The minimum uncertainty in the speed of an electron in an one dimensional region of length $2 \mathrm{a}_{\mathrm{o}}$ (Where $\mathrm{a}_{\mathrm{o}}=$ Bohr radius $52.9 \,\mathrm{pm}$) is _________ $\mathrm{km} \,\mathrm{s}^{-1}$.

(Given : Mass of electron = 9.1 $\times$ 10^$-$31 kg, Planck's constant h = 6.63 $\times$ 10^$-$34 Js)

Heisenberg's uncertainty principle <br/> $$ \begin{aligned} & \Delta \mathrm{x} \times \Delta \mathrm{P}_{\mathrm{x}} \geq \frac{h}{4 \pi} \\ & \Rightarrow 2 \mathrm{a}_{0} \times \mathrm{m} \Delta \mathrm{v}_{\mathrm{x}}=\frac{h}{4 \pi} \text { (minimum) } \\ & \Rightarrow \Delta \mathrm{v}_{\mathrm{x}}=\frac{h}{4 \pi} \times \frac{1}{2 a_{0}} \times \frac{1}{m} \\ & =\quad 6.63 \times 10^{-34} \end{aligned} $$<br/> <hr style="margin: 8px 0"/> $$ \begin{aligned} &4 \times 3.14 \times 2 \times 52.9 \times 10^{-12} \times 9.1 \times 10^{-31} \\ & =548273 \,\mathrm{~ms}^{-1} \\ & =548.273 \,\mathrm{kms}^{-1} \\ & =548 \,\mathrm{kms}^{-1} \end{aligned} $$