Particles in the atom and atomic radius

1. Overview

Every atom is built from three kinds of particle: protons and neutrons packed into a tiny nucleus, and electrons in shells around it. This section covers what those particles are, how to count them in any atom or ion, and how beams of them behave in an electric field. It ends with atomic and ionic radius: why atoms shrink across a period, grow down a group, and differ in size from their ions.

Key Definitions

• Nucleus: the very small, dense centre of an atom; it contains the protons and neutrons, so it holds almost all of the atom's mass and all of its positive charge.
• Proton number (atomic number), Z: the number of protons in the nucleus of an atom; it identifies the element.
• Nucleon number (mass number), A: the total number of protons and neutrons in the nucleus of an atom.
• Nuclear charge: the total positive charge on the nucleus, which is set by the number of protons it contains.
• Shielding: the reduction in the attraction between the nucleus and the outer electrons caused by repulsion from the electrons in the inner shells.
• Atomic radius: half the distance between the nuclei of two neighbouring atoms of the same element that are bonded together (covalently in a non-metal, by metallic bonding in a metal).
• Ionic radius: the radius of an ion in an ionic lattice, worked out from the distance between the nuclei of neighbouring ions.
• Isoelectronic species: atoms or ions that have the same number of electrons, for example the Ar atom and the K⁺ and Ca²⁺ ions, which each have 18 electrons.

Content

A tiny nucleus in a lot of empty space

An atom has two parts:

• The nucleus, at the centre, holds the protons and neutrons. It is very small and very dense.
• The electrons are found in shells in the space around the nucleus. The first shell is closest to the nucleus, and each later shell is further out.

The nucleus is about $10^{-15}$ to $10^{-14}$ m across, while a whole atom is about $10^{-10}$ m across. If the nucleus were a bead 1 cm wide, the atom would be between 100 m and 1 km across. The nucleus fills less than a millionth of a millionth of the atom's volume: an atom is mostly empty space, with electrons moving in it.

The evidence: when alpha particles (small, fast, positive particles) were fired at very thin gold foil, almost all passed straight through or were deflected only slightly. A tiny fraction were turned through large angles, a few of them straight back: only a very small, massive, positive nucleus could do that.

For the first 20 elements, the first shell holds up to 2 electrons, the second up to 8, and the third takes 8 before the fourth shell starts to fill. Section 1.3 adds sub-shells and orbitals.

Protons, neutrons and electrons

Particle	Where it is found	Relative mass	Relative charge
proton	nucleus	1	+1
neutron	nucleus	1	0
electron	shells around the nucleus	1/1836 (about 0.0005)	−1

• Relative means compared with the proton. A proton carries $+1.60 \times 10^{-19}$ C; the Data Booklet gives the electronic charge as $e = -1.60 \times 10^{-19}$ C.
• A neutron has very slightly more mass than a proton (about 0.1% more), but both count as 1.
• A proton has about 1836 times the mass of an electron (some books round this to 1840). The electron's mass is tiny, but not zero.
• Proton and electron charges are equal in size and opposite in sign, so a neutral atom has equal numbers of protons and electrons.

Proton number and nucleon number

• The proton number (atomic number), $Z$, is the number of protons in the nucleus. It identifies the element: every atom with 20 protons is a calcium atom.
• The nucleon number (mass number), $A$, is the total number of protons and neutrons in the nucleus. A nucleon is either kind of particle in the nucleus.

Both numbers can be written on the symbol of the element, with $A$ at the top and $Z$ at the bottom:

$${}^{A}_{Z}\text{X} \qquad \text{for example} \qquad {}^{32}_{16}\text{S}$$

This sulfur atom has 16 protons and $32 - 16 = 16$ neutrons. Atoms of one element can have different numbers of neutrons (isotopes, section 1.2), so $A$ belongs to a particular atom, not to the element as a whole.

The nucleon number is a count, so it is always a whole number. Do not confuse it with relative atomic mass, $A_{\text{r}}$ (section 2.1), which is an average and is often not a whole number (35.5 for chlorine).

Where the mass and the charge are

Mass. In a carbon-12 atom, the 12 nucleons have a relative mass of 12 and the 6 electrons only $6 \times \frac{1}{1836} = 0.00327$, about 0.03% of the total. In every atom, more than 99.9% of the mass is in the nucleus, so the nucleus is extremely dense.

