A) \[5.79\times {{10}^{6}}m{{s}^{-1}}\]
B) \[5.79\times 10\,m{{s}^{-1}}\]
C) \[5.79\times {{10}^{8}}\,m{{s}^{-1}}\]
D) \[579\,\times {{10}^{5}}\,m{{s}^{-1}}\]
Correct Answer: A
Solution :
By Heisenberg's uncertainty principle \[\Delta p\times \Delta x\ge \frac{h}{4\pi }\] or \[\Delta v\times \Delta x\ge \frac{h}{4\pi m}\] \[\Delta p\to \,\]uncertainty in momentum \[\Delta x\to \,\]uncertainty in position \[\Delta v\to \,\]uncertainty in velocity \[m\to \,\]mass of particle Given that, \[\Delta x=0.1\,{\AA}=0.1\times {{10}^{-10}}\,m\] \[\Delta x=0.1\,{\AA}=0.1\times {{10}^{-10}}\,m\] \[m=9.11\times {{10}^{-31}}\,kg\] \[h=Planck\,\text{constant}\,=6.626\times {{10}^{-34}}\,Js\] \[\pi =3.14\] In uncertain position \[\Delta v\times \Delta x=\frac{h}{4\pi m}\] \[\Delta v\times 0.1\times {{10}^{-10}}=\frac{6.626\times {{10}^{-34}}}{4\times 3.14\times 9.11\times {{10}^{-31}}}\] \[\Delta v=\frac{6.626\times {{10}^{-34}}}{4\times 3.14\times 9.11\times {{10}^{-31}}\times 0.1\times {{10}^{-10}}}\,m{{s}^{-1}}\] \[=5.785\times {{10}^{6}}\,m{{s}^{-1}}\] \[=5.79\times {{10}^{6}}\,m{{s}^{-1}}\]
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