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A body starts from rest and moves under constant acceleration for 20 s. It covers distance $S_1$ in the first 10 s and $S_2$ in the next 10 s. Then $S_2$ equals:
Using $s = \dfrac{1}{2}at^2$ (from rest):
$S_1 = \dfrac{1}{2}a(10)^2 = 50a$
Total in 20 s: $S_{\text{total}} = \dfrac{1}{2}a(20)^2 = 200a$
$S_2 = 200a - 50a = 150a = 3 \times 50a = \mathbf{3S_1}$
The unit of equilibrium constant \(K_c\) for \(\mathrm{PCl}_5 \rightleftharpoons \mathrm{PCl}_3 + \mathrm{Cl}_2\) is:
\(\Delta n_g = (1+1)-1 = 1\). \(K_c = \frac{[\mathrm{PCl}_3][\mathrm{Cl}_2]}{[\mathrm{PCl}_5]}\) β units: \((\mathrm{mol}\ \mathrm{dm}^{-3})^2 / (\mathrm{mol}\ \mathrm{dm}^{-3}) = \mathrm{mol}\ \mathrm{dm}^{-3} = \mathrm{dm}^3/\mathrm{mol}\).
Evaluate: $\displaystyle\lim_{x\to1^-}\dfrac{\sqrt{\pi}-\sqrt{2\sin^{-1}x}}{\sqrt{1-x}}$
Let $\sin^{-1}x = \pi/2-\epsilon$ where $\epsilon\to0^+$ as $x\to1^-$. Then $x=\sin(\pi/2-\epsilon)=\cos\epsilon\approx1-\epsilon^2/2$, so $1-x\approx\epsilon^2/2$.
$\sqrt{\pi}-\sqrt{2(\pi/2-\epsilon)}=\sqrt{\pi}-\sqrt{\pi-2\epsilon}=\sqrt{\pi}\left(1-\sqrt{1-\frac{2\epsilon}{\pi}}\right)\approx\sqrt{\pi}\cdot\dfrac{\epsilon}{\pi}=\dfrac{\epsilon}{\sqrt{\pi}}$
$\sqrt{1-x}\approx\sqrt{\epsilon^2/2}=\epsilon/\sqrt{2}$
Limit $=\dfrac{\epsilon/\sqrt{\pi}}{\epsilon/\sqrt{2}}=\sqrt{\dfrac{2}{\pi}}$
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