Charge. All the positive charge is on the protons in the nucleus; the electrons carry the negative charge, spread through the space around it.

	Nucleus	Space around the nucleus
Particles	protons and neutrons	electrons, in shells
Share of the mass	more than 99.9%	less than 0.1%
Charge	positive, +$Z$	negative, −$Z$ in a neutral atom
Share of the volume	tiny	almost all

Beams of particles in an electric field

Two parallel metal plates, one positive and one negative, set up an electric field between them. Beams of protons, neutrons and electrons are sent between the plates, parallel to them, all at the same velocity.

Beam	Relative charge	Force from the field	Path between the plates
protons	+1	towards the negative plate	bends slightly towards the negative plate
neutrons	0	none	carries straight on
electrons	−1	towards the positive plate	bends sharply towards the positive plate

Why the electron beam bends so much more than the proton beam:

1. The force on a particle in the field is proportional to its charge. A proton and an electron have charges of the same size, so the forces on them are equal in size and opposite in direction.
2. The same force gives a much larger acceleration to a smaller mass. An electron has 1/1836 of the proton's mass, so its sideways acceleration is 1836 times as large.
3. At the same velocity, both particles spend the same time between the plates. So the electron is pushed sideways 1836 times as far as the proton.

For beams at the same velocity in the same field, the deflection is proportional to charge ÷ mass. The sign of the charge gives the direction. Each charged beam follows a curve while it is between the plates, then carries on in a straight line once it leaves them.

This rule needs equal velocities. A slower beam spends longer between the plates and is deflected more, so beams at different speeds cannot be compared by charge ÷ mass alone.

Worked example 1: comparing deflections

Question: Beams of electrons, protons, neutrons and ²H⁺ ions (each ²H⁺ ion is one proton and one neutron) enter the same electric field at the same velocity. By the time it leaves the plates, the electron beam has moved 5.00 mm sideways. How far, and in which direction, does each of the other beams move?

Step 1: work out charge ÷ mass for each particle.

Particle	Relative charge	Relative mass	Charge ÷ mass
electron	−1	1/1836	−1836
proton	+1	1	+1
²H⁺ ion	+1	2	+0.5
neutron	0	1	0

Step 2: scale from the electron beam. The sideways movement is proportional to the size of charge ÷ mass.

$$\text{proton:} \quad 5.00 \times \frac{1}{1836} = 2.72 \times 10^{-3}\ \text{mm}$$

$${}^{2}\text{H}^{+}\ \text{ion:} \quad 5.00 \times \frac{0.5}{1836} = 1.36 \times 10^{-3}\ \text{mm}$$

Answer: by the time the beams leave the plates, the proton beam has moved $2.72 \times 10^{-3}$ mm towards the negative plate, and the ²H⁺ beam $1.36 \times 10^{-3}$ mm towards the negative plate (half as far as the protons, because it has the same charge but twice the mass). The neutron beam is not deflected. The electron beam goes the other way, towards the positive plate.

The proton beam moves only a few thousandths of a millimetre, so a sketch has to exaggerate it, but the electron curve must still look far sharper.

Counting protons, neutrons and electrons

• Protons = $Z$. If you are given only the element, read $Z$ from the Periodic Table.
• Neutrons = $A - Z$.
• Electrons in a neutral atom = $Z$.
• A positive ion has lost electrons: for a charge of 3+, electrons = $Z - 3$.
• A negative ion has gained electrons: for a charge of 2−, electrons = $Z + 2$.
• Forming an ion changes only the electrons. The nucleus, and so $Z$ and $A$, stay the same.
• For an ion made of several atoms, add up the protons and neutrons of every atom, then adjust the electrons for the overall charge.

Worked example 2: particles in atoms and ions

Question (a): Find the numbers of protons, neutrons and electrons in ⁸⁰Se²⁻, ⁵⁹Co³⁺ and the ion ¹H₃¹⁶O⁺.

Step 1: find each proton number from the Periodic Table: Se 34, Co 27, H 1, O 8.

Step 2: apply the rules.

Species	Protons	Neutrons	Electrons
⁸⁰Se²⁻	34	80 − 34 = 46	34 + 2 = 36
⁵⁹Co³⁺	27	59 − 27 = 32	27 − 3 = 24
¹H₃¹⁶O⁺	3 × 1 + 8 = 11	3 × 0 + 8 = 8	11 − 1 = 10

Answer (a): ⁸⁰Se²⁻ has 34 protons, 46 neutrons and 36 electrons; ⁵⁹Co³⁺ has 27 protons, 32 neutrons and 24 electrons; ¹H₃¹⁶O⁺ has 11 protons, 8 neutrons and 10 electrons.

