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If potential energy is taken to be zero for an electron in ground state of hydrogen atom, then calculate its total mechanical energy in the 1st excited state.

Asked by patra04011965 | 28 Mar, 2020, 11:54: AM

Expert Answer

potential energy U₁ of ground state electron in Hydrogen atom is given by,

begin mathsize 14px style U subscript 1 equals space fraction numerator negative e squared over denominator 4 pi epsilon subscript o a subscript o end fraction end style

where e is charge of electron, ε_o is permittivity of free space and a_o is Bohr radius

potential energy U₂ of first excited state electron in Hydrogen atom is given by,

begin mathsize 14px style U subscript 2 equals space fraction numerator negative e squared over denominator 4 pi epsilon subscript o space left parenthesis space 2 a subscript o space right parenthesis end fraction end style

Above expression for potential energy is written by considering orbit radius for first excited state is 2a_o

potential energy U of first excited state electron in Hydrogen atom with respect to ground state is given by,

begin mathsize 14px style U space equals space U subscript 2 space minus space U subscript 1 equals space fraction numerator negative e squared over denominator 4 pi epsilon subscript o space left parenthesis space 2 a subscript o space right parenthesis end fraction space plus space fraction numerator e squared over denominator 4 pi epsilon subscript o space a subscript o space end fraction space equals space fraction numerator e squared over denominator 4 pi epsilon subscript o space left parenthesis space 2 a subscript o space right parenthesis end fraction end style

Speed of electron when it is in first excited state of hydrogen atom is obtained by

balancing electrostatic force with centripetal force

begin mathsize 14px style space fraction numerator e squared over denominator 4 pi epsilon subscript o space 4 space a subscript o superscript 2 end fraction space equals space m space fraction numerator v squared over denominator 2 space a subscript o end fraction end style

....................(1)

From above expression we get kinetic energy as ,

begin mathsize 14px style 1 half m space v to the power of 2 to the power of space end exponent equals space fraction numerator e squared over denominator 16 pi epsilon subscript o space a subscript o end fraction space end style

Hence total mechanical energy = Kinetic energy + Potential energy =

begin mathsize 14px style space fraction numerator e squared over denominator 4 pi epsilon subscript o space a subscript o end fraction space open square brackets 1 half plus 1 fourth close square brackets space equals space 3 over 4 cross times fraction numerator e squared over denominator 4 pi epsilon subscript o space a subscript o end fraction end style

by substituting values e = 1.602 × 10^-19 , Coulomb's constant 1/(4πε_o ) = 9 × 10⁹ N m² C^-2

and Bohr radius = 0.0529 nm

we get Total mechanical energy = 3.275 × 10^-18 J

Answered by Thiyagarajan K | 28 Mar, 2020, 02:12: PM

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