Although your university chemistry course may have mentioned quarks, leptons, and gluons, the MCAT keeps things simpler. There are three subatomic particles you need to know cold: protons, neutrons, and electrons.
Protons
Protons live in the nucleus - the dense center of the atom. Each proton carries a charge of +1 and has a mass of approximately 1 atomic mass unit (amu).
The number of protons defines the element. Change the number of protons and you change the element entirely. This number is called the atomic number (Z). All carbon atoms have 6 protons. All oxygen atoms have 8. No exceptions.
Neutrons
Neutrons also live in the nucleus. They have no charge (0) and a mass of approximately 1 amu - just barely heavier than a proton.
Neutrons contribute to the atom’s mass but not its charge. The total number of protons plus neutrons is called the mass number (A). The number of neutrons can vary within the same element, creating isotopes - atoms with the same atomic number but different mass numbers. We will cover isotopes in detail in the next section.
Electrons
Electrons are fundamentally different from the particles in the nucleus. They move through the space surrounding the nucleus in regions called orbitals. Each electron carries a charge of -1 and has a mass that is approximately 20001 of a proton’s mass - so small that electrons contribute virtually nothing to the atom’s overall mass.
Because electrons are so light compared to protons and neutrons, the mass of an atom is almost entirely determined by its nucleus. This is why we can effectively ignore electron mass when calculating atomic mass.
Electrons closer to the nucleus are at lower energy levels and are held more tightly. Electrons farther out are at higher energy levels, are held less tightly, and interact most strongly with the surrounding environment. The outermost electrons are called valence electrons, and they are the electrons most involved in chemical bonding and reactions.
Charge and Ions
In a neutral atom, the number of protons equals the number of electrons, so the charges cancel out. But atoms can gain or lose electrons:
Losing electrons creates a cation (positive charge). Think: “cations are pawsitive.”
Gaining electrons creates an anion (negative charge). Anions are negative.
Atomic number (Z) = number of protons = defines the element
Mass number (A) = protons + neutrons
Number of neutrons = A - Z
In a neutral atom: protons = electrons
In an ion: electrons = protons - charge (for cations) or protons + |charge| (for anions)
Inside an atom, and how to read an isotope symbol
Atomic structure
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Reading the symbolThe bottom number is Z, the proton count, and it is what makes the element what it is. The top number is A, protons plus neutrons. Neutrons are the difference, A − Z, and they are never written directly.
Isotopes versus ionsChanging the neutron count gives an isotope: same element, same chemistry, different mass. Changing the electron count gives an ion: same element, same mass to three figures, different charge and very different behaviour.
Why atomic masses are not whole numbersThe mass on the periodic table is a weighted average over natural abundances. Chlorine sits at 35.45 because it is roughly three quarters chlorine-35 and one quarter chlorine-37, not because any single atom weighs 35.45 amu.
Mass lives in the nucleus, chemistry lives outside it. A proton and a neutron weigh about the same; an electron weighs about one eighteen-hundredth as much, which is why adding or removing electrons changes an atom's charge and reactivity without meaningfully changing its mass.
A nickel-60 cation has a +2 charge (60Ni2+). How many protons, neutrons, and electrons does it have?
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28 protons, 32 neutrons, 26 electrons. Nickel has atomic number 28 (always 28 protons). Neutrons = 60 - 28 = 32. The +2 charge means it lost 2 electrons: 28 - 2 = 26 electrons.
Which subatomic particle determines an atom’s identity as a particular element?
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The proton. The number of protons (atomic number, Z) uniquely identifies every element. Change the number of protons and you have a different element. Neutrons can vary (isotopes) and electrons can vary (ions), but protons define what element you have.
These terms sound similar, and the MCAT knows it. Questions that test whether you can distinguish between atomic mass and atomic weight, or between mass number and atomic number, appear regularly. Let’s make the distinctions crystal clear.
Isotope Notation
The standard way to represent an isotope is:
In practice, you will often see isotopes written as the element name followed by the mass number - “carbon-14” or “uranium-238.” Since the element name already tells you the atomic number (carbon is always 6), the mass number is the only additional information you need.
Isotopes
Isotopes are atoms of the same element (same number of protons) that have different numbers of neutrons, and therefore different mass numbers.
