Write the rate expression for the formation of CO2 and H2O and the disappearance of CH4 and O2 in the reaction CH4(g) + 2O2(g) -> CO2(g) + 2H2O(g).

Here are the solutions to the problems:

1. Reaction: $\text{CH}_4(\text{g}) + 2\text{O}_2(\text{g}) \rightarrow \text{CO}_2(\text{g}) + 2\text{H}_2\text{O}(\text{g})$

The general rate expression for a reaction $aA + bB \rightarrow cC + dD$ is given by:

$$\text{Rate} = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$$

where the negative sign indicates the disappearance of reactants and the positive sign indicates the formation of products.

a) The formation of $\text{CO}_2$ and $\text{H}_2\text{O}$

Step 1: Write the rate expression for the formation of $\text{CO}_2$.
The stoichiometric coefficient for $\text{CO}_2$ is 1.

$$\text{Rate} = \frac{1}{1}\frac{\Delta[\text{CO}_2]}{\Delta t} = \frac{\Delta[\text{CO}_2]}{\Delta t}$$

Step 2: Write the rate expression for the formation of $\text{H}_2\text{O}$.
The stoichiometric coefficient for $\text{H}_2\text{O}$ is 2.

$$\text{Rate} = \frac{1}{2}\frac{\Delta[\text{H}_2\text{O}]}{\Delta t}$$

b) The disappearance of $\text{CH}_4$ and $\text{O}_2$

Step 1: Write the rate expression for the disappearance of $\text{CH}_4$.
The stoichiometric coefficient for $\text{CH}_4$ is 1.

$$\text{Rate} = -\frac{1}{1}\frac{\Delta[\text{CH}_4]}{\Delta t} = -\frac{\Delta[\text{CH}_4]}{\Delta t}$$

Step 2: Write the rate expression for the disappearance of $\text{O}_2$.
The stoichiometric coefficient for $\text{O}_2$ is 2.

$$\text{Rate} = -\frac{1}{2}\frac{\Delta[\text{O}_2]}{\Delta t}$$

2. Reaction: $\text{N}_2(\text{g}) + 3\text{H}_2(\text{g}) \rightleftharpoons 2\text{NH}_3(\text{g})$

a) The formation of $\text{NH}_3$

Step 1: Write the rate expression for the formation of $\text{NH}_3$.
The stoichiometric coefficient for $\text{NH}_3$ is 2.

$$\text{Rate} = \frac{1}{2}\frac{\Delta[\text{NH}_3]}{\Delta t}$$

b) The disappearance of $\text{N}_2$ and $\text{H}_2$

Step 1: Write the rate expression for the disappearance of $\text{N}_2$.
The stoichiometric coefficient for $\text{N}_2$ is 1.

$$\text{Rate} = -\frac{1}{1}\frac{\Delta[\text{N}_2]}{\Delta t} = -\frac{\Delta[\text{N}_2]}{\Delta t}$$

Step 2: Write the rate expression for the disappearance of $\text{H}_2$.
The stoichiometric coefficient for $\text{H}_2$ is 3.

$$\text{Rate} = -\frac{1}{3}\frac{\Delta[\text{H}_2]}{\Delta t}$$

3. Reaction: $4\text{NO}_2(\text{g}) + \text{O}_2(\text{g}) \rightarrow 2\text{N}_2\text{O}_5(\text{g})$
Given: Molecular oxygen is reacting at the rate of $0.037 \text{ mol L}^{-1}\text{s}^{-1}$.
This means $-\frac{\Delta[\text{O}_2]}{\Delta t} = 0.037 \text{ mol L}^{-1}\text{s}^{-1}$.
From the stoichiometry, the overall reaction rate is $\text{Rate} = -\frac{1}{1}\frac{\Delta[\text{O}_2]}{\Delta t} = 0.037 \text{ mol L}^{-1}\text{s}^{-1}$.

a) At what rate is $\text{N}_2\text{O}_5$ being formed?

Step 1: Relate the rate of formation of $\text{N}_2\text{O}_5$ to the overall reaction rate.

$$\text{Rate} = \frac{1}{2}\frac{\Delta[\text{N}_2\text{O}_5]}{\Delta t}$$

Step 2: Calculate the rate of formation of $\text{N}_2\text{O}_5$.

$$\frac{\Delta[\text{N}_2\text{O}_5]}{\Delta t} = 2 \times \text{Rate} = 2 \times 0.037 \text{ mol L}^{-1}\text{s}^{-1} = \boxed{\text{0.074 mol L}^{-1}\text{s}^{-1}\text{}}$$

b) At what rate is $\text{NO}_2$ reacting?

Step 1: Relate the rate of reaction of $\text{NO}_2$ to the overall reaction rate.

$$\text{Rate} = -\frac{1}{4}\frac{\Delta[\text{NO}_2]}{\Delta t}$$

Step 2: Calculate the rate of reaction of $\text{NO}_2$.

$$-\frac{\Delta[\text{NO}_2]}{\Delta t} = 4 \times \text{Rate} = 4 \times 0.037 \text{ mol L}^{-1}\text{s}^{-1} = \boxed{\text{0.148 mol L}^{-1}\text{s}^{-1}\text{}}$$

4. Reaction: $4\text{PH}_3(\text{g}) \rightarrow \text{P}_4(\text{g}) + 6\text{H}_2(\text{g})$
Given: Molecular hydrogen is being formed at the rate of $0.078 \text{ mol L}^{-1}\text{s}^{-1}$.
This means $\frac{\Delta[\text{H}_2]}{\Delta t} = 0.078 \text{ mol L}^{-1}\text{s}^{-1}$.

Step 1: Calculate the overall reaction rate from the rate of formation of $\text{H}_2$.

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