IMAT Chemistry · Formula sheet

Atomic Structure formulas

5 formulas, 6 reference tables and 19 common traps for IMAT Chemistry Atomic Structure, grouped by subtopic.

Protons, Neutrons, Electrons and Ions

Counting protons, neutrons and electrons from a nuclide symbol

Neutrons and electrons

N = A - Z \qquad n_{e} = Z - q

$A$ mass number (protons + neutrons)
$Z$ atomic number (protons)
$N$ number of neutrons
$q$ charge on the particle, with its sign
$n_e$ number of electrons

The three subatomic particles: charge, mass and position

Particle	Relative charge	Relative mass	Where it is
Proton	+1	1	Nucleus
Neutron	0	1	Nucleus
Electron	−1	About 1/1840 (almost zero)	Shells (energy levels) around the nucleus

The proton and neutron masses are almost equal; the electron is about 1840 times lighter than either.

Common traps

Ions form by moving electrons, never protons

A positive ion has MORE protons than electrons because it has lost electrons. It has not gained protons: the nucleus never changes in a chemical reaction. Options that explain a positive charge by extra protons, or by a difference between protons and neutrons, are wrong.

Subtract a positive charge, add a negative one

Electrons equal

Z - q

. A 2+ ion has two FEWER electrons than protons, a 2− ion two MORE. Reversing the sign is the commonest slip, and IMAT always offers the reversed count as an option.

The mass number is not the number of neutrons

The mass number counts protons and neutrons together. The neutrons are

A - Z

. An option equal to the mass number itself is there to catch students who skip the subtraction.

Isoelectronic does not mean identical

\mathrm{O^{2-}}

and

\mathrm{Na^+}

have the same 10 electrons in the same arrangement, but different numbers of protons and neutrons. Options saying two isoelectronic particles have the same number of protons, or the same size, are wrong.

Two ions of the same element are not isoelectronic

\mathrm{O^-}

and

\mathrm{O^{2-}}

have the same nucleus but 9 and 10 electrons. Being the same element is a distractor: only the electron count matters.

Isotopes and Relative Atomic Mass

Learn this subtopic in the notes

Relative atomic mass from isotope abundances

Relative atomic mass

A_r = \frac{\sum (\text{isotope mass} \times \%\ \text{abundance})}{100}

isotope massrelative mass of each isotope, close to its mass number
% abundancepercentage of atoms of that isotope in the natural element

Isotopes: same element, different number of neutrons

Isotope	Protons	Neutrons	Note
Hydrogen-1 (protium)	1	0	Over 99.9% of natural hydrogen; the only atom with no neutron
Hydrogen-2 (deuterium, D)	1	1	Stable; $\mathrm{D_2O}$ is heavy water
Hydrogen-3 (tritium, T)	1	2	Radioactive (beta emitter)
Carbon-12	6	6	Defines the atomic mass scale: exactly 12
Carbon-14	6	8	Radioactive; used to date once-living material
Chlorine-35	17	18	About 75% of natural chlorine
Chlorine-37	17	20	About 25% of natural chlorine

Common traps

Same mass number does not make two atoms isotopes

Isotopes share the atomic number, not the mass number. Carbon-14 and nitrogen-14 have the same mass number but are different elements. Two atoms with the same

Z

and different

A

are isotopes.

A simple average ignores abundance

The relative atomic mass is a WEIGHTED average. Only when the two isotopes are equally common does it equal the simple average of their masses. An option exactly halfway between two isotope masses is a trap unless the abundances are equal.

Relative atomic mass is per atom, not per molecule

For hydrogen, chlorine or any diatomic gas, the relative atomic mass refers to one atom. Doubling it gives the relative molecular mass of the gas. Check which one the question asks for before choosing.

Shells, Orbitals and Electron Configurations

Learn this subtopic in the notes

Writing electron configurations with the Aufbau principle and Hund's rule

Order of filling

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p

$<$ lower in energy, so filled first

Shells, subshells and orbitals, and how many electrons each holds

Subshell	Orbitals in it	Maximum electrons	First shell that has it
s	1	2	n = 1
p	3	6	n = 2
d	5	10	n = 3
f	7	14	n = 4

Shell 3 has 3s, 3p and 3d: 1 + 3 + 5 = 9 orbitals, so up to 18 electrons.

Quantum numbers and the Pauli exclusion principle

Quantum number	Symbol	Allowed values	What it describes
Principal	$n$	1, 2, 3, ...	The shell: energy and size
Angular momentum (azimuthal)	$l$	0 to $n - 1$ (0 = s, 1 = p, 2 = d, 3 = f)	The subshell: orbital shape
Magnetic	$m_l$	$-l$ to $+l$ , whole numbers	Which orbital: its orientation
Spin	$m_s$	$+\tfrac{1}{2}$ or $-\tfrac{1}{2}$	The direction of the electron's spin

The chromium and copper exceptions to the filling order

Element	Predicted by the filling order	Actual ground state	Why
Chromium, Z = 24	$[\mathrm{Ar}]\,3d^4\,4s^2$	$[\mathrm{Ar}]\,3d^5\,4s^1$	Half-filled 3d subshell is extra stable
Copper, Z = 29	$[\mathrm{Ar}]\,3d^9\,4s^2$	$[\mathrm{Ar}]\,3d^{10}\,4s^1$	Full 3d subshell is extra stable
Molybdenum, Z = 42	$[\mathrm{Kr}]\,4d^4\,5s^2$	$[\mathrm{Kr}]\,4d^5\,5s^1$	Same pattern as chromium (same group)
Silver, Z = 47	$[\mathrm{Kr}]\,4d^9\,5s^2$	$[\mathrm{Kr}]\,4d^{10}\,5s^1$	Same pattern as copper (same group)

Common traps

Period 3 has 8 elements, but shell 3 holds 18 electrons

The 3d subshell only starts filling after 4s, in period 4. So the third row of the table stops at 8 elements, while the third shell can eventually hold

2 \times 3^2 = 18

electrons.

