In the bohr model of the hydrogen atom a single electron...

In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius $r .$ Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron's speed. (b) Obtain an expression for the electron's kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using $r=$ $5.29 \times 10^{-11} \mathrm{m} .$ Give your numerical result in joules and in electron volts.

Transcript

00:01 To derive the borer's model for a hydrogen atom.

00:07 So in the borne, there is a proton in the center.

00:12 There is a nucleus with a mass m capital and an electron rotating around this nuclear with the mass of m small.

00:21 And it charges, charge of an electron and charge of a nuclei of the nucleus charge of a proton.

00:28 So, it means that second newton's law for the system becomes the following.

00:34 It's electric force, which is kq, nuclear, multiplied by q, electron divided by squared r, where r is the radius of the shell, or it equals to k e squared divided by r squared, and it equals to the acceleration of an electron multiplied by its mass.

01:03 And it's here electron is rotating around therefore its acceleration equals to square velocity divided by the radius so now let's calculate the velocity from this expression so again i'm going to rewrite it velocity equals to the square root of k e squared multiplied by r divided by m r squared or we can simplify it equals to the square root of k e squared divided by m r where m is a mass of an electron so we can calculate velocity now it equals to the square root of 9 multiplied by 10 power by 9 nuten's meter square over column squared multiplied by 1 .6 times 10 power by negative 19 quorum squared divided by 9 .11 times 10 power by negative 31 kilograms multiplied by 5 .39 multiplied by 10 power by negative 11 meters.

02:49 So let's use a calculator to get the result.

03:35 It equals to 2 .19 times 10 power by 6 meters per second.

03:46 So we'll answer the question now we have to obtain the expression for kinetic energy and we have to show that kinetic energy equals to one half of the potential energy.

04:10 Let's do this.

04:15 So the kinetic energy equals to mass of electron multiplied by its velocity squared over 2 and potential energy equals to negative k multiply by e squared divided by r squared so k negative sign comes from the fact that both the electron and proton have positive charge have opposite charges so now let's use this expression for kinetic energy sorry for velocity to calculate to recalculate the kinetic energy.

05:33 So it becomes k e squared over 2r, so which equals to one half of the potential energy, as we were asked to show.

05:50 Now, lastly, we have to obtain an expression for a total energy and evaluated...

Question

Please give Ace some feedback