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Given: The mass of electron is $9.11 \times 10^{-31}$ kg, Planck constant is $6.626 \times 10^{-34}$ J s, the uncertainty involved in the measurement of velocity within a distance of 0.1 Å is :-
A
$5.79 \times 10^{6}$ m s$^{-1}$
B
$5.79 \times 10^{7}$ m s$^{-1}$
C
$5.79 \times 10^{8}$ m s$^{-1}$
D
$5.79 \times 10^{5}$ m s$^{-1}$
Detailed Solution
Heisenberg's uncertainty principle: $\Delta x \times \Delta v \geq \dfrac{h}{4\pi m}$
$\Delta x = 0.1$ Å $= 0.1 \times 10^{-10}$ m $= 1 \times 10^{-11}$ m
$\Delta v \geq \dfrac{h}{4\pi m \times \Delta x}$
$\Delta v \geq \dfrac{6.626 \times 10^{-34}}{4 \times 3.14 \times 9.11 \times 10^{-31} \times 10^{-11}}$
$\Delta v \geq \dfrac{6.626 \times 10^{-34}}{1.144 \times 10^{-40}}$
$\Delta v \geq 5.79 \times 10^{6}$ m s$^{-1}$
$\Delta x = 0.1$ Å $= 0.1 \times 10^{-10}$ m $= 1 \times 10^{-11}$ m
$\Delta v \geq \dfrac{h}{4\pi m \times \Delta x}$
$\Delta v \geq \dfrac{6.626 \times 10^{-34}}{4 \times 3.14 \times 9.11 \times 10^{-31} \times 10^{-11}}$
$\Delta v \geq \dfrac{6.626 \times 10^{-34}}{1.144 \times 10^{-40}}$
$\Delta v \geq 5.79 \times 10^{6}$ m s$^{-1}$