Inside an atom, and how to read an isotope symbol
Atomic structure
1
Scroll sideways to see the whole map.
Reading the symbolThe bottom number is Z, the proton count, and it is what makes the element what it is. The top number is A, protons plus neutrons. Neutrons are the difference, A − Z, and they are never written directly.
Isotopes versus ionsChanging the neutron count gives an isotope: same element, same chemistry, different mass. Changing the electron count gives an ion: same element, same mass to three figures, different charge and very different behaviour.
Why atomic masses are not whole numbersThe mass on the periodic table is a weighted average over natural abundances. Chlorine sits at 35.45 because it is roughly three quarters chlorine-35 and one quarter chlorine-37, not because any single atom weighs 35.45 amu.
Mass lives in the nucleus, chemistry lives outside it. A proton and a neutron weigh about the same; an electron weighs about one eighteen-hundredth as much, which is why adding or removing electrons changes an atom's charge and reactivity without meaningfully changing its mass.
Consider hydrogen - the simplest element. It has three isotopes:
Protium (¹H): 1 proton, 0 neutrons - by far the most abundant
Deuterium (²H): 1 proton, 1 neutron - stable, found in “heavy water”
Tritium (³H): 1 proton, 2 neutrons - radioactive
All three are hydrogen. All three have one proton. They differ only in neutron count. Because isotopes share the same number of protons and electrons, they have nearly identical chemical properties - they form the same bonds and undergo the same reactions. Their physical properties (mass, radioactive behavior) may differ.
Isotones, Isobars, and Allotropes
The MCAT occasionally tests whether you can distinguish isotopes from three related terms. These show up less frequently than isotopes, but knowing the differences prevents easy points from slipping away.
Isotones are atoms of different elements that have the same number of neutrons. Carbon-14 (6 protons, 8 neutrons) and nitrogen-15 (7 protons, 8 neutrons) are isotones - both have 8 neutrons, but they are entirely different elements.
Isobars are atoms of different elements that have the same mass number. Argon-40 (18 protons, 22 neutrons) and calcium-40 (20 protons, 20 neutrons) are isobars - both have a mass number of 40, but different numbers of protons and neutrons.
Allotropes are different structural forms of the same element in the same physical state. Diamond, graphite, and fullerene are all allotropes of carbon - same element, same number of protons and neutrons, but the atoms are arranged in completely different crystal structures, giving them wildly different physical properties.
| Term | Same Element? | Same Protons? | Same Neutrons? | Same Mass Number? |
|------|:---:|:---:|:---:|:---:|
| Isotopes | Yes | Yes | No | No |
| Isotones | No | No | Yes | No |
| Isobars | No | No | No | Yes |
| Allotropes | Yes | Yes | Yes | Yes |
Atomic Mass vs. Atomic Weight
These two terms are easily confused:
Atomic mass (or mass number, A) is the total count of protons and neutrons in a specific atom. It is always a whole number. Carbon-12 has an atomic mass of 12 amu. Carbon-14 has an atomic mass of 14 amu.
Atomic weight is the weighted average of the atomic masses of all naturally occurring isotopes of an element. It is the number printed on the periodic table, and it is almost never a whole number.
Calculating Atomic Weight
The atomic weight is calculated by multiplying each isotope’s mass by its natural abundance (as a decimal fraction), then summing:
Example: Chlorine has two main isotopes: Cl-35 (75.77%) and Cl-37 (24.23%).
Atomic weight = (35 x 0.7577) + (37 x 0.2423) = 26.52 + 8.97 = 35.49 amu
This is why the periodic table lists chlorine as 35.5, not 35 or 37.
The Mole and Molar Mass
One mole of any substance contains 6.022×1023 particles (Avogadro’s number, NA). The beauty of the atomic weight is that it connects the microscopic world to the lab bench:
The atomic weight of carbon is 12.01 amu
One atom of carbon-12 weighs exactly 12 amu
One mole of naturally occurring carbon weighs 12.01 grams
In other words, the atomic weight in amu for a single atom equals the molar mass in grams per mole (g/mol) for a mole of atoms. This equivalence is what makes the mole concept so powerful.
Element Q has three isotopes: A (40 amu, 60%), B (44 amu, 25%), and C (41 amu, 15%). What is the atomic weight of Q?
