Chapter 15, Problem 19QP

Write the equilibrium constant expressions for $K_c\text{ and for }{K}_p$ _p, if applicable, for the following reactions:

$\begin{array}{l}\text{(a)}\\ {\text{2NO}}_{\text{2}}\text{(}g{\text{)+7H}}_{\text{2}}\text{(}g\text{)}\underset{}{\rightleftarrows }{\text{2NH}}_3(g)+4H_{2}O(l)\\ (b)\\ \text{2ZnS}(s)+3O_2(g)\underset{}{\rightleftarrows }\text{2ZnO}(s)+{\text{2SO}}_2(g)\\ (c)\\ C(s)+C{O}_2(g)\underset{}{\rightleftarrows }\text{2CO}(g)\\ (d)\\ C_6{H}_{5}COOH(aq)\underset{}{\rightleftarrows }{\text{C}}_{\text{6}}{\text{H}}_{\text{5}}{\text{COO}}^{-}(aq)+{\text{H}}^{\text{+}}(aq)\end{array}$

Interpretation Introduction

Interpretation:

The equilibrium constant expressions for $K_c$ and $K_p$ in each reaction are to be determined.

Concept introduction:

The chemical equilibrium is a state of a chemical reaction when the rate of forward reaction becomes equal to the rate of reverse reaction and the concentration of the products and reactants becomes constant known as equilibrium concentrations.

According to Dalton’s law, the total pressure of the mixture of nonreactive gases is equal to the partial pressure of individual gases. The pressure utilized by the individual gas is called the partial pressure of that gas.

Equilibrium constant is the ratio of the concentration of reactants and products present in the chemical reaction.

For a general reaction: $a\text{A}\text{+}b\text{B}\rightleftharpoons c\text{C}\text{+}d\text{D}$

The general formula for writing equilibrium expression for the reaction is given as:

$K_C=\frac{{\left[\text{C}\right]}^c{}_{eqm}{\left[\text{D}\right]}^d{}_{eqm}}{{\left[\text{A}\right]}^a{}_{eqm}{\left[\text{B}\right]}^b{}_{eqm}}$

Here, $K_C$ is the equilibrium constant and $C$ in $K_C$ stands for the concentration. $\left[\text{A}\right]$, $\left[\text{B}\right]$, $\left[\text{C}\right]$, and $\left[\text{D}\right]$ are the equilibrium concentration of reactants A and B and product C and D, respectively.

A and B are reactants, C is products, and $x,y$, and $z$ are their respective stoichiometric coefficients.

For the general reaction: $a\text{A}\text{+}b\text{B}\rightleftharpoons c\text{C}\text{+}d\text{D}$

The general formula for writing equilibrium expression for the reaction is given as:

$K_p=\frac{P_C^c{P}_D^d}{P_A^a{P}_B^b}$

Here, $p$ in $K_p$ stands for the pressure. $P_A,P_B,P_C,\text{and}{P}_D$ are the equilibrium partial pressure of reactants A and B and products C and D, respectively.

A and B are reactants, C and D are products, and $a,b,c$, and $d$ are their respective stoichiometric coefficients.

Equilibrium constants of gas phase reaction are written in terms of partial pressures because concentration of gases is directly proportional to partial pressures.

The relationship between $K_P$ and $K_C$ is given as:

$\begin{array}{c}{K}_P=K_C{\left(RT\right)}^{\Delta n}\\ K_c=\frac{K_P}{{\left(RT\right)}^{\Delta n}}\end{array}$

Answer to Problem 19QP

Solution:

(a)

$K_c=\frac{{\left[{\text{NH}}_3\right]}^2}{{\left[{\text{NO}}_2\right]}^2{\left[{\text{H}}_2\right]}^7}$

$K_p=\frac{{\left[P_{{\text{NH}}_3}\right]}^2}{{\left[P_{{\text{NO}}_2}\right]}^2{\left[P_{{\text{H}}_2}\right]}^7}$

(b)

$K_c=\frac{{\left[{\text{SO}}_2\right]}^2}{{\left[{\text{O}}_2\right]}^3}$

$K_p=\frac{{\left[P_{{\text{SO}}_2}\right]}^2}{{\left[P_{{\text{O}}_2}\right]}^3}$

(c)

$K_c=\frac{{\left[\text{CO}\right]}^2}{\left[{\text{CO}}_2\right]}$

$K_p=\frac{{\left[P_{\text{CO}}\right]}^2}{\left[P_{{\text{CO}}_2}\right]}$

(d)

$K_c=\frac{\left[{\text{C}}_6{\text{H}}_5{\text{COO}}^{-}\right]\left[{\text{H}}^{+}\right]}{\left[{\text{C}}_6{\text{H}}_5\text{COOH}\right]}$

The $K_p$ expression for the reaction cannot be written as the equilibrium expression does not contain any gases.

Explanation of Solution

a) $2{\text{NO}}_2\left(g\right)+7{\text{H}}_2\left(g\right)\rightleftarrows 2{\text{NH}}_3\left(g\right)+4{\text{H}}_2\text{O}\left(l\right)$

The constant expression for the reaction that is applicable only for chemical reactants and products that are in the same section iscalled homogeneous equilibrium. Only gaseous or aqueous species are in equilibrium expression.

In the given reaction, only ${\text{NO}}_2,{\text{H}}_2$, and ${\text{NH}}_3$ will appear in the equilibrium expression.

