Chapter 22, Problem 22.84QP

Interpretation Introduction

Interpretation: The equilibrium concentrations of cysteine and glutathione for the given data are to be calculated.

Concept introduction: The equilibrium constant $\left(K_{\text{c}}\right)$ is calculated by the formula,

$K_{\text{c}}=\frac{{\left[\text{Product}\right]}^y}{{\left[\text{Reactant}\right]}^x}$

To determine: The equilibrium concentrations of cysteine and glutathione for the given data.

Answer to Problem 22.84QP

Solution

The equilibrium concentrations of cysteine and glutathione for the given data are $\underset{\_}{0.37mM}$ and $\underset{\_}{0.830mM}$.

Explanation of Solution

Explanation

Given

The reaction is,

${\text{CH}}_{\text{3}}\text{Hg}{\left(\text{glutathione}\right)}^{+}\left(aq\right)+\text{cysteine}\left(aq\right)\rightleftharpoons {\text{CH}}_{\text{3}}\text{Hg}{\left(\text{cysteine}\right)}^{+}\left(aq\right)+\text{glutathione}\left(aq\right)$

The initial concentration of cysteine solution is $1.20\text{mM}$.

The initial concentration of ${\text{CH}}_{\text{3}}\text{Hg}{\left(\text{glutathione}\right)}^{+}$ solution is $1.20\text{mM}$.

The equilibrium constant $\left(K_{\text{c}}\right)$ of given reaction is $5.0$.

The equilibrium constant $\left(K_{\text{c}}\right)$ for the given reaction is calculated by the formula,

$K_{\text{c}}=\frac{\left[{\text{CH}}_{\text{3}}\text{Hg}{\left(\text{cysteine}\right)}^{+}\left(aq\right)\right]\left[\text{glutathione}\left(aq\right)\right]}{\left[{\text{CH}}_{\text{3}}\text{Hg}{\left(\text{glutathione}\right)}^{+}\left(aq\right)\right]\left[\text{cysteine}\left(aq\right)\right]}$ (1)

Where,

• $\left[{\text{CH}}_{\text{3}}\text{Hg}{\left(\text{cysteine}\right)}^{+}\left(aq\right)\right]$ is the equilibrium concentration of product ${\text{CH}}_{\text{3}}\text{Hg}{\left(\text{cysteine}\right)}^{+}\left(aq\right)$.
• $\left[\text{glutathione}\left(aq\right)\right]$ is the equilibrium concentration of product $\text{glutathione}\left(aq\right)$.
• $\left[{\text{CH}}_{\text{3}}\text{Hg}{\left(\text{glutathione}\right)}^{+}\left(aq\right)\right]$ is the equilibrium concentration of reactant ${\text{CH}}_{\text{3}}\text{Hg}{\left(\text{glutathione}\right)}^{+}\left(aq\right)$.
• $\left[\text{cysteine}\left(aq\right)\right]$ is the equilibrium concentration of reactant $\text{cysteine}\left(aq\right)$.

The change in concentration that occurs during the reaction is considered to be $x$.

The table that shows the initial and the equilibrium concentrations of the reactants and products is stated as follows.

$\begin{array}{c}\\ \text{Initial}\\ \text{At}\text{equilibrium}\end{array}\begin{array}{ccccccc}{\text{CH}}_{\text{3}}\text{Hg}{\left(\text{glutathione}\right)}^{+}\left(aq\right)& +& \text{cysteine}\left(aq\right)& \rightleftharpoons & {\text{CH}}_{\text{3}}\text{Hg}{\left(\text{cysteine}\right)}^{+}\left(aq\right)& +& \text{glutathione}\left(aq\right)\\ 1.20\text{mM}& & 1.20\text{mM}& & 0& & 0\\ 1.20\text{mM}-x& & 1.20\text{mM}-x& & x& & x\end{array}$

Substitute the values of equilibrium concentration of reactants, product and equilibrium constant in equation (1).

$5.0=\frac{{\left(x\right)}^2}{\left(1.20\times {10}^{-3}-x\right)\left(1.20\times {10}^{-3}-x\right)}$

Take square root on both sides.

$\begin{array}{c}\sqrt{5.0}=\sqrt{\frac{{\left(x\right)}^2}{\left(1.20\text{mM}-x\right)\left(1.20\text{mM}-x\right)}}\\ 2.23=\frac{x}{1.20\text{mM}-x}\\ 2.23\left(1.20\text{mM}-x\right)=x\\ 2.68\text{mM}-2.23x=x\end{array}$

Simplify the above equation.

$\begin{array}{c}2.23x+x=2.68\text{mM}\\ \text{3}\text{.23}x=2.68\text{mM}\\ x=\frac{2.68\text{mM}}{\text{3}\text{.23}}\\ =0.830\text{mM}\end{array}$

The equilibrium concentration of cysteine from above table is expressed as,

$\text{cysteine}\left(aq\right)=1.20\text{mM}-x$ (2)

Substitute the value of $x$ in equation (2).

$\begin{array}{c}\text{cysteine}\left(aq\right)=1.20\text{mM}-0.830\text{mM}\\ =0.37\text{mM}\end{array}$

Thus, the equilibrium concentrations of cysteine and glutathione are $\underset{\_}{0.37mM}$ and $\underset{\_}{0.830mM}$.

Conclusion

The equilibrium concentrations of cysteine and glutathione are $\underset{\_}{0.37mM}$ and $\underset{\_}{0.830mM}$