Two electrons in one orbital have opposite spins

An orbital holds two electrons at most, and they must have antiparallel (opposite) spins, by the Pauli principle. Options with two parallel spins, one electron only, or three or four electrons per orbital are wrong.

4s fills before 3d

After 3p comes 4s, not 3d. Potassium is

[\mathrm{Ar}]\,4s^1

, not

[\mathrm{Ar}]\,3d^1

. The 3d subshell starts filling only at scandium,

Z = 21

Hund's rule: a p⁴ subshell has two unpaired electrons, not four

Three p orbitals take one electron each first; a fourth electron must pair. So

p^4

has 2 unpaired electrons and

p^3

has 3, the most a p subshell can have.

Electron Configurations of Ions and Valence Electrons

Learn this subtopic in the notes

Common traps

Transition metals lose 4s electrons before 3d

4s fills before 3d, but once 3d holds electrons the 4s electrons are the outermost ones, and they leave first.

\mathrm{Fe^{2+}}

[\mathrm{Ar}]\,3d^6

and

\mathrm{Cu^{2+}}

[\mathrm{Ar}]\,3d^9

. An option that keeps 4s electrons in such an ion is wrong.

An anion's valence electrons are not its group number

A halogen atom has 7 valence electrons, but its 1− ion has 8: the extra electron joins the same outer shell. Options giving 7, or giving the total number of electrons, are the usual traps.

A cation with no electrons left in its old outer shell still has an outer shell

When

\mathrm{Mg}

becomes

\mathrm{Mg^{2+}}

, shell 3 is empty and shell 2 becomes the outer shell, with 8 electrons. The ion has the same protons and neutrons as the atom, two fewer electrons, and a smaller radius.

Radioactive Decay, Nuclear Equations and Half-Life

Learn this subtopic in the notes

Balancing nuclear equations

Conservation in a nuclear equation

\sum A_{\text{before}} = \sum A_{\text{after}} \qquad \sum Z_{\text{before}} = \sum Z_{\text{after}}

$A$ mass number of each particle
$Z$ atomic number (charge number) of each particle

Half-life calculations

Amount left after time t

N = N_0 \left(\tfrac{1}{2}\right)^{t/T_{1/2}}

$N_0$ starting amount (nuclei, mass or activity)
$N$ amount left after time $t$
$T_{1/2}$ half-life, in the same unit as $t$

Alpha, beta and gamma radiation and what each does to the nucleus

Radiation	What it is	Change to the nucleus	Charge	Stopped by
Alpha (α)	Helium nucleus ${}^{4}_{2}\mathrm{He}$ : 2 protons and 2 neutrons	A falls by 4, Z falls by 2	+2	A sheet of paper or a few cm of air
Beta-minus (β⁻)	Fast electron ${}^{0}_{-1}e$ , made when a neutron becomes a proton	A unchanged, Z rises by 1	−1	A few mm of aluminium
Beta-plus (β⁺)	Positron ${}^{0}_{+1}e$ , made when a proton becomes a neutron	A unchanged, Z falls by 1	+1	A few mm of aluminium
Gamma (γ)	High-energy electromagnetic radiation	No change to A or Z	0	Only reduced, by thick lead or concrete

Common traps

The beta electron comes from the nucleus

In beta-minus decay a neutron in the nucleus changes into a proton and an electron, and the electron is emitted. The atom's shell electrons are not involved. That is why the atomic number rises by one.

Beta decay does not change the mass number

An electron has mass number 0, so beta emission leaves A unchanged and changes Z by one. Only alpha decay lowers the mass number (by 4). Options that lower A after a beta decay are wrong.

Two half-lives leave a quarter, not nothing

Each half-life halves what is LEFT, not the original amount. After two half-lives a quarter remains, after three an eighth. An option saying the sample is gone after two half-lives is wrong.

Heating or reacting a sample does not change its half-life

Radioactive decay happens in the nucleus, which chemical reactions and temperature do not reach. The half-life of an isotope is the same in any compound, at any temperature and for any sample size.

Atomic Structure formulas

Protons, Neutrons, Electrons and Ions

Counting protons, neutrons and electrons from a nuclide symbol

The three subatomic particles: charge, mass and position

Common traps

Isotopes and Relative Atomic Mass

Relative atomic mass from isotope abundances

Isotopes: same element, different number of neutrons

Common traps

Shells, Orbitals and Electron Configurations

Writing electron configurations with the Aufbau principle and Hund's rule

Shells, subshells and orbitals, and how many electrons each holds

Quantum numbers and the Pauli exclusion principle

The chromium and copper exceptions to the filling order

Common traps

Electron Configurations of Ions and Valence Electrons

Common traps

Radioactive Decay, Nuclear Equations and Half-Life

Balancing nuclear equations

Half-life calculations

Alpha, beta and gamma radiation and what each does to the nucleus

Common traps

More IMAT Chemistry formula sheets