Question

For the reaction, 3A + 2B $\to$ C + D, the differential rate law can be written as :

• A. $\frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m}$
• B. $- \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m}$
• C. $+\frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m}$
• D. $-\frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m}$

Answer 1

The Correct Option is D

The given chemical reaction is:

3A + 2B \to C + D

We need to determine the correct differential rate law for this reaction.

Understanding Rate Laws:

• The rate of a chemical reaction is expressed as the change in concentration of a reactant or product per unit time.
• For a general reaction of the form: aA + bB \to cC + dD, the rate can be written as: -\frac{1}{a} \frac{d\left[A\right]}{dt} = -\frac{1}{b} \frac{d\left[B\right]}{dt} = \frac{1}{c} \frac{d\left[C\right]}{dt} = \frac{1}{d} \frac{d\left[D\right]}{dt}
• The rate law is usually given as: Rate = k[A]^{n}[B]^{m}, where k is the rate constant and n and m are the orders of reaction with respect to A and B respectively.

Applying to the Given Reaction:

• For the reaction 3A + 2B \to C + D, according to stoichiometry, the rate can be expressed as: -\frac{1}{3} \frac{d[A]}{dt} = -\frac{1}{2} \frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt}
• The differential rate law can be therefore written as: -\frac{1}{3} \frac{d[A]}{dt} = \frac{d[C]}{dt} = k[A]^{n}[B]^{m}

Examining Options:

• Option 1: \frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m} - This is incorrect because the rate of change of reactant concentration must be negative.
• Option 2: - \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m} - This does not consider the stoichiometric coefficient 3 for A.
• Option 3: +\frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m} - This is incorrect due to the positive sign for the reactant rate.
• Option 4: -\frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m} - This correctly accounts for stoichiometry and the negative rate for consumption of a reactant.

Conclusion:

The correct differential rate law for the reaction is: -\frac{1}{3} \frac{d\left[A\right]}{dt} = \frac{d\left[C\right]}{dt} = k\left[A\right]^{n} \left[B\right]^{m}.