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Question 12 An atom cap be modelled by electrons in 'orbit' around a small posi...

Jun 10, 2024

Solution

Use the de Broglie wavelength formula:

\lambda = \frac{h}{mv}

where

\lambda

is the wavelength,

h

is Planck's constant,

m

is the mass of the electron, and

v

is the velocity of the electron. Given

\lambda = 314 \, \mathrm{pm}

, we can find the velocity

v

For an electron to form a stable orbit, the circumference of the orbit must be an integer multiple of the de Broglie wavelength. The circumference of the orbit is given by

2\pi r

. For

r = 100 \, \mathrm{pm}

, the circumference is

2\pi \times 100 \, \mathrm{pm} = 628 \, \mathrm{pm}

Check if the circumference is an integer multiple of the de Broglie wavelength:

\frac{628 \, \mathrm{pm}}{314 \, \mathrm{pm}} = 2

. Since 2 is an integer, the electron can form a stable orbit with a radius of 100 pm

Answer

Yes, the electron can form a stable orbit with a radius of 100 pm.

Key Concept

de Broglie Wavelength: The de Broglie wavelength of a particle is given by

\lambda = \frac{h}{mv}

, where

h

is Planck's constant,

m

is the mass of the particle, and

v

is its velocity. For an electron to form a stable orbit, the circumference of the orbit must be an integer multiple of its de Broglie wavelength.

Explanation

Given the de Broglie wavelength of the electron and the radius of the orbit, we can determine if the orbit is stable by checking if the circumference of the orbit is an integer multiple of the de Broglie wavelength. In this case, it is, so the orbit is stable.