In a first order reaction, $N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g)$, the initial concentration of $N_2O_5$ was $1.6 \times 10^{-3} mol\,lit^{-1}$ at $300\,K$. The concentration of $N_2O_5$ after 23 minutes was $0.8 \times 10^{-3} mol\,lit^{-1}$. ($\log 2 = 0.3010$). Find the rate constant.