1, write the rate law of the reaction 2 derived the rate law 3 derived the order of the reaction

To find $y$, we compare experiments where [A] and [C] are constant, but [B] changes.
Comparing Experiment 1 and Experiment 2:

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

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

To find $z$, we compare experiments where [A] and [B] are constant, but [C] changes.
Comparing Experiment 1 and Experiment 4:

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

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

To find $x$, we compare experiments where [B] and [C] are constant (or where changes in [B] don't affect the rate since $y=0$), but [A] changes.
Comparing Experiment 1 and Experiment 3, using $y=0$ and $z=1$:

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

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

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1. Write the rate law of the reaction
Based on the orders determined ($x=1, y=0, z=1$):

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

2. Derived the rate law
This involves finding the orders of reaction for each reactant (as shown in Steps 1-3) and then calculating the rate constant ($k$).
Using the rate law $\text{Rate} = k[A][C]$ and data from Experiment 1:

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

3. Derived the order of the reaction
The overall order of the reaction is the sum of the individual orders ($x+y+z$):

$$\text{Overall order} = 1 + 0 + 1 = 2$$

The overall order of the reaction is:

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