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41.15 amu. (40 x 0.60) + (44 x 0.25) + (41 x 0.15) = 24.00 + 11.00 + 6.15 = 41.15 amu. Notice the result is closest to 40, the most abundant isotope.
What is the difference between atomic mass and atomic weight?
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Atomic mass is the mass of a specific isotope, approximately equal to its mass number (protons + neutrons). Atomic weight is the weighted average of all naturally occurring isotopes, which is the number on the periodic table. Atomic mass is for one isotope; atomic weight is the average across all isotopes.
Carbon-14 (6 protons, 8 neutrons) and nitrogen-15 (7 protons, 8 neutrons) are examples of which relationship: isotopes, isotones, or isobars?
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Isotones. They have the same number of neutrons (8) but different numbers of protons, making them different elements entirely. Isotopes share the same protons (same element). Isobars share the same mass number. Isotones share the same neutron count - the “n” in isotone helps you remember “neutrons.”
In 1910, Ernest Rutherford demonstrated that atoms have a dense, positively charged nucleus with electrons surrounding it. But Rutherford’s model had a problem - classical physics predicted that orbiting electrons should continuously radiate energy and spiral into the nucleus within nanoseconds. Atoms should not be stable, yet here we are.
Three years later, Niels Bohr solved this problem by borrowing an idea from Max Planck: energy is quantized. It comes in discrete packets, not a continuous stream.
Planck’s Quantum Theory
Planck proposed that energy emitted as electromagnetic radiation comes in bundles called quanta. The energy of a single quantum is:
Since the speed of light relates frequency and wavelength (c = f x lambda), we can also write:
Bohr’s Key Postulates
Bohr proposed a model for the hydrogen atom with these critical features:
Hydrogen energy levels and the lines they emit
Bohr model
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Why the rungs crowdEnergy goes as 1/n², so the n = 1 to n = 2 gap is 10.2 eV while n = 5 to n = 6 is under 0.2 eV. Every question that asks which transition emits the shortest wavelength is really asking which gap is biggest, and the biggest gaps are always at the bottom.
Reading a series off the ladderA series is named for where the electron lands, not where it starts. Landing on n = 1 means falling the whole way, so Lyman lines are ultraviolet. Landing on n = 2 gives the four visible Balmer lines. Landing on n = 3 gives infrared.
The sign conventionEnergies are negative because the electron is bound: zero is the free electron at n = ∞. Absorbing a photon moves the electron up toward zero, and it takes the full 13.6 eV to ionise hydrogen from the ground state.
The ladder and the spectrum are the same fact. Each coloured line on the right is drawn at the wavelength its arrow on the left produces, through λ = hc/ΔE. A bigger drop makes a bluer photon, which is why H-α from n = 3 is red and H-δ from n = 6 is violet.
Electrons orbit the nucleus in circular paths at fixed distances - called orbits or energy levels.
Each orbit has a specific energy that depends on the principal quantum number n.
Electrons do not radiate energy while in a stable orbit - this was the radical break from classical physics.
Electrons can only jump between orbits by absorbing or emitting exactly the right amount of energy.
Energy of an Electron in the Bohr Model
The energy of an electron in the nth orbit of hydrogen is:
Key insights from this equation:
The ground state (n = 1) has the most negative energy - the electron is most tightly bound
As n increases, energy becomes less negative (increases toward zero) - the electron is less tightly bound
At n = infinity, E = 0: the electron has escaped the atom completely (ionization)
The energy levels get closer together as n increases (the gap between n = 1 and n = 2 is much larger than between n = 5 and n = 6)
Ground State vs. Excited State
The ground state is when all electrons occupy the lowest possible energy levels. This is the default state for atoms at room temperature.
An excited state occurs when one or more electrons have absorbed energy and jumped to a higher energy level. Excited states are unstable - the electron will quickly fall back to its ground state, releasing the excess energy as a photon.
Energy of a Transition
When an electron transitions between energy levels, the energy of the emitted or absorbed photon equals the difference between the two levels:
Limitations of the Bohr Model
The Bohr model works beautifully for hydrogen and other one-electron systems (He⁺, Li²⁺), but it fails for atoms with more than one electron. Why? Because it ignores the repulsion between multiple electrons orbiting the same nucleus. The quantum mechanical model, which we will cover next, handles multi-electron atoms correctly.
