Question

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

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

Hint:

Negative means left side (reactants), positive means right side (products), and the number under the derivative is the balancing coefficient. Look at the denominators directly: $x$ has 2, $y$ has 1, $z$ has 2 $\rightarrow$ $2x + y \rightarrow 2z$!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an instantaneous rate expression matching reactants and products. We need to reconstruct the balanced chemical equation based on the stoichiometric coefficients implied by the denominators and the signs.

Step 2: Key Formula or Approach:
For a generalized chemical reaction: $$ aA + bB \longrightarrow cC + dD $$ The unified rate of reaction is written as: $$ \text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} $$

• Negative signs ($-$) denote consumption of reactants.

• Positive signs ($+$) denote generation of products.

• The denominators represent the exact stoichiometric coefficients in the balanced equation.


Step 3: Detailed Explanation:
Let's systematically decode the given rate expression piece by piece: $$ \text{Rate} = -\frac{1}{2}\frac{d[x]}{dt} = -\frac{1}{1}\frac{d[y]}{dt} = \frac{1}{2}\frac{d[z]}{dt} $$

• For

$x$: The negative sign reveals it is a reactant. The fraction is $\frac{1}{2}$, meaning its stoichiometric coefficient is $2$.

• For

$y$: The negative sign reveals it is a reactant. The coefficient denominator is implicitly $1$, meaning its stoichiometric coefficient is $1$.

• For

$z$: The positive sign reveals it is a product. The fraction is $\frac{1}{2}$, meaning its stoichiometric coefficient is $2$.
Assembling the reactants and products together with their respective coefficients yields: $$ 2x + y \longrightarrow 2z $$

Step 4: Final Answer:
The balanced chemical equation matching the kinetic rate expression is $2x + y \rightarrow 2z$, which corresponds to option (B).