Chapter 17, Problem 17.85AP

Interpretation Introduction

Interpretation: The ratio of $\left[{\text{Ag}}^{+}\right]$ to $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ in a bath in which $\left[{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ is equal to $0.250\text{M}$ is to be calculated.

Concept introduction: The relation between equilibrium constant and concentration of reactants and product is given as,

$K_{\text{f}}=\frac{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}{\left[{\text{Ag}}^{+}\right]+\left[{\text{S}}_2{\text{O}}_3^{2-}\right]}$

To determine: The ratio of $\left[{\text{Ag}}^{+}\right]$ to $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ in a bath in which $\left[{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$.

Answer to Problem 17.85AP

The ratio of $\left[{\text{Ag}}^{+}\right]$ to $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ in a bath in which $\left[{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ equal to $0.250\text{M}$ is $\underset{\_}{3.20\times 10^{-13}}$.

Explanation of Solution

Given

The concentration of ${\text{S}}_2{\text{O}}_3^{2-}$ is $0.250\text{M}$.

The value of $K_{\text{f}}$ is $5\times {10}^{13}$.

The given reaction is,

${\text{Ag}}^{+}\left(aq\right)+2{\text{S}}_2{\text{O}}_3^{2-}\left(aq\right)\rightleftharpoons \left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]\left(aq\right)K_{\text{f}}=5\times {10}^{13}$

The expression for $K_{\text{f}}$ value is given as,

$K_{\text{f}}=\frac{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}{\left[{\text{Ag}}^{+}\right]+\left[{\text{S}}_2{\text{O}}_3^{2-}\right]}$ (1)

Where,

• $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ is the concentration of sodium thiosulphate.
• $\left[{\text{Ag}}^{+}\right]$ is the concentration of silver cation
• $\left[{\text{S}}_2{\text{O}}_3^{2-}\right]$ is the concentration of anion.
• $K_{\text{f}}$ is the equilibrium constant.

Substitute the values of $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right],\left[{\text{Ag}}^{+}\right]\text{and}\left[{\text{S}}_2{\text{O}}_3^{2-}\right]$ in the equation (1).

$\begin{array}{c}5\times {10}^{13}=\frac{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}{\left[{\text{Ag}}^{+}\right]+{\left[0.250\right]}^2}\\ 5\times {10}^{13}=\frac{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}{\left[{\text{Ag}}^{+}\right]+\left(0.0625\right)}\\ \frac{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}{\left[{\text{Ag}}^{+}\right]}=5\times {10}^{13}\left(0.0625\right)\\ \frac{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}{\left[{\text{Ag}}^{+}\right]}=3.12\times {10}^{12}\end{array}$

Simplify the above expression,

$\begin{array}{c}\frac{\left[{\text{Ag}}^{+}\right]}{\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]}=\frac{1}{3.12\times {10}^{12}}\\ =\underset{\_}{3.20\times 10^{-13}}\end{array}$

Therefore, the ratio of $\left[{\text{Ag}}^{+}\right]$ to $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ in a bath in which $\left[{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ equal to $0.250\text{M}$ is $\underset{\_}{3.20\times 10^{-13}}$.

Conclusion

The ratio of $\left[{\text{Ag}}^{+}\right]$ to $\left[\text{Ag}{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ in a bath in which $\left[{\left({\text{S}}_2{\text{O}}_3\right)}_2^{3-}\right]$ equal to $0.250\text{M}$ is $\underset{\_}{3.20\times 10^{-13}}$.