Q20.76P - Chemistry: Molecular Nature Of Matter And Change

Question

The K_SP of PbCl₂ is $\mathbf{1}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ at ${\mathbf{25}}^{\mathbf{°}}\mathbf{\text{C}}$. What is $\mathit{\Delta }{\mathit{G}}^{\mathbf{°}}$? Is it possible to prepare a solution that contains ${\mathbf{\text{Pb}}}^{\mathbf{2}\mathbf{+}}\mathbf{\left(}\mathit{a}\mathit{q}\mathbf{\right)}$ and ${\mathbf{\text{Cl}}}^{\mathbf{-}}\mathbf{\left(}\mathit{a}\mathit{q}\mathbf{\right)}$, at their standard-state concentrations?

Step-by-Step Solution

Answer

For this reaction, the $\Delta G_{rxn}^{°}$  is 27.21kJ/mol.

When the PbCl₂ salt is dissolved, the reaction is not spontaneous, and it is impossible to prepare 1M concentration solutions.

Step 1: Definition of standard-state concentrations

When the concentration of a solute is 1 M, it is considered to be under standard state conditions.

Step 2: Calculate Δ G r x n °

Because the solubility-product constant is known and the temperature is known, the reaction's Gibb's free energy can be computed as

$\Delta G^{°}=-RT\ln K_p$

Then, at  $298 \text{K}\left({25}^{°}\text{C}\right)$

$\begin{array}{c}\Delta G^{°}=-8.314\frac{\text{J}}{\text{mol}\cdot \text{K}}\cdot 298 \text{K}\cdot \ln (1.7\cdot {10}^{-5})\\ \Delta G^{°}=2.721\cdot {10}^4\frac{\text{J}}{\text{mol}}\\ =27.21\frac{\text{kJ}}{\text{mol}}.\end{array}$

For this reaction, the $\Delta G_{rxn}^{°}$ is 27.21kJ/mol.

Step 3: Whether reaction is spontaneous or not

At standard conditions, the reaction is non-spontaneous because $\Delta G^{°}>0$. As a result, preparing a solution of standard concentration - 1M would be impossible.

In ordinary conditions, the response quotient Q is

$\begin{array}{l}Q=\left[P{b}^{2+}\right]\cdot {\left[C{l}^{-}\right]}^2\\ Q=\left[1\text{M}\right]\text{Invalid <msup> element}\cdot \\ Q=1.\end{array}$

As a result, the forward reaction is non-spontaneous as $Q>>K_{sp\text{ p }}$. As a result, preparing 1M concentrations after dissolving PbCl₂ salt is difficult.

Therefore, reaction is not spontaneous and it is impossible to prepare 1M concentration solutions upon dissolution of PbCl₂ salt.