To determine the rate constant, we first need to determine the order of each reactant. The rate of a reaction is generally given by the equation:
rate = k[A]^x[B]^y[C]^z
where k is the rate constant, [A], [B], and [C] are the concentrations of the reactants, and x, y, and z are the orders of the reaction with respect to A, B, and C, respectively.
In order to determine the order of each reactant, we can compare the initial rates of the reaction for different sets of reactant concentrations.
Let's start by comparing runs 1 and 4, where only the concentration of reactant B changes.
Run 1: (A, B, C) = (0.200, 0.100, 0.600) -> Rate = 5.00
Run 4: (A, B, C) = (0.200, 0.100, 0.200) -> Rate = 5.00
Since the rate remains the same even though the concentration of B changes, we can conclude that the reaction is zero-order with respect to B.
Next, let's compare runs 1 and 5, where the concentration of reactant C changes.
Run 1: (A, B, C) = (0.200, 0.100, 0.600) -> Rate = 5.00
Run 5: (A, B, C) = (0.200, 0.200, 0.400) -> Rate = 20.0
Since the rate quadruples when the concentration of C is halved, we can conclude that the reaction is first-order with respect to C.
Now, let's compare runs 1 and 2, where the concentration of reactant B changes.
Run 1: (A, B, C) = (0.200, 0.100, 0.600) -> Rate = 5.00
Run 2: (A, B, C) = (0.200, 0.400, 0.400) -> Rate = 80.0
Since the rate increases eightfold when the concentration of B is doubled, we can conclude that the reaction is third-order with respect to B.
Finally, let's compare runs 1 and 3, where the concentration of reactant A changes.
Run 1: (A, B, C) = (0.200, 0.100, 0.600) -> Rate = 5.00
Run 3: (A, B, C) = (0.600, 0.100, 0.200) -> Rate = 15.0
Since the rate triples when the concentration of A triples, we can conclude that the reaction is first-order with respect to A.
Putting it all together, the rate equation becomes:
rate = k[A]^1[B]^3[C]^1
Now, we can use any of the given runs to determine the rate constant (k). Let's use run 1:
Rate = k[A]^1[B]^3[C]^1
5.00 = k(0.200)^1(0.100)^3(0.600)^1
5.00 = k(0.0002)(0.001)(0.36)
5.00 = k(0.000000072)
k = 5.00 / 0.000000072 ≈ 69444
Therefore, the rate constant (k) for this reaction is approximately 69444.
Looking at the answer choices provided, none of them match exactly, so it seems there might be a mistake in the calculations or the options provided.