`2N_2O_5 rarr 4 NO_2 +O_2`
If `-(D[N_2O_5])/(dt) =k_1[N_2O_5]`
`(d[NO_2])/(dt) =k_2[N_2O_5]`
` ([O_2])/(dt) =k_3[N_2O_5]`
What is the relation between ` k_1, k_2` and ` k_3 ?`.

Options

A. ` k_1 =k-2=k_3`
B. ` 2k_1= k_2 = 4k_3`
C. ` 2k_1 =4k_2 =k_3`
D. None

Explanation

To find the relationship between \( k_1 \), \( k_2 \), and \( k_3 \) based on the given reaction:

\[ 2N_2O_5 \rightarrow 4NO_2 + O_2 \]

We start by writing the rate expressions for the reaction.

### Step 1: Write the Rate of Reaction

The rate of disappearance of \( N_2O_5 \) can be expressed as:

\[ -\frac{d[N_2O_5]}{dt} = k_1 [N_2O_5] \]

The rate of appearance of \( NO_2 \) is given by:

\[ \frac{d[NO_2]}{dt} = k_2 [N_2O_5] \]

The rate of appearance of \( O_2 \) is given by:

\[ \frac{d[O_2]}{dt} = k_3 [N_2O_5] \]

### Step 2: Relate the Rates to Stoichiometry

From the stoichiometry of the reaction, we know:

- For every 2 moles of \( N_2O_5 \) that decompose, 4 moles of \( NO_2 \) are produced.
- For every 2 moles of \( N_2O_5 \) that decompose, 1 mole of \( O_2 \) is produced.

Thus, we can express the rates in terms of the stoichiometric coefficients:

\[ -\frac{1}{2} \frac{d[N_2O_5]}{dt} = \frac{1}{4} \frac{d[NO_2]}{dt} = \frac{1}{2} \frac{d[O_2]}{dt} \]

### Step 3: Substitute the Rate Expressions

Substituting the rate expressions into the stoichiometric relationships gives us:

1. From \( -\frac{1}{2} \frac{d[N_2O_5]}{dt} = \frac{1}{4} \frac{d[NO_2]}{dt} \):

\[ -\frac{1}{2} k_1 [N_2O_5] = \frac{1}{4} k_2 [N_2O_5] \]

Dividing both sides by \( [N_2O_5] \) (assuming \( [N_2O_5] \neq 0 \)):

\[ -\frac{1}{2} k_1 = \frac{1}{4} k_2 \]

Rearranging gives:

\[ k_2 = -2k_1 \]

2. From \( -\frac{1}{2} \frac{d[N_2O_5]}{dt} = \frac{1}{2} \frac{d[O_2]}{dt} \):

\[ -\frac{1}{2} k_1 [N_2O_5] = \frac{1}{2} k_3 [N_2O_5] \]

Dividing both sides by \( [N_2O_5] \):

\[ -\frac{1}{2} k_1 = \frac{1}{2} k_3 \]

Rearranging gives:

\[ k_3 = -\frac{1}{2} k_1 \]

### Step 4: Combine the Relationships

From the two relationships derived:

1. \( k_2 = -2k_1 \)
2. \( k_3 = -\frac{1}{2} k_1 \)

We can express all rates in terms of \( k_1 \):

\[ k_1 : k_2 : k_3 = 1 : -2 : -\frac{1}{2} \]

To eliminate the negative signs and make it more straightforward, we can multiply through by -1:

\[ k_1 : k_2 : k_3 = 1 : 2 : \frac{1}{2} \]

### Final Relation

Thus, the final relation between \( k_1 \), \( k_2 \), and \( k_3 \) can be expressed as:

\[ 2k_1 = k_2 = 4k_3 \]