Chapter 20, Problem 20.1E

Interpretation Introduction

Interpretation:

The three additional rate relationships like equation $20.5$ by the use of the definition of the rate from the equation $20.3$ or $20.4$ are to be stated.

Concept introduction:

The rate of a reaction is defined as the speed by which the reaction is proceeding. The rate of reaction depends on several factors such as the concentration of reactant and temperature. The rate of reaction is represented as,

$\text{rate}=-\frac{1}{a}\frac{d\left[\text{A}\right]}{dt}$

Answer to Problem 20.1E

The three additional rate relationships like equation $20.5$ by the use of the definition of the rate from the equation $20.3$ or $20.4$ are represented as,

$\begin{array}{l}\text{rate}=-\frac{d\left[\text{B}\right]}{dt}=-\frac{b}{a}\cdot \frac{d\left[\text{A}\right]}{dt}\\ \text{rate}=+\frac{d\left[\text{C}\right]}{dt}=+\frac{c}{d}\cdot \frac{d\left[\text{D}\right]}{dt}\\ \text{rate}=+\frac{d\left[\text{D}\right]}{dt}=+\frac{d}{c}\cdot \frac{d\left[\text{C}\right]}{dt}\end{array}$

Explanation of Solution

The given chemical reaction is represented as,

$a\text{A}+b\text{B}\to c\text{C}+d\text{D}$

The rate of reaction is represented in the equation $20.3$ as,

$\begin{array}{l}\text{rate}=\frac{1}{a}\cdot \left(-\frac{\Delta \left[\text{A}\right]}{\Delta \text{time}}\right)=\frac{1}{b}\cdot \left(-\frac{\Delta \left[\text{B}\right]}{\Delta \text{time}}\right)\\ \text{rate}=\frac{1}{c}\cdot \left(+\frac{\Delta \left[\text{C}\right]}{\Delta \text{time}}\right)=\frac{1}{d}\cdot \left(+\frac{\Delta \left[\text{D}\right]}{\Delta \text{time}}\right)\end{array}$

Where,

• $\Delta \left[\text{A}\right]$ represents the change in concentration of reactant $\text{A}$.

• $\Delta \left[\text{B}\right]$ represents the change in concentration of reactant $\text{B}$.

• $\Delta \left[\text{C}\right]$ represents the change in concentration of product $\text{C}$.

• $\Delta \left[\text{D}\right]$ represents the change in concentration of product $\text{D}$.

• $a$ represents the stoichiometric coefficient of reactant $\text{A}$.

• $b$ represents the stoichiometric coefficient of reactant $\text{B}$.

• $c$ represents the stoichiometric coefficient of product $\text{C}$.

• $d$ represents the stoichiometric coefficient of product $\text{D}$.

• $\Delta \text{time}$ represents change in time.

The rate of reaction is represented in the equation $20.4$ as,

$\begin{array}{rllll}\text{rate}& =& -\frac{1}{a}\cdot \frac{d\left[\text{A}\right]}{dt}& =& -\frac{1}{b}\cdot \frac{d\left[\text{B}\right]}{dt}\\ \text{rate}& =& +\frac{1}{c}\cdot \frac{d\left[\text{C}\right]}{dt}& =& +\frac{1}{d}\cdot \frac{d\left[\text{D}\right]}{dt}\end{array}$

The rate of reaction with respect to reactant $\text{A}$ is represented in the equation $20.5$ as,

$\text{rate}=-\frac{d\left[\text{A}\right]}{dt}=-\frac{a}{b}\cdot \frac{d\left[\text{B}\right]}{dt}$

Similarly, the rate of reaction with respect to reactant $\text{B}$ is represented as,

$\text{rate}=-\frac{d\left[\text{B}\right]}{dt}=-\frac{b}{a}\cdot \frac{d\left[\text{A}\right]}{dt}$

The rate of reaction with respect to the product $\text{C}$ is represented as,

$\text{rate}=+\frac{d\left[\text{C}\right]}{dt}=+\frac{c}{d}\cdot \frac{d\left[\text{D}\right]}{dt}$

The rate of reaction with respect to the product $\text{D}$ is represented as,

$\text{rate}=+\frac{d\left[\text{D}\right]}{dt}=+\frac{d}{c}\cdot \frac{d\left[\text{C}\right]}{dt}$

Conclusion

The three additional rate relationships like equation $20.5$ by the use of the definition of the rate from the equation $20.3$ or $20.4$ are represented as,

$\begin{array}{l}\text{rate}=-\frac{d\left[\text{B}\right]}{dt}=-\frac{b}{a}\cdot \frac{d\left[\text{A}\right]}{dt}\\ \text{rate}=+\frac{d\left[\text{C}\right]}{dt}=+\frac{c}{d}\cdot \frac{d\left[\text{D}\right]}{dt}\\ \text{rate}=+\frac{d\left[\text{D}\right]}{dt}=+\frac{d}{c}\cdot \frac{d\left[\text{C}\right]}{dt}\end{array}$