Chapter 12, Problem 12.27PAE

Interpretation Introduction

(a) Interpretation:

Examine for the following given equations and answer the questions

$$\begin{array}{l}{\text{CO}}_3^{2-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{HCO}}_3{}^{-}{\text{(aq), K}}_1\text{ = ?}\\ {\text{HCO}}_3{}^{-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3{\text{(aq), K}}_2\text{ = ?}\end{array}$$

The value of K₁ and K₂

Concept Introduction: Equilibrium mass action expression is the ratio of the concentration of product and reactant when equilibrium is attained. Let us consider a general equation:-

$$\begin{array}{l}\text{a A + b B }\rightleftarrows \text{ c C + d D}\\ \text{Q=}\frac{{\text{[C]}}^{\text{c}}{\text{[D]}}^{\text{d}}}{{\text{[A]}}^{\text{a}}{\text{[B]}}^{\text{b}}}\\ \text{Rate constant}\\ \text{K = }\frac{{\text{[C]}}_{eq}{}^{\text{c}}{\text{[D]}}_{eq}{}^{\text{d}}}{{\text{[A]}}_{eq}{}^{\text{a}}{\text{[B]}}_{eq}{}^{\text{b}}}\end{array}$$

Answer to Problem 12.27PAE

Solution:

$${\text{K}}_1\text{=}\frac{[{\text{HCO}}_3{}^{-}]}{[{\text{CO}}_3^{2-}][{\text{H}}^{+}]}$$, $${\text{K}}_2\text{=}\frac{[{\text{H}}_2{\text{CO}}_3]}{[{\text{HCO}}_3{}^{-}]\left[{\text{H}}^{+}\right]}$$

Explanation of Solution

Using the formula $$\text{K = }\frac{{\text{[C]}}_{eq}{}^{\text{c}}{\text{[D]}}_{eq}{}^{\text{d}}}{{\text{[A]}}_{eq}{}^{\text{a}}{\text{[B]}}_{eq}{}^{\text{b}}}$$

we can calculate the values of equilibrium constants for both the reactions:-

$${\text{CO}}_3^{2-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{HCO}}_3{}^{-}\text{(aq)}$$

$${\text{K}}_1\text{=}\frac{[{\text{HCO}}_3{}^{-}]}{[{\text{CO}}_3^{2-}][{\text{H}}^{+}]}$$

For the second reaction: $${\text{HCO}}_3{}^{-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3\text{(aq)}$$

$${\text{K}}_2\text{=}\frac{[{\text{H}}_2{\text{CO}}_3]}{[{\text{HCO}}_3{}^{-}]\left[{\text{H}}^{+}\right]}$$

Interpretation Introduction

(b) Interpretation:

Examine for the following given equations and answer the questions

$$\begin{array}{l}{\text{CO}}_3^{2-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{HCO}}_3{}^{-}{\text{(aq), K}}_1\text{ = ?}\\ {\text{HCO}}_3{}^{-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3{\text{(aq), K}}_2\text{ = ?}\end{array}$$

The equation by adding the given two equations

Concept Introduction: Equilibrium mass action expression is the ratio of the concentration of product and reactant when equilibrium is attained. Let us consider a general equation:-

$$\begin{array}{l}\text{a A + b B }\rightleftarrows \text{ c C + d D}\\ \text{Q=}\frac{{\text{[C]}}^{\text{c}}{\text{[D]}}^{\text{d}}}{{\text{[A]}}^{\text{a}}{\text{[B]}}^{\text{b}}}\\ \text{Rate constant}\\ \text{K = }\frac{{\text{[C]}}_{eq}{}^{\text{c}}{\text{[D]}}_{eq}{}^{\text{d}}}{{\text{[A]}}_{eq}{}^{\text{a}}{\text{[B]}}_{eq}{}^{\text{b}}}\end{array}$$

Answer to Problem 12.27PAE

Solution:

The overall reaction is $${\text{CO}}_3^{2-}{\text{(aq)+2H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3\text{(aq)}$$

Explanation of Solution

Overall reaction (Sum of both reactions) is given below:-

$${\text{CO}}_3^{2-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{HCO}}_3{}^{-}\text{(aq)}$$$${\text{HCO}}_3{}^{-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3\text{(aq)}$$$${\text{CO}}_3^{2-}{\text{(aq)+2H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3\text{(aq)}$$

Hence, this is the result.

$${\text{CO}}_3^{2-}{\text{(aq)+2H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3\text{(aq)}$$

This is the overall reaction.

Interpretation Introduction

(c) Interpretation:

Examine for the following given equations and answer the questions

$$\begin{array}{l}{\text{CO}}_3^{2-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{HCO}}_3{}^{-}{\text{(aq), K}}_1\text{ = ?}\\ {\text{HCO}}_3{}^{-}{\text{(aq)+H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3{\text{(aq), K}}_2\text{ = ?}\end{array}$$

The equilibrium constant for the overall equation.

Concept Introduction: Equilibrium mass action expression is the ratio of the concentration of product and reactant when equilibrium is attained. Let us consider a general equation:-

$$\begin{array}{l}\text{a A + b B }\rightleftarrows \text{ c C + d D}\\ \text{Q=}\frac{{\text{[C]}}^{\text{c}}{\text{[D]}}^{\text{d}}}{{\text{[A]}}^{\text{a}}{\text{[B]}}^{\text{b}}}\\ \text{Rate constant}\\ \text{K = }\frac{{\text{[C]}}_{eq}{}^{\text{c}}{\text{[D]}}_{eq}{}^{\text{d}}}{{\text{[A]}}_{eq}{}^{\text{a}}{\text{[B]}}_{eq}{}^{\text{b}}}\end{array}$$

Answer to Problem 12.27PAE

Solution:

$${\text{K}}_3\text{=}\frac{[{\text{HCO}}_3]}{[{\text{CO}}_3{}^{2-}]{\left[{\text{H}}^{+}\right]}^2}$$

Explanation of Solution

The new equilibrium constant for the overall equation is:-

$${\text{CO}}_3^{2-}{\text{(aq)+2H}}^{+}\text{(aq)}\rightleftarrows {\text{H}}_2{\text{CO}}_3\text{(aq)}$$$${\text{K}}_3\text{=}\frac{[{\text{HCO}}_3]}{[{\text{CO}}_3{}^{2-}]{\left[{\text{H}}^{+}\right]}^2}$$