Chapter 15, Problem 15.1VP

(a)

Interpretation Introduction

Interpretation: The mass action expression for the reaction is to be derived. The value of $K_{\text{c}}$ is to be calculated.

Concept introduction: The division of the ratio of the concentration of product raised to suitable stoichiometric coefficient with the concentrations of the reactants raised to suitable stoichiometric coefficient is called equilibrium constant for a reaction.

To determine: The mass action expression for the reaction given in the graph shown.

Answer to Problem 15.1VP

Solution

The mass action expression for the reaction given in the graph is $3{\text{A}}_{\text{2}}+{\text{B}}_2\to 2{\text{A}}_{\text{3}}\text{B}$.

Explanation of Solution

Explanation

The mass action expression for the reaction between ${\text{A}}_{\text{2}}$ and ${\text{B}}_{\text{2}}$ will be expressed with the help of Figure $1$ given below.

Figure1

In Figure1 it is shown that there is a difference in the number of moles of ${\text{A}}_{\text{2}}$, ${\text{B}}_{\text{2}}$ and ${\text{A}}_{\text{3}}\text{B}$. The graph in Figure1 describes that the concentration of ${\text{A}}_{\text{2}}$ changes from $\text{1}\text{.0}\text{M}$ to $\text{0}\text{.52}\text{M}$ whereas the concentration of ${\text{B}}_{\text{2}}$ changes from $\text{1}\text{.0}\text{M}$ to $\text{0}\text{.84}\text{M}$.

The time taken is assumed to be $\Delta \text{t}$.

The formula used to calculate the rate of change of concentration of ${\text{A}}_{\text{2}}$ is ,

$\text{Rate}\text{of}\text{change}\text{of}\text{concentration}\text{of}{\text{A}}_{\text{2}}\text{with}\text{time}=\frac{\Delta {\text{A}}_{\text{2}}}{\Delta \text{t}}$ (1)

Where,

• $\Delta {\text{A}}_{\text{2}}$ is change in concentration of ${\text{A}}_{\text{2}}$.

The value of change in concentration of ${\text{A}}_{\text{2}}$ is calculated by subtracting initial value from final value of ${\text{A}}_2$,

$\begin{array}{c}\Delta {\text{A}}_{\text{2}}=1.0\text{M}-0.52\text{M}\\ =\text{0}\text{.48}\text{M}\end{array}$

Substitute the value of $\Delta {\text{A}}_{\text{2}}$ in equation (1),

$\text{Rate}\text{of}\text{change}\text{of}\text{concentration}\text{of}{\text{A}}_{\text{2}}\text{with}\text{time}=\frac{0.48\text{M}}{\Delta \text{t}}$ (2)

The formula used to calculate the rate of change of concentration of ${\text{B}}_{\text{2}}$ is ,

$\text{Rate}\text{of}\text{change}\text{of}\text{concentration}\text{of}{\text{B}}_{\text{2}}\text{with}\text{time}=\frac{\Delta {\text{B}}_{\text{2}}}{\Delta \text{t}}$ (3)

Where,

• $\Delta {\text{B}}_{\text{2}}$ is change in concentration of ${\text{B}}_{\text{2}}$.

The value of change in concentration of ${\text{B}}_{\text{2}}$ is calculated by subtracting initial value from final value of ${\text{B}}_2$,

$\begin{array}{c}\Delta {\text{B}}_{\text{2}}=1.0\text{M}-0.84\text{M}\\ =\text{0}\text{.16}\text{M}\end{array}$

Substitute the value of $\Delta {\text{B}}_{\text{2}}$ in equation (3),

$\text{Rate}\text{of}\text{change}\text{of}\text{concentration}\text{of}{\text{B}}_{\text{2}}\text{with}\text{time}=\frac{0.16\text{M}}{\Delta \text{t}}$ (4)

The formula used to calculate the rate of change of concentration of ${\text{A}}_3\text{B}$ is,

$\text{Rate}\text{of}\text{change}\text{of}\text{concentration}\text{of}{\text{A}}_3\text{B}\text{with}\text{time}=\frac{\Delta {\text{A}}_3\text{B}}{\Delta \text{t}}$ (5)

Where,

• $\Delta {\text{A}}_3\text{B}$ is change in concentration of ${\text{A}}_3\text{B}$.

