Step 1: Understanding the Concept:
The "order" of a reaction with respect to a reactant is the power to which its concentration is raised in the rate law equation.
It indicates how sensitively the reaction rate responds to changes in the concentration of that specific reactant.
A first-order dependency means the rate is directly proportional to concentration.
A zero-order dependency means the rate is independent of concentration.
By observing experimental data (how the rate changes as concentrations are manipulated), we can algebraically solve for these orders.
Step 2: Key Formula or Approach:
Rate Law: \(\text{Rate} = k [A]^x [B]^y\).
Where \(x\) is order w.r.t. \(A\) and \(y\) is order w.r.t. \(B\).
Step 3: Detailed Explanation:
Let the initial rate be \(R_0 = k [A]^x [B]^y\).
From the first observation: Doubling \([A]\) doubles the rate.
\[ 2R_0 = k (2[A])^x [B]^y \implies 2 = 2^x \implies x = 1 \]
The reaction is first order with respect to \(A\).
From the second observation: Doubling both \([A]\) and \([B]\) doubles the rate (compared to the original \(R_0\)).
\[ 2R_0 = k (2[A])^1 (2[B])^y \]
Since \(R_0 = k [A]^1 [B]^y\), we can substitute:
\[ 2R_0 = 2 \cdot (k [A]^1) \cdot 2^y [B]^y = 2 \cdot 2^y \cdot R_0 \]
\[ 2 = 2 \cdot 2^y \implies 1 = 2^y \implies y = 0 \]
The reaction is zero order with respect to \(B\).
Step 4: Final Answer:
The order with respect to \(A\) is \(1\) and with respect to \(B\) is \(0\).