Despite its limitations, the Bohr model remains essential for understanding quantized energy levels, emission spectra, and the fundamental idea that electrons can only exist at specific energies - not anywhere they please.
An electron in a hydrogen atom jumps from n = 2 to n = 4. Is energy absorbed or emitted? Is this an excited state transition?
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Energy is absorbed. The electron is moving to a higher energy level (n = 2 to n = 4), so it must absorb a photon with energy exactly equal to the difference between these levels. Yes, the atom is now in an excited state because the electron is no longer in its lowest possible energy level.
Why does the Bohr model fail for multi-electron atoms?
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It ignores electron-electron repulsion. The Bohr model only accounts for the attraction between one electron and the nucleus. In atoms with multiple electrons, the repulsion between electrons significantly affects their energies and positions, making the simple circular orbit model inaccurate.
The Bohr model was a giant leap forward, but it treated electrons as tiny billiard balls traveling in neat circular orbits. Modern quantum mechanics tells us something far stranger: electrons behave as both particles and waves, and we can never know exactly where an electron is and how fast it is moving at the same time.
Wave-Particle Duality
In 1924, Louis de Broglie proposed that if light can behave as both a wave and a particle (photons), then matter - including electrons - should also have wave-like properties. This was confirmed experimentally when electron beams were shown to produce diffraction patterns, just like light waves.
The Heisenberg Uncertainty Principle
Werner Heisenberg showed that there is a fundamental limit to what we can know about a particle’s position and momentum simultaneously:
It is impossible to determine both the exact position and the exact momentum of an electron at the same time.
This is not a limitation of our instruments - it is a feature of nature itself. To measure an electron’s position precisely, you must interact with it (hit it with a photon, for example), which changes its momentum. To measure its momentum precisely, you need a long wavelength photon that cannot pinpoint its position.
Orbits vs. Orbitals
This distinction is critical and frequently tested:
| Feature | Bohr Orbit | Quantum Orbital |
|---------|-----------|----------------|
| Shape | Fixed circular path | 3D region of probability |
| Electron position | Known exactly (on the path) | Probability distribution |
| Works for | Hydrogen only | All atoms |
| Based on | Classical + quantization | Full quantum mechanics |
An orbital is a region of space around the nucleus where there is a high probability (typically 90%) of finding an electron. The shape of this region depends on the quantum numbers, which we will explore in the next section.
The Electron Cloud
Instead of a planet orbiting a sun, picture a fuzzy cloud surrounding the nucleus. The cloud is denser where the probability of finding the electron is higher and thinner where the probability is lower. For an s orbital, this cloud is spherical - densest near the nucleus and fading out with distance. For a p orbital, the cloud is shaped like a dumbbell with two lobes on either side of the nucleus.
Orbital shapes and the four quantum numbers
Quantum model
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Where the capacities come fromA subshell has 2ℓ + 1 orbitals because mℓ runs from −ℓ to +ℓ, and each orbital holds two electrons because ms has two values. So s holds 2, p holds 6, d holds 10, f holds 14. Nothing there needs memorising separately.
What a node isA node is a surface where the probability of finding the electron is zero. The pinch at the centre of every p orbital is one; the two planes that separate a d cloverleaf's four lobes are two more. Higher ℓ means more nodal planes, which is most of why higher-ℓ subshells sit higher in energy.
Lobes are not pathsThe surfaces drawn here are probability contours, usually the boundary enclosing about 90 % of the electron density, not orbits. The plus and minus signs are the phase of the wavefunction, and they are what decides whether two orbitals overlap constructively into a bond.
ℓ is the shape and mℓ is the orientation. Read the quantum numbers as a set of nested choices: n picks the shell, ℓ picks the shape within it, mℓ picks which copy of that shape, and ms picks which of the two electrons in it. Every capacity in the periodic table falls out of that counting.
