Question
Consider that an electron is revolving in an excited state of Hydrogen atom with velocity $\sqrt{25.6} \times 10^{5}{\text{ }ms}^{- 1}$. The radius of the orbit is $x \times 10^{- 9}\text{ }m$. The value of $x$ is:
[Take the mass of electron to the $9 \times 10^{- 31}\text{ }kg$, charge of electron $= - 1.6 \times 10^{- 19}C$ and $\frac{1}{4\pi\varepsilon_{0}} = 9 \times 10^{9}{Nm}^{2}C^{- 2}$ ]
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Solution
Sol. [Key Concept: Bohr model — properties of an excited-state electron]
$$\begin{matrix} & F_{c} = \frac{mv^{2}}{r} \\ & \ \Rightarrow \frac{kq_{1}q_{2}}{r^{2}} = \frac{mv^{2}}{r} \\ & \ \Rightarrow \frac{ke^{2}}{r} = mv^{2} \Rightarrow r = \frac{ke^{2}}{mv^{2}} = \frac{9 \times 10^{9} \times \left( 1.6 \times 10^{- 19} \right)^{2}}{9 \times 10^{- 31} \times 25.6 \times 10^{10}} \\ & \ \Rightarrow r = \frac{2.56 \times 10^{9} \times 10^{- 38}}{25.6 \times 10^{- 21}} \\ & r = 1 \times 10^{- 9}\text{ }m \\ & \ \therefore x = 1 \end{matrix}$$
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