Question 18.3: The decomposition of N2O5 is an important process in troposp...

Thermodynamics, Statistical Thermodynamics, & Kinetics

The decomposition of $N_{2}O_{5}$ is an important process in tropospheric chemistry. The half-life for the first-order decomposition of this compound is $2.05×10^{4} s$ . How long will it take for a sample of $N_{2}O_{5}$ to decay to $60\%$ of its initial value?

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Using Equation (18.29), the rate constant for the decay reaction is determined using the half-life as follows:

$\ln\left[ A \right]=\ln\left[ A \right]_{0}-kt$ (18.29)

k=\frac{\ln2}{t_{1/2}}=\frac{\ln 2}{2.05 ×10^{4}s}=3.38 ×10^{-5}s^{-1}

The time at which the sample has decayed to $60\%$ of its initial value is then determined using Equation (18.27):

$\left[ A \right]=\ln\left[ A \right]_{0}e^{-kt}$ (18.27)

\left[ N_{2}O_{5} \right]=0.6\left[ N_{2}O_{5} \right]_{0}=\left[ N_{2}O_{5} \right]_{0}e^{-\left( 3.38×10^{-5} s^{-1}\right)t}

0.6=e^{-\left( 3.38×10^{-5} s^{-1}\right)t}

\frac{-\ln\left( 0.6 \right)}{3.38×10^{-5} s^{-1}}=t=1.51 ×10^{4} s

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