# Why Use Nuclear Charge In Finding Energy Value of Singular Electron?

• member 659869
In summary, the formula for finding energy values for electron shells, as given by the textbook, is E = (-1312kJ/n^2). However, when dividing by 1 mol to get the energy value for each electron, the actual equation is E = (-2.178 x 10^-18 / n^2) J. This equation does not take into account the nuclear charge, which is only necessary for multi-electron atoms or hydrogen atoms. When calculating the net change in energy for an electron between two energy levels, there is no simple formula for multi-electron atoms, and numerical methods must be used. Approximate formulas can be used for atoms with a single valence electron, using quantum defect theory.
member 659869
My Textbook says this is the formula to find energy values for electron shells:

$$E_{mol of electrons} = \frac{-1312kJ}{n^2}$$

where $n$ is in electron shell number

But when we divide by 1 mol to get the energy value for each electron we get

$$E_{electron} = \frac{-2.178 \cdot 10^{-18}}{n^2} J$$

but the actual equation is (as given by textbook) rather

$$E_{electron} = \frac{-2.178 \cdot 10^{-18} \cdot Z^2}{n^2} J$$

Where $Z$ is nuclear charge. Why must we introduce $Z$ and why is it not used in the first equation?

Usually, if an equation for the energy an electron doesn't take into account the nuclear charge, it is because it is an equation for a hydrogen (##Z=1##) or for a multi-electron atom where only the outermost electron is taken into account, such that one can take the nucleus plus the other electrons as a single entity of net charge +1 (which is always an approximation).

Thanks! Second question: if i wanted to find the net change in an electron's energy for an arbitrary atom when going from say, ##n=4## to ##n=2##, would I use Rydberg's Equation? I have asked this question on another site where people have said no closed form exists, and you would have to use numerical methods, but that was for finding the electron energy values of ##n=4## and ##n=2##, not for finding the net difference.

It doesn't matter whether you want to difference or the absolute value. There is no simple formula for multi-electron atoms. In addition, the energy does not depend only on ##n##, but also on ##l##.

There are some approximate formulas that can be used, especially for atoms with a single valence electron (alkali atoms), using quantum defect theory.

## 1. Why is nuclear charge used to find the energy value of a singular electron?

Nuclear charge is used because it is the force that holds the electrons in an atom. The energy of an electron is determined by its distance from the nucleus and the strength of the nuclear charge. Therefore, nuclear charge is a crucial factor in determining the energy value of a singular electron.

## 2. How does nuclear charge affect the energy of an electron?

Nuclear charge has a direct impact on the energy of an electron. The higher the nuclear charge, the stronger the force holding the electron in place, resulting in a higher energy level for the electron. This is because the electron has to overcome a stronger force to move further away from the nucleus.

## 3. Can nuclear charge be used to find the energy of any electron in an atom?

No, nuclear charge can only be used to find the energy of a singular electron in a hydrogen atom. This is because in atoms with more than one electron, the nuclear charge is shielded by the other electrons, making it less effective in determining the energy of a specific electron.

## 4. How is nuclear charge calculated in finding the energy value of a singular electron?

Nuclear charge is calculated by multiplying the charge of the nucleus by the number of protons. In the case of a hydrogen atom, the nuclear charge is simply +1, as there is only one proton in the nucleus. However, in atoms with more than one electron, the nuclear charge is reduced by the shielding effect of the other electrons.

## 5. What is the significance of using nuclear charge in finding the energy value of a singular electron?

Using nuclear charge allows us to understand the energy levels and behavior of electrons in an atom. It also helps us to predict the reactivity and chemical properties of elements. Without considering nuclear charge, our understanding of atomic structure and electron behavior would be incomplete.

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