Question

If the rate with respect to \(\mathrm{O}_{2}, \mathrm{NO}\) and \(\mathrm{NO}_{2}\) are respectively $$ \frac{-\Delta\left[\mathrm{O}_{2}\right]}{\Delta \mathrm{t}}, \frac{-1}{2}, \frac{\Delta[\mathrm{NO}]}{\Delta \mathrm{t}}, \frac{+1}{2} \frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta \mathrm{t}} $$ then the corresponding chemical equation is \(2 \mathrm{NO}\) \(+\mathrm{O}_{2} \rightarrow 2 \mathrm{NO}_{2}\)

Short answer

Answer: The balanced chemical equation is 2 NO + O₂ → 2 NO₂.

Step-by-step solution

Identify the rate of change expressions for each species

We are given the rates of change for \(\mathrm{O}_{2}\), \(\mathrm{NO}\), and \(\mathrm{NO}_{2}\) as follows: 1. \(\frac{-\Delta\left[\mathrm{O}_{2}\right]}{\Delta \mathrm{t}}\) 2. \(\frac{-\Delta[\mathrm{NO}]}{2\Delta \mathrm{t}}\) 3. \(\frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta \mathrm{t}}\)

Determine the stoichiometric coefficients

We can use these rate expressions to determine the stoichiometric coefficients in the balanced equation. 1. For \(\mathrm{O}_2\) : The rate expression is \(\frac{-\Delta\left[\mathrm{O}_{2}\right]}{\Delta \mathrm{t}}\). The negative sign indicates that it is being consumed in the reaction, and its coefficient is 1 (since there is no other factor in the expression). 2. For \(\mathrm{NO}\) : The rate expression is \(\frac{-\Delta[\mathrm{NO}]}{2\Delta \mathrm{t}}\). The negative sign indicates that it is being consumed in the reaction, and its coefficient is 2 (since there is a factor of 2 in the denominator). 3. For \(\mathrm{NO}_2\) : The rate expression is \(\frac{\Delta\left[\mathrm{NO}_{2}\right]}{\Delta \mathrm{t}}\). The positive sign indicates that it is being produced in the reaction, and its coefficient is 1 (since there is no other factor in the expression).

Write the balanced chemical equation

Using the stoichiometric coefficients determined in Step 2, we can write the balanced chemical equation as: $$2 \mathrm{NO} \ + \ \mathrm{O}_2 \ \rightarrow \ 2 \mathrm{NO}_2$$

Key concepts

Stoichiometry

Stoichiometry is like the blueprint of a chemical reaction. It tells us the exact proportions of each reactant and product involved in a chemical process. This allows us to understand how substances interact in a reaction, and how much of each substance we need.
At its core, stoichiometry involves using the coefficients from a balanced chemical equation to determine the relationship between the quantities of reactants and products. The coefficients indicate the moles of each substance involved in the reaction. For instance, if a balanced equation is written as \(2 \text{A} + 3 \text{B} \rightarrow 1 \text{C}\), the coefficients (2, 3, and 1) show that 2 moles of A react with 3 moles of B to form 1 mole of C.
In essence, stoichiometry is a tool that helps us calculate the amounts needed or produced in chemical reactions. Whether it’s predicting the yield of a product or determining how much reactant is needed, stoichiometry is foundational to understanding chemistry.

• Helps in determining required reactant amounts.
• Assists in calculating expected product volumes.
• Uses coefficients from balanced equations.

Balanced Chemical Equation

A balanced chemical equation is crucial because it reflects the law of conservation of mass. This means that atoms are neither created nor destroyed during a chemical reaction. As a result, the number of atoms for each element on the reactant side (left) should match the number of atoms on the product side (right) of the equation.
To balance a chemical equation, you adjust the coefficients in front of each compound to ensure equal numbers of atoms for each element. For instance, in our given exercise, the balanced equation is \(2 \text{NO} + \text{O}_2 \rightarrow 2 \text{NO}_2\). Here, we start by counting the number of nitrogen and oxygen atoms before and after the reaction to make sure each side is equal.
Balancing equations ensures reactions are described accurately. Without balancing, calculations derived from stoichiometry, such as molar ratios and reactant-product quantifications, would be inaccurate.

• Ensures the conservation of mass is followed.
• Necessary for accurate stoichiometric calculations.
• Keeps chemical equations precise and meaningful.

Reaction Mechanism

A reaction mechanism provides a step-by-step description of how a chemical reaction occurs. It breaks down a complex reaction into simpler steps, showing the sequence in which bonds are made and broken. This helps us understand the path from reactants to products, including the formation of intermediate compounds.
Knowing the reaction mechanism helps chemists to predict the rate of a reaction and to devise methods for optimizing conditions to get desired products more efficiently. For instance, in our reaction \(2 \text{NO} + \text{O}_2 \rightarrow 2 \text{NO}_2\), understanding the mechanism helps in identifying whether the reaction proceeds through a direct collision of molecules or involves intermediates.
A deeper insight into the reaction mechanism helps in controlling and speeding up reactions in industrial processes and is essential for creating new drugs and materials.

• Describes each step of a chemical reaction.
• Reveals the formation of intermediates.
• Essential for predicting and controlling reactions.