The \"size\" of the atom in Rutherford\'s model is about 1.0 times 10^-10 m. (a)

Question

The "size" of the atom in Rutherford's model is about 1.0 times 10^-10 m. (a) Determine the speed of an electron moving about the proton using the attractive electrostatic force between an electron and a proton separated by this distance. (b) Does this speed suggest that Einsteinian relativity must be considered in studying the atom? (c) Compute the de Broglie wavelength of the electron as it moves about the proton. (d) Does this wavelength suggest that wave effects, such as diffraction and interference, must be considered in studying the atom?

Explanation / Answer

part a:

mass of electron=me=9.1*10^(-31) kg

charge on proton=q=1.6*10^(-19) C

charge on electron=-q

radius of the circular path=r=10^(-10) m

equating the forces:

electrical force=centripetal force

=>k*q*q/r^2=m*v^2/r

==>v=sqrt(k*q^2/(m*r))

where k=coloumb's constant

so v=sqrt(9*10^9*(1.6*10^(-19))^2/(9.1*10^(-31)*10^(-10)))

=1.5912*10^6 m/s

part b:

v/speed of light=0.0053

so theory of relativity can be ignored here.

part c:

de broglie wavelength=h/p

=h/(m*v)

=6.626*10^(-34)/(9.1*10^(-31)*1.5912*10^6)

=4.576*10^(-10) m

part d:

as wavelength and size of the atom are comparable, the wave effect needs to be taken into consideration.

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