# Energy Needed to Remove an Electron

## Homework Statement

If the total energy needed to remove an electron from a hydrogen atom is 24.6 eV, what is the distance that the electron is from the nucleus?

## Homework Equations

I'm not really sure....I know that the answer is $$r = 5.85\times10^-11$$.

## The Attempt at a Solution

As you can see, I know the answer, but I'm just not sure how to get to it. So far, I have thought of applying this process:

$$E_{e} = \frac{kq_{1}q_{2}}{r}$$

Technically, I get the answer if I sub in:$$q_{1}=q_{2}=e$$, $$E_{e}=24.6e$$, and solve for r.

Is this the correct process? I'm not really sure about the rationale behind it...lol I'm almost positive that there is another way.

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dynamicsolo
Homework Helper
What you've written is the expression for the electrostatic potential energy between two charges q_1 and q_2 separated by a distance r. Yes, this is what you want to use.

Make sure you put everything into SI units, which means you'll need to convert that energy of 24.6 electron-volts into Joules...

Thanks for the help!

Andrew Mason
Homework Helper

## Homework Statement

If the total energy needed to remove an electron from a hydrogen atom is 24.6 eV, what is the distance that the electron is from the nucleus?

## Homework Equations

I'm not really sure....I know that the answer is $$r = 5.85\times10^-11$$.

## The Attempt at a Solution

As you can see, I know the answer, but I'm just not sure how to get to it. So far, I have thought of applying this process:

$$E_{e} = \frac{kq_{1}q_{2}}{r}$$

Technically, I get the answer if I sub in:$$q_{1}=q_{2}=e$$, $$E_{e}=24.6e$$, and solve for r.

Is this the correct process? I'm not really sure about the rationale behind it...lol I'm almost positive that there is another way.
Your answer is correct. You can figure out the rationale from the definition of work. Work = Force x distance. Since the force changes with distance, you have to integrate (from beginning and end points, r to infinity):

$$W = \int_r^\infty F\cdot ds = \int_r^\infty -\frac{kQq}{r^2} dr = kQq/r - kQq/\infty$$

AM

Andrew Mason
$$W = \int_r^\infty F\cdot ds = \int_r^\infty \frac{kQq}{r^2} dr = \frac{kQq}{r} - \frac{kQq}{\infty}$$