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(D) According to Bohr's theory, for an electron in a hydrogen atom, the electrostatic force provides the necessary centripetal force:$\frac{1}{4\pi \e

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$1$

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(D) According to Bohr's theory, for an electron in a hydrogen atom, the electrostatic force provides the necessary centripetal force:$\frac{1}{4\pi \epsilon_0} \frac{e^2}{r^2} = \frac{m v^2}{r}$Rearranging for radius $r$:$r = \frac{1}{4\pi \epsilon_0} \frac{e^2}{m v^2}$Given values:$v = \sqrt{25.6} 	imes 10^5 	ext{ m s}^{-1} \implies v^2 = 25.6 	imes 10^{10} 	ext{ m}^2 	ext{ s}^{-2}$$m = 9 	imes 10^{-31} 	ext{ kg}$$e = 1.6 	imes 10^{-19} 	ext{ C}$$\frac{1}{4\pi \epsilon_0} = 9 	imes 10^9 	ext{ N m}^2 	ext{ C}^{-2}$Substituting these values:$r = \frac{9 	imes 10^9 	imes (1.6 	imes 10^{-19})^2}{9 	imes 10^{-31} 	imes 25.6 	imes 10^{10}}$$r = \frac{9 	imes 10^9 	imes 2.56 	imes 10^{-38}}{9 	imes 25.6 	imes 10^{-21}}$$r = \frac{2.56 	imes 10^{-29}}{25.6 	imes 10^{-21}} = 0.1 	imes 10^{-8} 	ext{ m} = 1 	imes 10^{-9} 	ext{ m}$Comparing with $x 	imes 10^{-9} 	ext{ m}$, we get $x = 1$.

$A$. Speed of revolution of electron	$i$. $\frac{1}{4 \pi \varepsilon_0} \frac{2 \pi Z e^2}{n h}$
$B$. Kinetic energy	$ii$. $-\left(\frac{1}{4 \pi \varepsilon_0}\right)^2 \frac{2 \pi^2 m e^4 Z^2}{n^2 h^2}$
$C$. Total energy	$iii$. $\left(\frac{1}{4 \pi \varepsilon_0}\right)^2 \frac{2 \pi^2 m e^4 Z^2}{n^2 h^2}$
$D$. Frequency	$iv$. $\left(\frac{1}{4 \pi \varepsilon_0}\right)^2 \frac{4 \pi^2 Z^2 e^4 m}{n^3 h^3}$

Similar Questions

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