Help with determining the rate constant of a reaction

The question asked was,



> 
> The initial concentration of $\ce{N2O5}$ in the first order reaction $$\ce{N2O5 -> 2NO2 + 1/2O2}$$ is $1.24\cdot 10^{-2}~\mathrm{mol\,L^{-1}}$ at 318K.
> 
> 
> The concentration of $\ce{N2O5}$ decreases to $0.20\cdot 10^{-2}~\mathrm{mol\,L^{-1}}$ after 1 hour.  
> 
> Calculate the rate constant at this temperature.
> 
> 
> 


How i attempted to solve this is as follows:- [![Attempted Solution](https://i.stack.imgur.com/midcT.jpg)](https://i.stack.imgur.com/midcT.jpg)


The answer given, however, is $4.82 \times 10^{-4}\ \mathrm{s{-1}}$. What have I understood incorrectly?




This [Khan Academy Video](https://www.khanacademy.org/science/chemistry/chem-kinetics/copy-of-kinetics/v/first-order-reaction-with-calculus) explains it very nicely.


So since this is a first order reaction
$$\ce{ -\Delta N2O5/\Delta t = k[N2O5]}$$


This gives you the rate of reaction at the very start. A second later however, $\ce{[N2 O5]}$ decreases which will in turn change the rate of reaction. So lets apply calculus. We can write this as


$$\ce{ \frac{-d [N2O5]}{d t} = k[N2O5]}$$


Then we can integrate to find out the total change over a period of time.


$$\int\_{[N2O5]\_0}^{[N2O5]t} \frac{d[N2O5]}{[N2O5]}= \int\_0^t-k dt$$


since $-k$ is constant, we can take it outside of the integral


$$\int\_{[N2O5]\_0}^{[N2O5]t} \frac{d[N2O5]}{[N2O5]}= -k\int\_0^tdt$$


So then we get


$$ \ln([N2O5]\_t) - \ln([N2O5]\_0) =-kt$$


So putting in your values


$$\ln(0.20\cdot 10^{-2}) - \ln(1.24\cdot 10^{-2}) = -k\cdot 60\cdot 60$$


So I'm pretty sure the answer is $$5.068\cdot10^{-4}~\pu{s^{-1}}$$


 3

$k=\frac{2.303}{t}\*\log{\frac{a}{a-x}}$


For first order reaction . here $a,(a-x)$ are initial and final reactant concentration.


$k\to rate constant$


$t\to time$


Instantaneous rate is proportional to instantaneous concentration of the reactant . So, u can't take the value of conc. as ini conc. and $dx$ is small change in conc. so u can't take it as $C\_{1}-C\_{2}$


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