The Bohr model of a hydrogen-like ion predicts that the total energy E is given by

E = −Z²R/n²

where Z is the number of protons in the nucleus, n is the principal quantum number and R is a constant equal to approximately 13.6 electron volts (eV). This formula is the result of the total energy being equal to

E = − Ze²/(2r_n)

where e is the charge of the electron and r_n is the orbit radius when the principal quantum number is n. The orbit radius is given by

r_n = n²h/(Zm_ee²)

where h is Planck's constant divided by 2π and m_e is the mass of the electron.

Shell Structure

Electrons in atoms are organized in shells whose capacities are equal to 2m², where m is an integer. Thus there can be at most 2 electrons in the first shell, 8 in the second shell and 18 in the third shell.

Shielding by Electrons

The term hydrogen-like ion means that there is but one electron in a shell, the inner shells being filled. The value of Z in the above formulas can be considered to be the number of protons p in the nucleus less the number of electrons in inner shells. ε₀; i.e., Z=p−ε₀. Thus ε₀ positive charges of the nucleus are said to be shielded by the ε₀ electrons which are closer to the center of the atom than the electron being considered. On the other hand, there is no shielding by any electrons which are farther from the center of the atom.

Shielding by Electrons in the Same Shell

When a charge is distributed uniformally on a spherical surface it has the effect on another charge outside of the spherical equal to what that same charge would have concentrated at the center of the sphere. The effect on a charge within the sphere is zero. The effect on a charge located on the sphere is equal to what half the charge would have located at the center of the sphere. Thus if the number of electrons in a shell is denoted as ε₁ then

Z = p − ε₀ − ½(ε₁-1)

Thus the energy required to remove an electron from a shell should decrease with the number of electrons in that shell. This is due to the shielding of some of the positive charge of the nucleus by electrons in the same shell.

The negatively charged ions are created when an atom acquires enough electrons to complete a shell. For example, the fluoring atom has nine protons and nine electrons. There are two electrons in the first shell and seven in the second shell. The capacity of the second shell is eight. The fluorine ion F^- has a net negative charge yet the electrons are some how clinging to it. Often chemistry students believe that the completion of the shell of eight involves some sort of attraction for the electron. The notion of shielding of electrons in the same shell provides a differents sort of justification for the F^-. The two electrons in the inner shell shield fully two protons. For any electron in the second shell there are seven other electrons in the same shell, each shielding a half unit of positive charge each. That make the charge experienced by each of the electrons in the second shell equal to (9-2-½(7))=3.5 positive charges. That is sufficient to hold each of the electrons in the second shell. As will be seen below the reality is more complicated but this computation explains how the electrons in the F^- ion could be clinging to a system with no net positive charge.

The Ionization Energies of the Elements

The ionization energies for the first eighteen elements from the CRC Handbook of Physics and Chemistry 82nd Edition (2001-2002) are shown below.

Z	Element	I		II		III		IV		V		VI		VII		VIII
1	H	13.59844							
2	He	24.58741	54.41778						
3	Li	5.39172		75.64018	122.45429					
4	Be	9.3227		18.21116	153.89661	217.71865				
5	B	8.29803		25.15484	37.93064	259.37521	340.22580				
6	C	11.2603		24.38332	47.8878		64.4939		392.087		489.99334		
7	N	14.53414	29.6013		47.44924	77.4735		97.8902		552.0718	667.046	
8	O	13.61806	35.1173		54.9355		77.41353	113.899		138.1197	739.29		871.4101
9	F	17.42282	34.97082	62.7084		87.1398		114.2428	157.1651	185.186		953.9112
10	Ne	21.5646		40.96328	63.45		97.12		126.21		157.93		207.2759	239.0989
11	Na	5.13908		47.2864		71.62		98.91		138.4		172.18		208.5		264.25
12	Mg	7.64624		15.03528	80.1437		109.2655	141.27		186.76		225.02		265.96
13	Al	5.98577		18.82856	28.44765	119.992		153.825		190.49		241.76		284.66
14	Si	8.15169		16.34585	33.49302	45.14181	166.767		205.27		246.5		303.54
15	P	10.48669	19.7694		30.2027		51.4439		65.0251		220.421		263.57		309.6
16	S	10.36001	23.3379		34.79		47.222		72.5945		88.053		280.948		328.75
17	Cl	12.96764	23.814		39.61		53.4652		67.8		97.03		114.1958	348.28
18	Ar	15.75962	27.62967	40.74		59.81		75.02		91009		124.323		143.46

Column I gives the energy in electron volts (eV) required to separate one electron from the neutral atom. Column II is the energy required to separate a second electron; i.e., to separate one more electron from the singly ionized element. And so forth.

