta S_{fus}}\\ =\frac{-2.1\times {10}^3}{-7.4}\end{array}$

By the above calculation, it is found that the value of temperature is 282.78k