What the Quantum Model Preserves from Bohr
Despite replacing orbits with orbitals, the quantum mechanical model keeps several key ideas from Bohr:
Energy is still quantized - electrons can only exist at specific energy levels
The principal quantum number (n) still describes the overall energy level
Transitions between levels still involve absorbing or emitting photons with specific energies
The ground state is still the lowest energy configuration
The quantum model adds more detail through additional quantum numbers (l, ml, and ms) that describe the shape, orientation, and spin within each energy level. These quantum numbers are the subject of the next section.
What is the key difference between a Bohr orbit and a quantum orbital?
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A Bohr orbit is a fixed, circular path at a defined distance from the nucleus. A quantum orbital is a three-dimensional region of space where there is a high probability of finding an electron. Bohr orbits give an exact location; orbitals give a probability distribution.
State the Heisenberg uncertainty principle in one sentence.
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It is impossible to simultaneously determine both the exact position and the exact momentum of an electron. This is not a measurement limitation - it is a fundamental property of nature. The more precisely you know position, the less precisely you can know momentum, and vice versa.
Every electron in an atom has a unique address described by exactly four quantum numbers. No two electrons in the same atom can share all four - this is the Pauli exclusion principle. Understanding these four numbers is essential because they determine everything about where an electron lives and how it behaves.
1. Principal Quantum Number (n)
The principal quantum number describes the energy level (shell) of the electron.
Values: n = 1, 2, 3, 4, … (any positive integer)
Physical meaning: Determines the average distance from the nucleus and the overall energy. Higher n = higher energy, larger orbital, farther from the nucleus.
Maximum electrons in shell n: 2n²
Shell (n)
Max Electrons (2n²)
1
2
2
8
3
18
4
32
The energy gaps between levels decrease as n increases. The jump from n = 1 to n = 2 is enormous compared to the jump from n = 5 to n = 6.
2. Azimuthal (Angular Momentum) Quantum Number (l)
The azimuthal quantum number describes the shape and subshell within a given energy level.
Values: l = 0, 1, 2, …, (n - 1)
Physical meaning: Determines the shape of the orbital and the subshell name
Number of subshells in shell n: n (equal to the value of n)
Maximum electrons in subshell l: 4l + 2
l value
Subshell letter
Shape
Max electrons
0
s
Sphere
2
1
p
Dumbbell
6
2
d
Cloverleaf
10
3
f
Complex
14
Spectroscopic notation combines n and l: an electron in the n = 3 shell, l = 2 subshell is described as being in the 3d subshell.
3. Magnetic Quantum Number (ml)
The magnetic quantum number specifies the orientation of the orbital in space - which specific orbital within a subshell the electron occupies.
Values: ml = -l, -(l-1), …, 0, …, (l-1), l (integers from -l to +l)
Physical meaning: Determines the spatial orientation of the orbital
Number of orbitals in subshell l: 2l + 1
Subshell
l
ml values
Number of orbitals
s
0
0
1
p
1
-1, 0, +1
3
d
2
-2, -1, 0, +1, +2
5
f
3
-3, -2, -1, 0, +1, +2, +3
7
For the p subshell, the three orbitals correspond to the three axes in space: px, py, and pz. Each orbital can hold two electrons, so the p subshell holds 3 x 2 = 6 electrons total.
4. Spin Quantum Number (ms)
The spin quantum number describes the intrinsic angular momentum of the electron - its “spin.”
Values: ms = +21 or -21
Physical meaning: Two electrons in the same orbital must have opposite spins (one “spin up,” one “spin down”)
This is a direct consequence of the Pauli exclusion principle. Since each orbital can hold at most two electrons, and they must differ in at least one quantum number, they must have opposite spins.
Putting It All Together
For any electron, the four quantum numbers (n, l, ml, ms) provide a complete description. Here is the full set for the second principal energy level:
n
l
Subshell
ml
ms
Total electrons
2
0
2s
0
+21, -21
2
2
1
2p
-1, 0, +1
+21, -21 each
6
Total
8 = 2(2)²
An electron has quantum numbers n=3, l=2, ml=-1, ms=+21. What subshell is it in, and is this a valid set of quantum numbers?
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The electron is in the 3d subshell, and yes, this is valid. n=3 allows l values 0, 1, 2. l=2 is the d subshell. ml=-1 is within the range -2 to +2. ms=+21 is valid. All four quantum numbers follow the rules.
How many orbitals are in the n=3 shell? How many total electrons can it hold?