The data can be rearranged and displayed in terms of the number of electrons in the shell, which is the more relevant quantity:

                              Number of Electrons in the Second Shell
Element	8		7		6		5		4		3		2		1
O					13.61806	35.1173		54.9355		77.41353	113.899		138.1197
F			17.42282	34.97082	62.7084		87.1398		114.2428	157.1651	185.186
Ne	21.5646		40.96328	63.45		97.12		126.21		157.93		207.2759	239.0989
Na	47.2864		71.62		98.91		138.4		172.18		208.5		264.25	
Mg	80.1437		109.2655	141.27		186.76		225.02		265.96		
Al	119.992		153.825		190.49		241.76		284.66			
Si	166.767		205.27		246.5		303.54				

The number of electrons in a shell is the complement of the separation order. That is to say, the figure of 113.899 eV for oxygen in the V column is the energy required for the separation of an electron when there are two electron in the second shell. (There are also two electrons in the first shell.)

It was indicated above that the shielding by electrons in the same shell implies the ionization potential should decrease with the number of electrons in the shell. Here are the graphs for Oxygen, Fluorine and Neon.

IE = 160.3169813 − 25.18089514(#e-2)
r² = 0.989295149

IE = 208.2926057 − 28.54330357(#e-2)
R² = 0.988004837

IE = 262.0676493 − 31.74801429(#e-2)
R² = 0.986066674

The shielding effect of electrons in the same shell is thus borne out.

The regularity of the effect is revealed when the curves for the three elements are displayed in the same graph, as below.

The Relative Magnitude of the Shielding Effect
for Electrons in the Same Shell

According to the notion that an electron in the same shell shields a half unit of positive charge in the nucleus the effect from an increase in the positive charge of the nucleus should have twice the magnitude of the the effect on ionization energy of a unit increase in the number of electrons in a shell. For oxygen for only one electron in the second shell the ionization energy is 138.1197 eV, so the increase to two electrons in the second shell decreased the ionization energy by 24.2207 eV. On the other hand the increase in the ionization energy when the proton number is increased from 8 for oxygen to 9 for fluorine and number of electrons in the second shell is maintained as two is (157.1651-113.899)=43.2661 eV. The ratio of these two numbers is 0.560. It is not 0.500 but it is reasonably close. When the same computation is made for the fluorine ion with two electrons in the second shell the ratio is 0.559. For neon the same computation produces a figure of 0.559.

When the computation is carried out for different numbers of electrons in the second shell the ratio are quite different from 0.5. The ratio vary with the number of electrons in the shell but the values are very close for the different elements.

                              Number of Electrons in the Second Shell

Element	8		7		6		5		4		3		2		
O					1.006860003	0.718282345	0.697982257	0.990665034	0.559807794
F			0.745439979	0.973959924	0.709975706	0.693700058	0.982491439	0.55917886
Ne	0.754172725	0.733500518	0.949520587	0.704699612	0.690015227	0.975793949	0.558552044
Na	0.740584284	0.724920641	0.932247403	0.698511166	0.687358062	0.970240167	 
Mg	0.730816622	0.718241901	0.924217798	0.695636364	0.686452046		
Al	0.723313736	0.712702887	0.915372255	0.694399482			

 

This ratio, which could be described as the shielding factor for electrons, has the averages shown below:

              Number of electrons in the shell
8		7		6		5		4		3		2
0.737221842	0.726961185	0.950362995	0.703584112	0.69110153	0.979797647	0.559179566

The graph of the data is as follows:

The values of the shielding factor suggest that when a second electron is added to the shell it is at nearly the same distance from the center of the atom as the first. When a third is added it is closer to the center than the first two and its shielding factor is one. For the fourth its distance is slightly less than the first two but greater than the third so its charge is spread over a range that puts 70 percent closer to the center than the first two electrons in the shell, thus shielding 0.7 of a positive charge. The same occurs for the fifth electron in the shell. The sixth electron is like the third in that its charge is closer to the center than the first two and it has a shielding factor of nearly one. The seventh and eight are like the fourth and fifth, being closer to the center than the first two but not so much closer that they have a shielding factor of one; instead their shielding factor is about 0.73.

(To be continued.)