For a rxn A+B+C -> D, the following results were obtained from 4 experiments. Write the rate law of the rxn, derive the rate law, derive the order of the reaction

Compare Experiment 1 and Experiment 2, where [A] and [C] are constant.

$$\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{k[A]_2^x[B]_2^y[C]_2^z}{k[A]_1^x[B]_1^y[C]_1^z}$$ $$\frac{0.006 \text{ M s}^{-1}}{0.006 \text{ M s}^{-1}} = \frac{k(0.01 \text{ M})^x(0.2 \text{ M})^y(0.02 \text{ M})^z}{k(0.01 \text{ M})^x(0.01 \text{ M})^y(0.02 \text{ M})^z}$$ $$1 = \left(\frac{0.2}{0.01}\right)^y$$ $$1 = (20)^y$$

This implies $y = 0$.
The reaction is zero order with respect to B.

Compare Experiment 1 and Experiment 4, where [A] and [B] are constant.

$$\frac{\text{Rate}_1}{\text{Rate}_4} = \frac{k[A]_1^x[B]_1^y[C]_1^z}{k[A]_4^x[B]_4^y[C]_4^z}$$ $$\frac{0.006 \text{ M s}^{-1}}{0.003 \text{ M s}^{-1}} = \frac{k(0.01 \text{ M})^x(0.01 \text{ M})^y(0.02 \text{ M})^z}{k(0.01 \text{ M})^x(0.01 \text{ M})^y(0.01 \text{ M})^z}$$ $$2 = \left(\frac{0.02}{0.01}\right)^z$$ $$2 = (2)^z$$

This implies $z = 1$.
The reaction is first order with respect to C.

Compare Experiment 1 and Experiment 3. We know $y=0$ and $z=1$.

$$\frac{\text{Rate}_3}{\text{Rate}_1} = \frac{k[A]_3^x[B]_3^y[C]_3^z}{k[A]_1^x[B]_1^y[C]_1^z}$$ $$\frac{0.018 \text{ M s}^{-1}}{0.006 \text{ M s}^{-1}} = \frac{k(0.03 \text{ M})^x(0.02 \text{ M})^0(0.02 \text{ M})^1}{k(0.01 \text{ M})^x(0.01 \text{ M})^0(0.02 \text{ M})^1}$$ $$3 = \left(\frac{0.03}{0.01}\right)^x \left(\frac{0.02}{0.02}\right)^1$$ $$3 = (3)^x (1)$$

This implies $x = 1$.
The reaction is first order with respect to A.

Using the determined orders $x=1, y=0, z=1$:

$$\text{Rate} = k[A]^1[B]^0[C]^1$$ $$\text{Rate} = k[A][C]$$

The rate law is:

$$\boxed{\text{Rate} = k[A][C]}$$

Using data from Experiment 1:
Rate = $0.006 \text{ M s}^{-1}$
$[A] = 0.01 \text{ M}$
$[C] = 0.02 \text{ M}$

$$0.006 \text{ M s}^{-1} = k(0.01 \text{ M})(0.02 \text{ M})$$ $$0.006 \text{ M s}^{-1} = k(0.0002 \text{ M}^2)$$ $$k = \frac{0.006 \text{ M s}^{-1}}{0.0002 \text{ M}^2}$$ $$k = 30 \text{ M}^{-1}\text{s}^{-1}$$

The rate constant is:

$$\boxed{k = 30 \text{ M}^{-1}\text{s}^{-1}}$$

The overall order of the reaction is the sum of the individual orders:
Overall order = $x + y + z = 1 + 0 + 1 = 2$.
The overall order of the reaction is:

$$\boxed{\text{2nd order}}$$