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9 orbitals, 18 electrons. The n=3 shell has subshells s (1 orbital), p (3 orbitals), and d (5 orbitals) = 9 orbitals total (or n² = 3² = 9). Each orbital holds 2 electrons, so 9 x 2 = 18 (or 2n² = 2 x 9 = 18).
An electron configuration tells you exactly where every electron in an atom lives. It uses spectroscopic notation - the principal quantum number (n) followed by the subshell letter, with a superscript showing how many electrons are in that subshell. For example, 1s²2s²2p⁶3s² describes magnesium (12 electrons total).
Writing electron configurations is a skill the MCAT expects you to have, and it boils down to three rules.
The Three Rules
1. Aufbau Principle (“Building Up”)
Electrons fill from the lowest energy subshell to the highest. “Aufbau” is German for “building up.” Each subshell fills completely before the next one begins to fill.
2. Pauli Exclusion Principle
No two electrons in the same atom can have the same four quantum numbers. In practice, this means each orbital holds at most 2 electrons, and those two electrons must have opposite spins.
3. Hund’s Rule
Within a subshell that has multiple orbitals (p, d, or f), electrons fill each orbital singly with parallel spins before any orbital gets a second electron.
The n + l Rule (Filling Order)
The energy of a subshell is determined by the sum n + l. Lower n + l = lower energy = fills first. If two subshells have the same n + l value, the one with the lower n fills first.
Filling order: the diagonal rule, Hund, and the two exceptions
Electron configuration
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The three rules, in the order they applyAufbau: fill the lowest-energy subshell that still has room. Hund: within a subshell, put one electron in each orbital before pairing any. Pauli: an orbital takes at most two, and they must have opposite spins.
Why 4s fills before 3dThe diagonal rule ranks subshells by n + ℓ: 4s has 4 + 0 = 4 and 3d has 3 + 2 = 5, so 4s goes first. Once the d electrons are in, though, 3d drops below 4s, which is why transition metals ionise out of 4s first. Fe²⁺ is [Ar]3d⁶, not [Ar]4s²3d⁴.
Chromium and copperBoth borrow one 4s electron to reach a half-filled or filled d subshell: Cr is [Ar]4s¹3d⁵ and Cu is [Ar]4s¹3d¹⁰, not the 4s²3d⁴ and 4s²3d⁹ the plain rule predicts. These two are the only ones the MCAT expects by name.
Read the diagonals from the top right, downward and to the left. Each arrow collects the subshells with the same n + ℓ, and taking the arrows in order gives the filling sequence. It is the same rule twice, drawn once as a path and once as a list.
You can also determine this from the diagonal rule diagram - write out the subshells in rows and draw diagonal arrows from upper right to lower left.
Noble Gas (Shorthand) Notation
Writing out the full electron configuration for heavier elements gets tedious. The shorthand uses the symbol of the preceding noble gas in brackets to represent all the core electrons:
Sodium (Z = 11): Full = 1s²2s²2p⁶3s¹; Shorthand = [Ne] 3s¹
Iron (Z = 26): Full = 1s²2s²2p⁶3s²3p⁶4s²3d⁶; Shorthand = [Ar] 4s²3d⁶
Orbital Diagrams
An orbital diagram shows each orbital as a line or box with arrows representing electrons. Spin-up electrons are shown as upward arrows and spin-down electrons as downward arrows.
For nitrogen (Z = 7): 1s²2s²2p³
The 2p subshell has 3 orbitals. By Hund’s rule, each gets one electron (all spin-up) before any pairing occurs. You would see three half-filled p orbitals with parallel spins.
The Chromium and Copper Exceptions
Two elements have electron configurations that violate the expected filling order because half-filled and fully filled d subshells are extra stable:
Chromium (Z = 24): Expected [Ar] 4s²3d⁴, actual [Ar] 4s¹3d⁵ - promotes one 4s electron to achieve a half-filled d subshell
Copper (Z = 29): Expected [Ar] 4s²3d⁹, actual [Ar] 4s¹3d¹⁰ - promotes one 4s electron to achieve a fully filled d subshell
Paramagnetic vs. Diamagnetic
The presence of unpaired electrons determines an atom’s magnetic behavior:
Paramagnetic: Has at least one unpaired electron. Weakly attracted to magnetic fields. Unpaired spins align with the field.