$K_C=\frac{{\left[\text{C}\right]}^c{}_{eqm}{\left[\text{D}\right]}^d{}_{eqm}}{{\left[\text{A}\right]}^a{}_{eqm}{\left[\text{B}\right]}^b{}_{eqm}}$

$K_c=\frac{{\left[{\text{NH}}_3\right]}^2}{{\left[{\text{NO}}_2\right]}^2{\left[{\text{H}}_2\right]}^7}$

Here, $K_c$ is the equilibrium constant, $\left[{\text{NH}}_3\right]$ is the concentration of ammonia, $\left[{\text{NO}}_2\right]$ is the concentration of ${\text{NO}}_2$, and $\left[{\text{H}}_2\right]$ is the concentration of ${\text{H}}_2$.

An equilibrium expression contains only gases. The concentration of gases is expressed as partial pressure. The $K_p$ expression for the reaction is as follows:

$K_p=\frac{P_C^c{P}_D^d}{P_A^a{P}_B^b}$

$K_p=\frac{{\left[P_{{\text{NH}}_3}\right]}^2}{{\left[P_{{\text{NO}}_2}\right]}^2{\left[P_{{\text{H}}_2}\right]}^7}$

Here, $K_p$ is the equilibrium constant pressure, $\left[P_{{\text{NH}}_3}\right]$ is the partial pressure of ammonia, $\left[P_{{\text{NO}}_2}\right]$ is the partial pressure of ${\text{NO}}_2$, and $\left[P_{{\text{H}}_2}\right]$ is the partial pressure of ${\text{H}}_2$.

b) $2\text{ZnS}\left(s\right)+3O_2\left(g\right)\rightleftarrows 2\text{ZnO}\left(s\right)+2{\text{SO}}_2\left(g\right)$

The equilibrium expression for the reaction is as follows:

$K_C=\frac{{\left[\text{C}\right]}^c{}_{eqm}{\left[\text{D}\right]}^d{}_{eqm}}{{\left[\text{A}\right]}^a{}_{eqm}{\left[\text{B}\right]}^b{}_{eqm}}$

$K_c=\frac{{\left[{\text{SO}}_2\right]}^2}{{\left[{\text{O}}_2\right]}^3}$

Here, $K_c$ is the equilibrium constant, $\left[{\text{SO}}_2\right]$ is the concentration of ${\text{SO}}_2$, and $\left[{\text{O}}_2\right]$ is the concentration of ${\text{O}}_2$.

The $K_p$ expression for the reaction is as follows:

$K_p=\frac{P_C^c{P}_D^d}{P_A^a{P}_B^b}$

$K_p=\frac{{\left[P_{{\text{SO}}_2}\right]}^2}{{\left[P_{{\text{O}}_2}\right]}^3}$

Here, $K_p$ is the equilibrium constant pressure, $\left[P_{{\text{SO}}_2}\right]$ is the partial pressure of ${\text{SO}}_2$, and $\left[P_{{\text{O}}_2}\right]$ is the partial pressure of ${\text{O}}_2$.

c) $\text{C}\left(s\right)+{\text{CO}}_2\left(g\right)\rightleftarrows 2\text{CO}\left(g\right)$

The equilibrium expression for the reaction is applicable for only $\text{CO}$ and ${\text{CO}}_2$.

The equilibrium expression for the reaction is as follows:

$K_C=\frac{{\left[\text{C}\right]}^c{}_{eqm}{\left[\text{D}\right]}^d{}_{eqm}}{{\left[\text{A}\right]}^a{}_{eqm}{\left[\text{B}\right]}^b{}_{eqm}}$

$K_c=\frac{{\left[\text{CO}\right]}^2}{\left[{\text{CO}}_2\right]}$

Here, $K_c$ is the equilibrium constant, $\left[\text{CO}\right]$ is the concentration of $\text{CO}$, and $\left[{\text{CO}}_2\right]$ is the concentration of ${\text{CO}}_2$.

The $K_p$ expression for the reaction is as follows:

$K_p=\frac{P_C^c{P}_D^d}{P_A^a{P}_B^b}$

$K_p=\frac{{\left[P_{\text{CO}}\right]}^2}{\left[P_{{\text{CO}}_2}\right]}$

Here, $K_p$ is the equilibrium constant pressure, $\left[P_{\text{CO}}\right]$ is the partial pressure of $\text{CO}$, and $\left[P_{{\text{CO}}_2}\right]$ is the partial pressure of ${\text{CO}}_2$.

d) ${\text{C}}_6{\text{H}}_5\text{COOH}\left(aq\right)\rightleftarrows {\text{C}}_6{\text{H}}_5{\text{COO}}^{-}\left(aq\right)+{\text{H}}^{+}\left(aq\right)$

The equilibrium expression for the reaction is applicable for both reactants and products because both are in homogeneous equilibrium.

$K_C=\frac{{\left[\text{C}\right]}^c{}_{eqm}{\left[\text{D}\right]}^d{}_{eqm}}{{\left[\text{A}\right]}^a{}_{eqm}{\left[\text{B}\right]}^b{}_{eqm}}$

$K_c=\frac{\left[{\text{C}}_6{\text{H}}_5{\text{COO}}^{-}\right]\left[{\text{H}}^{+}\right]}{\left[{\text{C}}_6{\text{H}}_5\text{COOH}\right]}$

Here, $K_c$ is the equilibrium constant, $\left[{\text{C}}_6{\text{H}}_5{\text{COO}}^{-}\right],\left[{\text{H}}^{+}\right]$ is the concentration of products, and $\left[{\text{C}}_6{\text{H}}_5\text{COOH}\right]$ is the concentration of reactant.

The $K_p$ expression for the reaction cannot be written as the equilibrium expression does not contain any gases.