How Do We Calculate Electron Speed and Radial Acceleration in Atomic Models?

[Shackleford]

Calculate the speed and radial acceleration for an electron in the hydrogen atom. Do the same for the Li++ ion.

v = e (4pie0mr)^(-1/2)

ar = (v^2)/r

r ~ 10^-10 m

v = 2.24 x 10^6

The book makes note that we can allow a nonrelativistic treatment since the velocity ~ .007c.

And for the lithium ion I simply multiplied this by the square root of three since the charge would be +3e from the three-proton lithium atom.

However, calculating the radial acceleration yields phenomenally high accelerations: 1.012 x 10^23. Now, I know I'm doing something wrongly here.

 

[vela]

You're not doing anything wrong. Your sense of "normal" is just off because the speed and radius of the electron's orbit in the Bohr model are not typical values you encounter. Even though the electron is moving at a non-relativistic speed, it's still moving really fast, and the radius of its orbit is very small. Combined, they result in an acceleration that only seems ridiculous because your intuition is based on everyday experiences.

 

[Shackleford]

You're not doing anything wrong. Your sense of "normal" is just off because the speed and radius of the electron's orbit in the Bohr model are not typical values you encounter. Even though the electron is moving at a non-relativistic speed, it's still moving really fast, and the radius of its orbit is very small. Combined, they result in an acceleration that only seems ridiculous because your intuition is based on everyday experiences.

Well, I base it on the c speed limit. lol. At that acceleration, even with the theoretical 10^-10 s lifespan the velocity would still far exceed c.

 

[vela]

Radial acceleration only changes the direction an object moves, not its speed.

 

[gabbagabbahey]

At that acceleration, even with the theoretical 10^-10 s lifespan the velocity would still far exceed c.

Would it?

Remember, velocity and acceleration are vectors. They have both a magnitude and a direction. In this case, only the direction of the velocity changes, not its magnitude.

 

[Shackleford]

Radial acceleration only changes the direction an object moves, not its speed.

http://bluejaunte.files.wordpress.com/2010/01/double_facepalm.jpg

I know that. Crap. Now, I feel silly. Sorry. It only changes the direction of the velocity vector, not its magnitude.

 

[gabbagabbahey]

Radial acceleration only changes the direction an object moves, not its speed.

In this case, yes. But in general, this isn't true of course.

 

[vela]

In this case, yes. But in general, this isn't true of course.

Yes, you're right. When I was posting, I was thinking "radial" wasn't the best word here, but I was just being a bit lazy.