Diamagnetic: All electrons are paired. Weakly repelled by magnetic fields. No unpaired spins to align.
To determine if an atom is paramagnetic, write out its electron configuration and check for unpaired electrons using Hund’s rule.
Write the electron configuration of iron (Z = 26) in noble gas notation. Is iron paramagnetic or diamagnetic?
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[Ar] 4s²3d⁶. Iron is paramagnetic. The 3d subshell has 5 orbitals and 6 electrons. By Hund’s rule, 5 electrons fill singly (one per orbital), and the 6th pairs up in one orbital. This leaves 4 unpaired electrons, making iron strongly paramagnetic.
Why does chromium have the configuration [Ar] 4s¹3d⁵ instead of [Ar] 4s²3d⁴?
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Half-filled d subshells have extra stability. Moving one electron from 4s to 3d gives chromium a half-filled d subshell (3d⁵), where all five d orbitals are singly occupied with parallel spins. The extra stability of this symmetric arrangement outweighs the energetic cost of leaving 4s half-filled.
The shape of an orbital is not random - it is determined by the azimuthal quantum number (l). Each type of orbital has a characteristic shape that tells you where an electron is most likely to be found. You do not need to draw these shapes from memory on the MCAT, but you must recognize them and understand what they mean.
s Orbitals (l = 0)
s orbitals are spherical. The electron density is evenly distributed in all directions from the nucleus. There is one s orbital per shell.
As n increases, s orbitals get larger (the electron is farther from the nucleus on average), and they develop radial nodes - spherical shells where the probability of finding the electron is zero. The 1s orbital has no nodes, the 2s has one radial node, and the 3s has two.
p Orbitals (l = 1)
p orbitals are dumbbell-shaped (or peanut-shaped), with two lobes on opposite sides of the nucleus and a nodal plane through the center where the probability of finding the electron is zero.
Orbital shapes and the four quantum numbers
Quantum model
1
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Where the capacities come fromA subshell has 2ℓ + 1 orbitals because mℓ runs from −ℓ to +ℓ, and each orbital holds two electrons because ms has two values. So s holds 2, p holds 6, d holds 10, f holds 14. Nothing there needs memorising separately.
What a node isA node is a surface where the probability of finding the electron is zero. The pinch at the centre of every p orbital is one; the two planes that separate a d cloverleaf's four lobes are two more. Higher ℓ means more nodal planes, which is most of why higher-ℓ subshells sit higher in energy.
Lobes are not pathsThe surfaces drawn here are probability contours, usually the boundary enclosing about 90 % of the electron density, not orbits. The plus and minus signs are the phase of the wavefunction, and they are what decides whether two orbitals overlap constructively into a bond.
ℓ is the shape and mℓ is the orientation. Read the quantum numbers as a set of nested choices: n picks the shell, ℓ picks the shape within it, mℓ picks which copy of that shape, and ms picks which of the two electrons in it. Every capacity in the periodic table falls out of that counting.
There are three p orbitals per shell (starting at n = 2), oriented along the x, y, and z axes: px, py, and pz. They are identical in shape and energy but point in different directions.
d Orbitals (l = 2)
d orbitals have more complex shapes, generally described as cloverleaf patterns with four lobes. There are five d orbitals per shell (starting at n = 3).
Four of the five d orbitals have four lobes arranged in a cloverleaf pattern but oriented differently in space. The fifth (dz²) looks different - it has two lobes along the z-axis with a donut (torus) in the xy-plane.
The MCAT will not ask you to draw d orbital shapes, but you should know they exist starting at n = 3, there are five of them, and they hold up to 10 electrons total.
f Orbitals (l = 3)
f orbitals have even more complex shapes with multiple lobes. There are seven f orbitals per shell (starting at n = 4), holding up to 14 electrons.
The shapes of f orbitals are not tested on the MCAT. What matters is knowing that f subshells exist, they appear in the lanthanide and actinide series, and they hold up to 14 electrons.
Nodes
A node is a region where the probability of finding the electron is exactly zero. There are two types:
Radial (spherical) nodes: Spherical shells within the orbital where electron density is zero. Number = n - l - 1.