Question (b): An ion has a charge of 2+. It contains 26 electrons and 30 neutrons. Identify the ion and write its full symbol.

Step 1: protons. The ion has lost 2 electrons, so protons = 26 + 2 = 28. The element with proton number 28 is nickel.

Step 2: nucleon number. $A = 28 + 30 = 58$.

Answer (b): the ion is ${}^{58}_{28}\text{Ni}^{2+}$.

Atomic radius

An atom has no sharp edge, so its radius is found from the distance between the nuclei of two bonded atoms of the same element. Half of that distance is the atomic radius: the covalent radius for a non-metal and the metallic radius for a metal. Atomic radii range from about 0.03 nm to about 0.3 nm (1 nm = $10^{-9}$ m = 1000 pm).

Noble gas atoms do not bond to each other, so their radius has to be measured between neighbouring atoms that are not bonded. That gives a larger value than a bonded radius would, so leave the noble gases out when you compare atomic radii across a period.

Three factors decide the size of an atom:

Factor	Effect on the outer electrons	Effect on the radius
More occupied shells	further from the nucleus	larger
More protons (greater nuclear charge), same shells	pulled more strongly towards the nucleus	smaller
More shielding by inner shells	less strongly attracted	larger

Across a period, atomic radius decreases. From sodium to chlorine:

• Each step adds one proton, so the nuclear charge rises from +11 to +17.
• Each step adds one electron to the same outer (third) shell, so the number of occupied shells does not change.
• The inner shells (2 and 8 electrons) stay the same, so the shielding is similar. Electrons in the same shell shield each other only a little.
• So the attraction between the nucleus and the outer electrons increases, and the outer shell is pulled closer to the nucleus.

Down a group, atomic radius increases. From lithium to sodium to potassium, for example:

• Each element has one more occupied shell, so the outer electron is further from the nucleus.
• There are more inner shells, so there is more shielding.
• The nuclear charge also increases, but the greater distance and extra shielding outweigh it. The outer electrons are held less strongly, and the atom is larger.

Ionic radius

A positive ion is smaller than its atom. A sodium atom (2, 8, 1) loses its outer electron to form Na⁺ (2, 8). The whole third shell is gone, so the outermost electrons are now in the second shell, closer to the nucleus. The 11 protons also now hold only 10 electrons, so there is less repulsion between the electrons.

A negative ion is larger than its atom. A chlorine atom (2, 8, 7) gains an electron to form Cl⁻ (2, 8, 8). The number of shells and the nuclear charge are unchanged, but the extra electron increases the repulsion between the electrons. The same 17 protons hold more electrons, so the outer shell spreads out.

In isoelectronic ions, more protons means a smaller ion. The electrons and shielding are the same, so the only difference is the nuclear charge.

Across Period 3, the ions fall into two isoelectronic sets:

Ion	Na⁺	Mg²⁺	Al³⁺	P³⁻	S²⁻	Cl⁻
Protons	11	12	13	15	16	17
Electrons	10	10	10	18	18	18
Occupied shells	2	2	2	3	3	3

• From Na⁺ to Al³⁺, the ionic radius decreases: the same 10 electrons are pulled in by more protons.
• From P³⁻ to Cl⁻, the ionic radius also decreases, for the same reason with 18 electrons.
• Each negative ion is much larger than each positive ion in the period, because it has one more occupied shell. That explains the large jump from Al³⁺ to P³⁻.
• Silicon is usually left out, because its compounds are mainly covalent, so it rarely forms a simple ion.

Down a group, ionic radius increases for ions with the same charge: Li⁺ < Na⁺ < K⁺ and F⁻ < Cl⁻ < Br⁻. Each ion has one more occupied shell than the ion above it.

Worked example 3: putting ions in order of size

Question: Put the ions Se²⁻, Br⁻, Rb⁺ and Sr²⁺ in order of increasing ionic radius, and explain the order.

Step 1: count protons and electrons.

