The rate constant k , for the reaction `N_(2)O_(5) to 2NO_(2) (g) + (1)/(2) O_(2)(g)` correspond to concentration of `N_(2)O_(5)` initially and at time t .

Options

A. `[N_(2)O_(5)]_(t) [N_(2)O_(5)]_(0) + kt`
B. `[N_(2)O_(5)]_(0) = [N_(2)O_(5)]_(t) e^(kt)`
C. `log_(10) [N_(2)O_(5)]_(t) = log_(10)[N_(2)O_(5)]_(0) - kt`
D. ln `([N_(2)O_(5)]_(0))/([N_(2)O_(5)]_(t))= kt`

Explanation

To solve the problem regarding the rate constant \( k \) for the reaction

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

we will derive the relationship between the concentration of \( N_2O_5 \) at initial time and at time \( t \).

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The given reaction shows the decomposition of dinitrogen pentoxide (\( N_2O_5 \)) into nitrogen dioxide (\( NO_2 \)) and oxygen (\( O_2 \)). This is a first-order reaction with respect to \( N_2O_5 \).

2. **Writing the Rate Law**:
For a first-order reaction, the rate law can be expressed as:
\[ -\frac{d[N_2O_5]}{dt} = k[N_2O_5]. \]

3. **Integrating the Rate Law**:
To find the concentration of \( N_2O_5 \) at time \( t \), we integrate the rate equation:
\[ \int \frac{1}{[N_2O_5]} d[N_2O_5] = -k \int dt. \]
This gives us:
\[ \ln[N_2O_5] = -kt + C, \]
where \( C \) is the integration constant.

4. **Applying Initial Conditions**:
At \( t = 0 \), let the initial concentration of \( N_2O_5 \) be \( [N_2O_5]_0 \). Thus, we have:
\[ \ln[N_2O_5]_0 = C. \]
Substituting back into our equation gives:
\[ \ln[N_2O_5] = -kt + \ln[N_2O_5]_0. \]

5. **Rearranging the Equation**:
We can rewrite this equation in terms of concentrations:
\[ \ln\left(\frac{[N_2O_5]}{[N_2O_5]_0}\right) = -kt. \]

6. **Exponentiating Both Sides**:
To eliminate the natural logarithm, we exponentiate both sides:
\[ \frac{[N_2O_5]}{[N_2O_5]_0} = e^{-kt}. \]

7. **Final Expression**:
Rearranging gives us the final expression for the concentration of \( N_2O_5 \) at time \( t \):
\[ [N_2O_5] = [N_2O_5]_0 e^{-kt}. \]

### Conclusion:
The rate constant \( k \) can be determined from the change in concentration of \( N_2O_5 \) over time using the derived formula.

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