Question

{"given":"Reaction: $2A + B \\rightarrow$ Products. When concentration of B alone was doubled, the half-life did not change. When concentration of A alone was doubled, the rate increased by two times.","key_observation":"When half-life is independent of concentration, the reaction is first order with respect to that reactant. When rate doubles upon doubling concentration, the reaction is first order with respect to that reactant. The overall reaction order determines the units of the rate constant.","option_analysis":[{"label":"(A)","text":"$L \\, mol^{-1} \\, s^{-1}$","verdict":"correct","explanation":"Step 1: Analyze the effect of doubling [B]:\nSince half-life doesn't change when [B] is doubled, the reaction is first order w.r.t. B\n$$t_{1/2} = \\frac{0.693}{k} \\text{ (independent of concentration for first order)}$$\nStep 2: Analyze the effect of doubling [A]:\nWhen [A] is doubled, rate doubles, so reaction is first order w.r.t. A\n$$\\text{Rate} \\propto [A]^1$$\nStep 3: Determine overall order and units:\nOverall order = 1 + 1 = 2\n$$\\text{Rate} = k[A][B]$$\n$$\\text{Units of } k = \\frac{mol \\, L^{-1} \\, s^{-1}}{(mol \\, L^{-1})^2} = L \\, mol^{-1} \\, s^{-1}$$"},{"label":"(B)","text":"no unit","verdict":"incorrect","explanation":"This would be correct for a zero-order reaction where rate = k. However, our analysis shows this is a second-order reaction overall, so the rate constant must have units."},{"label":"(C)","text":"$mol \\, L^{-1} \\, s^{-1}$","verdict":"incorrect","explanation":"These are the units of reaction rate, not the rate constant. The rate constant units depend on the overall reaction order, which is 2 in this case."},{"label":"(D)","text":"$s^{-1}$","verdict":"incorrect","explanation":"This would be correct for a first-order reaction. However, our reaction is second-order overall (first order w.r.t. both A and B), so different units are required."}],"answer":"(A)","formula_steps":[]}