# Estimating Distances Between Electrons in He Atom

• strangequark
Your name]In summary, the question asks for the distance between two electrons in a helium atom in the ground state and the first excited state. The calculation for the ground state results in an undefined distance, as the two electrons are in the same orbital. The calculation for the first excited state involves solving for the atomic number of helium, which is then used to calculate the distance between the two electrons. The final result is r~\frac{a_{0}}{2}.
strangequark
b]1. Homework Statement [/b]
Given the observed spectrum of helium, estimate the distance between two electrons in a helium atom (a) in the ground state and (b) in the first excited state. Neglect the exchange energy.

## Homework Equations

$$E_{1}=-78eV$$
$$E_{2}=-58eV$$

Given in my textbook,
"Hartree theory predicts that the radius of the n=1 shell is smaller than that of the n=1 shell of hydrogen by approximately a factor of $$\frac{1}{Z-2}$$"

$$r~\frac{r_{hydrogen}}{Z-2}$$

## The Attempt at a Solution

I'm confused by the text above. Doesn't $$Z_{helium}=2$$, and hence $$Z-2=0$$?

Another source I found says that,

$$E_{n}=\frac{n^{2}(-13.6eV)}{(Z-2)^{2}}$$

and,

$$r~\frac{a_{0}}{Z-2}$$

I can solve the first eq. for Z-2, but again this doesn't really make sense to me because I get Z-2=2.39 (for ground state) which would imply that Z=4.39? Also using this, I don't get the answers that are in the key.

Please point me in the right direction... thanks.

Thank you for your question. The text you referenced from your textbook is correct. The Z in the equation refers to the atomic number of the element, which in this case is 2 for helium. Therefore, the calculation for the distance between two electrons in a helium atom would be as follows:

(a) In the ground state:
Using the equation r~\frac{r_{hydrogen}}{Z-2}, we can calculate the distance between two electrons in a helium atom as:
r~\frac{a_{0}}{Z-2} = \frac{a_{0}}{2-2} = \frac{a_{0}}{0} = undefined

This result makes sense, as the ground state of helium is a singlet state where the two electrons are in the same orbital, thus the distance between them is undefined.

(b) In the first excited state:
Using the equation E_{n}=\frac{n^{2}(-13.6eV)}{(Z-2)^{2}}, we can calculate the energy of the electron in the first excited state as:
E_{2}=\frac{2^{2}(-13.6eV)}{(2-2)^{2}} = -13.6eV

Now we can use this energy value in the equation E_{n}=\frac{n^{2}(-13.6eV)}{(Z-2)^{2}} to solve for the distance between two electrons in the first excited state:

-13.6eV = \frac{2^{2}(-13.6eV)}{(Z-2)^{2}}
(Z-2)^{2} = 2^{2}
Z-2 = 2
Z = 4

Therefore, the distance between two electrons in the first excited state of helium is:
r~\frac{a_{0}}{Z-2} = \frac{a_{0}}{4-2} = \frac{a_{0}}{2}

I hope this helps clarify the confusion you had. Let me know if you have any further questions. Keep up the good work in your studies!

## 1. How do you estimate the distance between electrons in a helium atom?

The distance between electrons in a helium atom can be estimated using the Bohr model, which states that the electrons orbit the nucleus in specific energy levels. By calculating the radius of each energy level, the distance between the electrons can be estimated.

## 2. Why is it important to estimate the distance between electrons in a helium atom?

Understanding the distance between electrons in a helium atom is important for understanding the behavior and properties of atoms. It can also help to predict and explain various chemical reactions and interactions between atoms.

## 3. What factors affect the distance between electrons in a helium atom?

The distance between electrons in a helium atom is primarily affected by the number of protons in the nucleus and the number of electrons in the atom. Other factors that may influence the distance include the energy level of the electrons, the presence of other atoms or molecules, and external forces such as electric fields.

## 4. How accurate are estimates of the distance between electrons in a helium atom?

The estimates of the distance between electrons in a helium atom are based on theoretical models and may not be entirely accurate. However, they provide a useful approximation for understanding the behavior of atoms and molecules.

## 5. Can the distance between electrons in a helium atom change?

The distance between electrons in a helium atom is not fixed and can change depending on the energy level of the electrons. When an atom gains or loses energy, the electrons can move to different energy levels, resulting in a change in the distance between them.

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