# How to Calculate Effective Nuclear Charge

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Effective nuclear charge refers to the charge felt by the outermost (valence) electrons of a multi-electron atom after the number of shielding electrons that surround the nucleus is taken into account. The trend on the periodic table is to increase across a period and increase down a group.

## Effective Nuclear Charge Formula

The formula for calculating the effective nuclear charge for a single electron is:

Zeff *=* Z S

• Zeff is the effective nuclear charge, or Z effective
• Z is the number of protons in the nucleus, the atomic number
• S is the average amount of electron density between the nucleus and the electron

## Calculating Effective Nuclear Charge

Calculating effective nuclear charge involves understanding the Z and S values. Z is atomic number, and S requires the use of Slater’s Rules to determine an electron cloud shielding value between the nucleus and the electron under consideration.

## Step 1: Find Atomic Number to Determine Z Value

Example problem: What is the effective nuclear charge for the valence electron in sodium?

Z is the number of protons in the nucleus of the atom, and this determines the positive charge of the nucleus. The number of protons in the nucleus of an atom is also known as the atomic number.

Using a periodic table of elements, locate the desired atomic number. In the example above, sodium, symbol Na, has atomic number 11.

## Step 2: Write the Electron Configuration

Write the electron configuration of the element in the following order and groupings:

(1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d), (4f), (5s, 5p), (5d), (5f). . .

Recall that the numbers (1, 2, 3. . .) correspond to the principal quantum number or energy shell level of the electrons in the atom, and this designates how far away the electrons are from the nucleus. The letters (s, p, d, f) correspond to the given shape of an electron's orbital. For example, "s” is a spherical orbital shape, and "p" resembles a figure 8.

For sodium, the electron configuration is (1s2) (2s2, 2p6) (3s1).

In the example above, sodium has 11 electrons: two electrons in the first energy level (1), eight electrons in the second energy level (2), and one electron in the third energy level. The electron in the 3s1 orbital is the focus of the example.

## Step 3: Attribute a Shielding Value to Each Electron

The value S may be calculated using Slater’s Rules, named after the scientist John C. Slater who developed them. These rules give shielding values to each electron. Do not include a value of the electron of interest. Assign the following values:

1. Any electrons to the right of the electron of interest contain no shielding value.
2. Electrons in the same group (as found in the electron configuration grouping in Step 2) as the electron of interest shields 0.35 nuclear charge units.
3. For s or p electrons: electrons with one less value of the principal quantum number (energy level: 1, 2, 3. . .) are assigned 0.85 units of nuclear charge. Electrons found two or more energy levels lower shield 1.00 unit.
4. For d or f electrons: all electrons shield 1.00 unit.

For the example above, the answers for Na would be:

1. 0; there are no electrons higher (or to the right in the electronic configuration)
2. 0; there are no other electrons in the 3s orbital of Na.
3. 8.8; Requires two calculations: first, there are eight electrons in the energy level 2 shell, two in the s shell, and six in the p; 8 × 0.85 = 6.8. Plus, since the 1s2 electrons are two levels from the electron of interest: 2 × 1.
4. 0; there are no d or f electrons.

## Step 4: Sum the S Values

Add all the shielding charges calculated using Slater’s Rules.

In the sample problem, the shielding values sum to 8.8 (0 + 0 + 8.8 + 0).

## Step 5: Find Z Effective Using Formula

Place the values for Z and S into the effective nuclear charge formula:

Zeff *=* Z S

In the above example for Na: 11 − 8.8 = 2.2

The effective nuclear charge of the 3s1 electron in the sodium atom is 2.2. Note the value is a charge and contains no units.