Dinitrogen pentoxide decomposes as follows
`N_2O_5 to 2NO_2 +1///2 O_2`, If `-(d[N_2O_5])/(dt)=k'[N_2O_5],(d[NO_2])/(dt)=k''[N_2O_5], (d[O_2])/(dt)=k''[N_2O_5]`. Derive a relation between in k',k'' and k''':

Options

A. `(k')/2=(k'')/3=(k'')/4`
B. `(k')^2=k''=k'''`
C. `2k'=k''=2k'''`
D. `k'=(k'')/2=2k''`

Answer

To derive a relation between \( k' \), \( k'' \), and \( k''' \) for the decomposition of dinitrogen pentoxide (\( N_2O_5 \)), we start with the reaction:

\[ N_2O_5 \rightarrow 2 NO_2 + \frac{1}{2} O_2 \]

### Step 1: Write the rate expressions for each species

From the given information, we have the following rate expressions:

1. For \( N_2O_5 \):
\[ -\frac{d[N_2O_5]}{dt} = k'[N_2O_5] \]

2. For \( NO_2 \):
\[ \frac{d[NO_2]}{dt} = k''[N_2O_5] \]

3. For \( O_2 \):
\[ \frac{d[O_2]}{dt} = k'''[N_2O_5] \]

### Step 2: Relate the rates using stoichiometry

According to the stoichiometry of the reaction:

- The change in concentration of \( N_2O_5 \) is related to the formation of \( NO_2 \) and \( O_2 \) as follows:
- For every 1 mole of \( N_2O_5 \) that decomposes, 2 moles of \( NO_2 \) are produced.
- For every 1 mole of \( N_2O_5 \) that decomposes, \( \frac{1}{2} \) mole of \( O_2 \) is produced.

### Step 3: Write the stoichiometric relationships

Using stoichiometry, we can express the rates in terms of \( N_2O_5 \):

1. For \( NO_2 \):
\[ \frac{d[NO_2]}{dt} = -\frac{1}{2} \left(-\frac{d[N_2O_5]}{dt}\right) = 2 \left(-\frac{d[N_2O_5]}{dt}\right) \]
Thus, we can write:
\[ k''[N_2O_5] = 2k'[N_2O_5] \]
This simplifies to:
\[ k'' = 2k' \]

2. For \( O_2 \):
\[ \frac{d[O_2]}{dt} = \frac{1}{2} \left(-\frac{d[N_2O_5]}{dt}\right) = \frac{1}{2} \left(-\frac{d[N_2O_5]}{dt}\right) \]
Thus, we can write:
\[ k'''[N_2O_5] = \frac{1}{2} \left(-\frac{d[N_2O_5]}{dt}\right) \]
This simplifies to:
\[ k''' = \frac{1}{2}k' \]

### Step 4: Final relations

From the above steps, we have derived the following relationships:

1. \( k'' = 2k' \)
2. \( k''' = \frac{1}{2}k' \)

### Summary of Relations

The relations between the rate constants are:
- \( k'' = 2k' \)
- \( k''' = \frac{1}{2}k' \)