Ion	Se²⁻	Br⁻	Rb⁺	Sr²⁺
Protons	34	35	37	38
Electrons	34 + 2 = 36	35 + 1 = 36	37 − 1 = 36	38 − 2 = 36

Step 2: compare the electrons. All four ions are isoelectronic, with 36 electrons arranged 2, 8, 18, 8. They have the same number of occupied shells and the same shielding.

Step 3: compare the nuclear charge. The only difference is the number of protons. More protons attract the same 36 electrons more strongly and pull them closer.

Answer: Sr²⁺ < Rb⁺ < Br⁻ < Se²⁻. The ion with the most protons, Sr²⁺, is the smallest.

Key Equations

Number of neutrons ($A$ = nucleon number; $Z$ = proton number): $$\text{number of neutrons} = A - Z$$

Number of electrons in an atom or ion (charge with its sign: +3 for a 3+ ion, −2 for a 2− ion, 0 for an atom): $$\text{number of electrons} = Z - \text{charge}$$

Deflection in an electric field (beams moving at the same velocity through the same field; the sign of the charge gives the direction): $$\text{deflection} \propto \frac{\text{relative charge}}{\text{relative mass}}$$

Relative masses and charges: proton 1 and +1; neutron 1 and 0; electron 1/1836 and −1. The size of the charge on a proton or an electron is $1.60 \times 10^{-19}$ C.

Units for radii: 1 nm = $10^{-9}$ m; 1 pm = $10^{-12}$ m; 1 nm = 1000 pm.

Common Mistakes to Avoid

• Mistake: taking the nucleon number as the number of neutrons. Fix: subtract the proton number. ⁸⁰Se has 80 nucleons but only 80 − 34 = 46 neutrons.
• Mistake: adding electrons for a positive ion, or taking them away for a negative ion. Fix: a positive charge means electrons were lost. ⁵⁹Co³⁺ has 27 − 3 = 24 electrons, not 30, and Se²⁻ has 34 + 2 = 36.
• Mistake: drawing the electron and proton beams bending by similar amounts, or the electrons bending less. Fix: at the same velocity, deflection depends on charge ÷ mass. The electron beam is pushed sideways 1836 times as far, so draw a gentle curve towards the negative plate for protons, a much sharper curve towards the positive plate for electrons, and a straight line for neutrons.
• Mistake: describing the inside of an atom vaguely ("the protons are in the middle"). Fix: say where the mass and each charge are: over 99.9% of the mass and all the positive charge are in the tiny nucleus, and the electrons' negative charge is spread around it.
• Mistake: explaining the decrease across a period by "more electrons, so more shielding". Fix: the extra electrons go into the same shell and shield each other only a little, and the inner 10 electrons stay the same, so the shielding is similar. The nuclear charge rises from +11 (Na) to +17 (Cl), so the attraction for the outer electrons increases.
• Mistake: explaining the increase down a group by the change in nuclear charge alone. Fix: the extra shell decides it. Potassium has 8 more protons than sodium, but its outer electron is in a fourth shell, further away and more shielded, so the potassium atom is larger.
• Mistake: saying the nuclear charge increases when an atom forms a positive ion. Fix: the nucleus does not change. Rb and Rb⁺ both have 37 protons; the ion is smaller because it has lost its fifth shell and has less repulsion between its electrons.
• Mistake: writing "it has a larger radius" without naming the species. Fix: name the atom or ion every time, for example "the Sr²⁺ ion has a smaller ionic radius than the Rb⁺ ion", not "strontium is smaller".

Exam Tips

• State a trend in one sentence, with its direction and range: "atomic radius decreases from sodium to chlorine". No reason is needed unless the question also says explain.
• Explain a difference in radius with linked points: the number of protons (the nuclear charge, not "nucleon charge"), the number of occupied shells and the shielding, then the result for the attraction between the nucleus and the outer electrons. Say "shell", not "orbital" or "sub-shell", for these points.
• Determine particle numbers in a small table of protons, neutrons and electrons, showing each subtraction or addition, so every number can be checked.
• Complete a diagram of beams in a field by starting all the beams from the same point, labelling each beam, and making the electron curve clearly sharper than the proton curve and in the opposite direction. If asked why, refer to both charge and mass.
• Describe the distribution of mass and charge with four facts: the nucleus is tiny and dense; it holds almost all the mass; it holds all the positive charge; the electrons, with the negative charge, occupy the space around it.
• Give the electron's relative mass as 1/1836 (or about 0.0005), not 0, and its relative charge as −1, not just "negative".