The minimum uncertainty in the speed of an electron in a one dimensional region of length \(2a_0\) (where \(a_0=\)Bohar radius \(52.9\) pm) is \(km~s^{-1}\).

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

Heisenberg's uncertainty principle

\(\Delta x\times \Delta P_x\geq \dfrac{h}{4\pi}\)

\(\Rightarrow 2a_0\times m\Delta v_x=\dfrac{h}{4\pi}\) (minimum)

\(\Rightarrow \Delta V_x=\dfrac{h}{4\pi}\times \dfrac{1}{2a_0}\times \dfrac{1}{m}\)

\(=\dfrac{6.63\times 10^{-34}}{4\times 3.14\times 2\times 52.9\times 10^{-12}\times 9.1\times 10^{-31}}\)

\(=548273~ms^{-1}\)

\(=548.273~kms^{-1}\)

\(=548~kms^{-1}\)