186.

(a) The decomposition of N₂O₅(g) is a first order reaction with a rate constant of 5 x 10^–4 sec^–1 at 45°C.
i.e., 2N₂O₅(g) = 4NO₂(g) + O(g)
If initial concentration of N₂O₅ is 0.25, calculate its concentration after two minutes. Also calculate half life for the decomposition of N₂O₅(g).
(b) For an elementary reaction: 2A + B → 3C The rate of appearance of C at time ‘t’ is 1.3 x 10^–4 mol l^–1 s^–1. Calculate at this time:
(i) Rate of reaction (ii) Rate of disappearance of A.

Rate constant K = 5 x 10^–4 sec. Initial concentration [A]₀ = 0.25 M Final concentration [A]_t =? Time taken by the reaction, t = 2 min.
For a first order reaction, rate constant (K) is given by
                        $\mathrm{K} = \frac{2.303}{\mathrm{t}}\log \frac{\left[{\mathrm{A}}_0\right]}{{\left[\mathrm{A}\right]}_{\mathrm{t}}}$


$$\begin{gathered} 5\times {10}^{-4} = \frac{2.303}{\mathrm{t}}\log \frac{0.25}{{\left[\mathrm{A}\right]}_{\mathrm{t}}} \\ \frac{5\times {10}^{-4}\times 2}{2.303} = \log 0.25 - \log {\left[\mathrm{A}\right]}_{\mathrm{t}} \\ 0.0004 = \log 0.25 - \log {\left[\mathrm{A}\right]}_{\mathrm{t}} \\ 0.0004 = 0.3979 - \log {\left[\mathrm{A}\right]}_{\mathrm{t}} \\ 0.0004 - 0.3979 = -\log {\left[\mathrm{A}\right]}_{\mathrm{t}} \\ -0.3975 = -\log {\left[\mathrm{A}\right]}_{\mathrm{t}} \\ {\left[\mathrm{A}\right]}_{\mathrm{t}} = \mathrm{antilog} (0.3975) \\ {\left[\mathrm{A}\right]}_{\mathrm{t}} = 2.6 \end{gathered}$$


(b) $2\mathrm{A}+\mathrm{B} \to \mathrm{C}$

(i) The rate of appearance of C at time t

                              $= 1.3 \times {10}^{-4} \mathrm{mol} {\mathrm{L}}^{-1} {\mathrm{s}}^{-1}$

$-\frac{\mathrm{d}}{\mathrm{dt}}\left[\mathrm{C}\right] = \frac{1}{2}\frac{\mathrm{d}}{\mathrm{dt}}\left[\mathrm{A}\right] = +\frac{1}{2}\frac{\mathrm{d}}{\mathrm{dt}}\left[\mathrm{B}\right]$

(ii) Rate of disappearance of A

                  $$\begin{gathered} = 2 \times 1.3 \times {10}^{-4} \mathrm{mol} {\mathrm{L}}^{-1} {\mathrm{S}}^{-1} \\ =2.6 \times {10}^{-4} \mathrm{mol}\hspace{0.17em}{\mathrm{L}}^{-1} {\mathrm{S}}^{-1}. \end{gathered}$$