Question

Find the de broglie wavelength of an electron in the ground state of the hydrogen atom.

Answer 1

de Broglie wavelength (λ) is given by the equation

λ = h/p

where h=Planck’s constant whose value is 6.62 x 10^(−34) joule-seconds and

p = momentum of the particle(here electron)

In terms of kinetic energy(E) momentum(p) can be written as,

p=(2mE)^1/2

where m=mass of the particle.

Hence λ becomes

1 λ = h(2mE)^-1/2

Given here, E = 13.6 eV = 13.6×1.6×10^-19 joule

m(mass of electron)= 9.1×10^-31 kg

Putting these values in equation (1) we get ,

λ =0.332×10^(-9) meter

=3.32×10^(-10) meter

=3.32 Å

Answer 2

De Broglie wavelength of an electron in the ground state of the hydrogen atom is about 3.33 × 10⁻¹⁰ m

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Further explanation

The term of package of electromagnetic wave radiation energy was first introduced by Max Planck. He termed it with photons with the magnitude is :

$\large {\boxed {E = h \times f}}$

where:

E = Energi of A Photon ( Joule )

h = Planck's Constant ( 6.63 × 10⁻³⁴ Js )

f = Frequency of Eletromagnetic Wave ( Hz )

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Let's recall De Broglie's Wavelength Formula as follows:

$\boxed{\lambda = \frac{h}{mv}}$

where:

λ = wavelength ( m )

h = Planck's Constant ( 6.63 × 10⁻³⁴ Js )

m = mass of object ( kg )

v = velocity of object ( m/s )

Let us now tackle the problem !

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Given:

energy of the ground state of the hydrogem atom = E = 13.6 eV = 2.176 × 10⁻¹⁸ J

Asked:

wavelength of electron = λ = ?

Solution:

Firstly , we will calculate the speed of the electron :

$E = E_k$

$E = \frac{1}{2}mv^2$

$v^2 = 2E \div m$

$\boxed{v = \sqrt{ 2E \div m } }$ → Equation A

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Next, we will use the formula of The Broglie's Wavelength:

$\lambda = \frac{h}{mv}$

$\lambda = \frac{h}{m\sqrt{ 2E \div m }}$ ← Equation A

$\lambda = \frac{6.63 \times 10^{-34}}{9.11 \times 10^{-31} \sqrt{ 2 \times 2.176 \times 10^{-18} \div 9.11 \times 10^{-31} }}$

$\boxed{\lambda = 3.33 \times 10^{-10} \texttt{ m}}$

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Learn more

• Photoelectric Effect : brainly.com/question/1408276
• Statements about the Photoelectric Effect : brainly.com/question/9260704
• Rutherford model and Photoelecric Effect : brainly.com/question/1458544

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Answer details

Grade: College

Subject: Physics

Chapter: Quantum Physics