For `2NO+O_(2)to 2NO_(2)` change if the volume of the reaction vessel is doubled , the rate of the reaction

Options

A. Will diminsh to 1/4 of initial value
B. Will diminsh to 1/8 of initial value
C. Will grow 4 times
D. will grow 8 times

Explanation

To solve the problem regarding the effect of doubling the volume of the reaction vessel on the rate of the reaction \(2NO + O_2 \rightarrow 2NO_2\), we can follow these steps:

### Step 1: Understand the reaction and its order
The given reaction is:
\[ 2NO + O_2 \rightarrow 2NO_2 \]
This is a third-order reaction because the rate law can be expressed as:
\[ \text{Rate} = k [NO]^2 [O_2]^1 \]
Where \(k\) is the rate constant, and the exponents correspond to the stoichiometric coefficients of the reactants.

### Step 2: Define the initial concentrations
Let’s assume the initial concentrations of \(NO\) and \(O_2\) are:
- \([NO] = a\) (in moles per liter)
- \([O_2] = b\) (in moles per liter)

### Step 3: Write the initial rate of the reaction
The initial rate of the reaction (\(R_1\)) can be expressed as:
\[ R_1 = k [NO]^2 [O_2] = k a^2 b \]

### Step 4: Analyze the effect of doubling the volume
When the volume of the reaction vessel is doubled, the concentrations of the reactants will change. If the volume is doubled, the new concentrations will be:
- \([NO] = \frac{a}{2}\)
- \([O_2] = \frac{b}{2}\)

### Step 5: Write the new rate of the reaction
The new rate of the reaction (\(R_2\)) after doubling the volume is:
\[ R_2 = k \left(\frac{a}{2}\right)^2 \left(\frac{b}{2}\right) \]
This simplifies to:
\[ R_2 = k \cdot \frac{a^2}{4} \cdot \frac{b}{2} = k \cdot \frac{a^2 b}{8} \]

### Step 6: Compare the rates
Now we can compare the new rate (\(R_2\)) with the initial rate (\(R_1\)):
\[ R_2 = \frac{1}{8} R_1 \]

### Conclusion
Thus, when the volume of the reaction vessel is doubled, the rate of the reaction decreases to \(\frac{1}{8}\) of its initial value.

### Summary of the steps:
1. Identify the reaction and its order.
2. Define the initial concentrations of the reactants.
3. Write the initial rate of the reaction.
4. Determine the new concentrations after doubling the volume.
5. Write the new rate of the reaction.
6. Compare the new rate with the initial rate.