Question

If instantaneous rate of reaction is stated as $-\frac{1}{2} \frac{d[x]}{dt} = -\frac{d[y]}{dt} = \frac{1}{2} \frac{d[z]}{dt}$, identify the reaction.

• A. $x - 2y \rightarrow 2z$
• B. $2x + y \rightarrow 2z$
• C. $x + y \rightarrow z$
• D. $2x - 2y \rightarrow z$

Hint:

Read the denominators carefully. The denominator in the rate expression fraction is always the stoichiometric coefficient. A missing fraction simply means the coefficient is 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a mathematical expression representing the rate of a reaction in terms of the disappearance of reactants and appearance of products. We must work backward to determine the balanced stoichiometric chemical equation.

Step 2: Key Formula or Approach:
For a general reaction $aA + bB \rightarrow cC$, the universal rate of reaction is defined as:
$$\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt}$$
- A negative ($-$) sign indicates a reactant being consumed.
- A positive ($+$) sign indicates a product being formed.
- The denominator in the fraction is the exact stoichiometric coefficient of that species in the balanced chemical equation.

Step 3: Detailed Explanation:
Let's analyze the given rate expression part by part:
$$-\frac{1}{2} \frac{d[x]}{dt}$$: The negative sign means $x$ is a reactant. The fraction $\frac{1}{2}$ means its coefficient is 2.
$$-\frac{d[y]}{dt}$$: (which is $-\frac{1}{1} \frac{d[y]}{dt}$). The negative sign means $y$ is a reactant. Its coefficient is 1.
$$+\frac{1}{2} \frac{d[z]}{dt}$$: The positive sign means $z$ is a product. The fraction $\frac{1}{2}$ means its coefficient is 2.
Putting it all together, the reactants are $2x$ and $y$, and the product is $2z$.
The balanced equation is: $2x + y \rightarrow 2z$.

Step 4: Final Answer:
The reaction is $2x + y \rightarrow 2z$, matching option (b).