Problem 23 - Chemistry: The Central Science in SI Units, Global Edition

Question

For each of the following gas-phase reactions, indicate how the rate of disappearance of each reactant is related to the rate of appearance of each product: (a) \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g})\) (b) \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\) (c) \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\) (d) \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\)

Step-by-Step Solution

Answer

For each of the given gas-phase reactions, the rate of disappearance of each reactant is related to the rate of appearance of each product as follows: (a) \(-\frac{d[\mathrm{CO}]}{dt} = +\frac{d[\mathrm{CO}_{2}]}{dt} = -\frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt} = +\frac{d[\mathrm{H}_{2}]}{dt}\) (b) \(-\frac{1}{2} \frac{d[\mathrm{NO}]}{dt} = -\frac{d[\mathrm{Cl}_{2}]}{dt} = +\frac{1}{2} \frac{d[\mathrm{NOCl}]}{dt}\) (c) \(-\frac{d[\mathrm{CH}_{4}]}{dt} = -\frac{1}{2} \frac{d[\mathrm{O}_{2}]}{dt} = +\frac{d[\mathrm{CO}_{2}]}{dt} = +\frac{1}{2} \frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt}\) (d) \(-\frac{1}{2} \frac{d[\mathrm{N}_{2} \mathrm{O}_{4}]}{dt} = +\frac{d[\mathrm{NO}_{2}]}{dt}\)

Step 1: (a) Reaction: \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g})\)

The rate of disappearance of reactants CO and H₂O, and rate of appearance of products CO₂ and H₂ can be related as follows: \[ -\frac{d[\mathrm{CO}]}{dt} = +\frac{d[\mathrm{CO}_{2}]}{dt} = -\frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt} = +\frac{d[\mathrm{H}_{2}]}{dt} \]

Step 2: (b) Reaction: \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\)

The rate of disappearance of reactants NO and Cl₂, and rate of appearance of product NOCl can be related as follows: \[ -\frac{1}{2} \frac{d[\mathrm{NO}]}{dt} = -\frac{d[\mathrm{Cl}_{2}]}{dt} = +\frac{1}{2} \frac{d[\mathrm{NOCl}]}{dt} \]

Step 3: (c) Reaction: \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)\)

The rate of disappearance of reactants CH₄ and O₂, and rate of appearance of products CO₂ and H₂O can be related as follows: \[ -\frac{d[\mathrm{CH}_{4}]}{dt} = -\frac{1}{2} \frac{d[\mathrm{O}_{2}]}{dt} = +\frac{d[\mathrm{CO}_{2}]}{dt} = +\frac{1}{2} \frac{d[\mathrm{H}_{2} \mathrm{O}]}{dt} \]

Step 4: (d) Reaction: \(\mathrm{N}_{2} \mathrm{O}_{4}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\)

The rate of disappearance of reactant N₂O₄, and rate of appearance of product NO₂ can be related as follows: \[ -\frac{1}{2} \frac{d[\mathrm{N}_{2} \mathrm{O}_{4}]}{dt} = +\frac{d[\mathrm{NO}_{2}]}{dt} \]

Key Concepts

Rate of Reaction

The rate of reaction tells us how quickly a chemical reaction takes place. It is the speed at which reactants are converted into products. Imagine baking a cake: the faster you mix and heat the ingredients, the quicker the cake bakes. In a chemical reaction, this is similar. We can measure the rate of reaction by seeing how fast the concentration of a reactant decreases or how fast the concentration of a product increases, usually over time.

• Mathematically, the rate of reaction can be described using the concentration of the substances involved and the time taken.
• The rate can be expressed as the change in concentration of a reactant/product per unit time.

For example, if you have the reaction (\( \mathrm{A} + \mathrm{B} \longrightarrow \mathrm{C} \)), you can write its rate as:
\[\text{Rate} = - \frac{d[\mathrm{A}]}{dt} = - \frac{d[\mathrm{B}]}{dt} = \frac{d[\mathrm{C}}{dt}\]Negative signs appear with reactants because their concentrations decrease over time, indicating disappearance.

Stoichiometry

Stoichiometry is like the recipe of a chemical reaction. It tells us the proportion of reactants and products involved in the reaction, very similar to how recipes tell you how many eggs to use or how much flour is needed. In the chemistry world, stoichiometry helps us balance equations by making sure that the number of atoms for each element is the same on both sides of the reaction.

• Stoichiometry provides coefficients that represent the number of moles of each substance involved.
• These coefficients are crucial as they ensure that mass is conserved during the reaction.
• For instance, in the equation \(2H_2 + O_2 \longrightarrow 2H_2O\), one molecule of \(O_2\) pairs with two molecules of \(H_2\) to form water.

If we look at sample reaction (a): \(\mathrm{CO}(g)+\mathrm{H}_{2}O(g) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g)\)
Here, all substances have a coefficient of one, simplifying the stoichiometric relationships.

Reaction Rate Expressions

The reaction rate expression, sometimes called the rate law, is a mathematical equation that describes the speed of a reaction based on the concentration of its reactants. Knowing this expression allows scientists to predict how changes in concentration will influence the reaction speed.

• It incorporates the concentration of each reactant raised to a power.
• The power is called the order of the reaction and it indicates how the concentration of that reactant affects the rate.
• For example, a simple rate expression is \(\text{Rate} = k[A]^m[B]^n\), where \(k\) is the rate constant, and \(m\) and \(n\) are the orders of the reaction with respect to \(A\) and \(B\).

In reaction (b) \(2 \mathrm{NO}(g) + \mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g)\), one can relate the disappearance rates accordingly, for instance:
\[-\frac{1}{2} \frac{d[\mathrm{NO}]}{dt} = -\frac{d[\mathrm{Cl}_{2}]}{dt} = +\frac{1}{2} \frac{d[\mathrm{NOCl}]}{dt}\]These expressions become essential for calculating proper rates and understanding the chemical dynamics involved.