The value of change in concentration of ${\text{A}}_3\text{B}$ will be calculated by subtracting initial value from final value of ${\text{A}}_3\text{B}$,

$\begin{array}{c}\Delta {\text{A}}_3\text{B}=0.32\text{M}-0.0\text{M}\\ =\text{0}\text{.32}\text{M}\end{array}$

Substitute the value of ${\text{A}}_3\text{B}$ in equation (5).

$\text{Rate}\text{of}\text{change}\text{of}\text{concentration}\text{of}{\text{A}}_3\text{B}\text{with}\text{time}=\frac{0.32\text{M}}{\Delta \text{t}}$ (6)

The ratio of rates is calculated as,

$\text{Ratio}\text{of}\text{rates}=\frac{{\text{ΔA}}_{\text{2}}}{\text{Δt}}:\frac{{\text{ΔB}}_{\text{2}}}{\text{Δt}}:\frac{{\text{ΔA}}_3\text{B}}{\text{Δt}}$

Substitute the values of $\frac{\Delta {\text{A}}_{\text{2}}}{\Delta \text{t}}$, $\frac{\Delta {\text{B}}_{\text{2}}}{\Delta \text{t}}$ and $\frac{{\text{ΔA}}_3\text{B}}{\text{Δt}}$ in the above expression,

$\begin{array}{c}\text{Ratio}\text{of}\text{rates}=\frac{0.48}{\text{Δt}}:\frac{0.16}{\text{Δt}}:\frac{0.32}{\text{Δt}}=\\ =3:1:2\end{array}$

As the ratio of the rates is equal to the order of the reaction for the same value of $\Delta \text{t}$ then,

$\left[{\text{A}}_{\text{2}}\right]:\left[{\text{B}}_{\text{2}}\right]:\left[{\text{A}}_3\text{B}\right]=3:1:2$

Substitute the values of ratio as the order of the reaction that gives the reaction as,

$3{\text{A}}_{\text{2}}+{\text{B}}_2\to 2{\text{A}}_{\text{3}}\text{B}$

Hence, the mass action expression for the reaction given in the graph is $3{\text{A}}_{\text{2}}+{\text{B}}_2\to 2{\text{A}}_{\text{3}}\text{B}$.

(b)

Interpretation Introduction

To determine: The value of $K_{\text{c}}$ is to be calculated.

Answer to Problem 15.1VP

Solution

The value of equilibrium constant $K_{\text{c}}=\frac{{\left[{\text{A}}_{\text{3}}\text{B}\right]}^2}{{\left[A_2\right]}^3}\frac{}{\left[{\text{B}}_{\text{2}}\right]}$

Explanation of Solution

Explanation

The value of equilibrium constant $K_{\text{c}}$ is calculated by the expression,

$K_{\text{c}}=\frac{\left[\text{Product}\right]}{\left[\text{Reactant}\text{A}\right]}\frac{}{\left[\text{Reactant}\text{B}\right]}$

Substitute the reactants and products at their respective places in the above expression,

$K_{\text{c}}=\frac{{\left[{\text{A}}_{\text{3}}\text{B}\right]}^2}{{\left[A_2\right]}^3}\frac{}{\left[{\text{B}}_{\text{2}}\right]}$

Therefore the value of $K_{\text{c}}=\frac{{\left[{\text{A}}_{\text{3}}\text{B}\right]}^2}{{\left[A_2\right]}^3}\frac{}{\left[{\text{B}}_{\text{2}}\right]}$

Conclusion

1. a) The mass action expression for the reaction given in the graph is $3{\text{A}}_{\text{2}}+{\text{B}}_2\to 2{\text{A}}_{\text{3}}\text{B}$.
2. b) The value of equilibrium constant $K_{\text{c}}=\frac{{\left[{\text{A}}_{\text{3}}\text{B}\right]}^2}{{\left[A_2\right]}^3}\frac{}{\left[{\text{B}}_{\text{2}}\right]}$