Angular (planar) nodes: Flat planes or cones through the nucleus. Number = l.
| Subshell | l | Shape | Orbitals | Max electrons | First appears at n = |
|----------|---|-------|----------|--------------|---------------------|
| s | 0 | Sphere | 1 | 2 | 1 |
| p | 1 | Dumbbell | 3 | 6 | 2 |
| d | 2 | Cloverleaf | 5 | 10 | 3 |
| f | 3 | Complex | 7 | 14 | 4 |
How many radial nodes does a 3p orbital have? How many angular nodes?
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1 radial node, 1 angular node. Total nodes = n - 1 = 3 - 1 = 2. Angular nodes = l = 1. Radial nodes = total - angular = 2 - 1 = 1.
What shape is a d orbital, and at what principal quantum number do d orbitals first appear?
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Cloverleaf shape (four lobes), first appearing at n = 3. Since l can range from 0 to n-1, the d subshell (l = 2) requires at least n = 3. There are five d orbitals, holding up to 10 electrons.
Not all electrons are created equal. The electrons buried deep inside an atom (core electrons) are tightly bound and chemically inert. The electrons in the outermost energy shell (valence electrons) are the ones that interact with other atoms - forming bonds, getting transferred, and determining chemical behavior.
What Makes an Electron a Valence Electron?
Valence electrons are the electrons in the outermost (highest n) energy shell. They are the least tightly held and the most available for bonding.
For main group elements (Groups 1-2 and 13-18), identifying valence electrons is straightforward:
The group number tells you the number of valence electrons
Group 1 (alkali metals): 1 valence electron (in the s subshell)
Group 2 (alkaline earth metals): 2 valence electrons (in the s subshell)
Group 13: 3 valence electrons (2 in s, 1 in p)
Group 14: 4 valence electrons (2 in s, 2 in p)
Group 15: 5 valence electrons
Group 16: 6 valence electrons
Group 17 (halogens): 7 valence electrons
Group 18 (noble gases): 8 valence electrons (except He with 2)
Transition Metals
For transition metals (d-block elements), valence electrons include electrons in both the highest s subshell AND the highest d subshell, even though they have different principal quantum numbers. For example:
For lanthanides and actinides (f-block elements), valence electrons include those in the highest s subshell AND the f subshell.
Core Electrons
All electrons that are NOT valence electrons are core electrons. They are in completely filled inner shells and do not participate in chemical reactions. Core electrons shield the valence electrons from the full charge of the nucleus - this concept of electron shielding is critical for understanding periodic trends (covered in Chapter 2).
The Octet Rule
Most atoms bond to achieve eight valence electrons (an octet) in their outermost shell, mimicking the electron configuration of the nearest noble gas. This is the octet rule, and it drives the vast majority of chemical bonding covered on the MCAT.
Elements in period 3 and below can exceed eight valence electrons by using their empty d orbitals. This is called an expanded octet and is covered in Chapter 3 (Bonding).
How many valence electrons does selenium (Group 16) have? What about its core electrons?
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6 valence electrons, 28 core electrons. Selenium (Z = 34) is in Group 16, so it has 6 valence electrons (4s²4p⁴). The remaining 34 - 6 = 28 electrons are core electrons. Note: selenium's 3d electrons are core electrons, not valence electrons, because the d subshell is fully filled and in a lower principal energy level.
Why do noble gases not form bonds under normal conditions?
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Noble gases already have a full octet (8 valence electrons, or 2 for helium). They have no driving force to gain, lose, or share electrons. Their electron configuration is maximally stable - every subshell in their outermost shell is filled. This is why the octet rule exists: other elements bond to achieve this same stable arrangement.
Neutral atoms gain or lose electrons to form ions. Writing the electron configuration of an ion follows different rules for anions and cations, and there is one critical trap that the MCAT loves to test.
Anions (Negative Ions)
Anions are formed when atoms gain electrons. The extra electrons fill into the next available subshell following the same Aufbau rules as neutral atoms.
Example: Fluorine (Z = 9) has the configuration 1s²2s²2p⁵. The fluoride ion (F⁻) gains one electron: 1s²2s²2p⁶ - the same configuration as neon.
Example: Oxygen (Z = 8) is 1s²2s²2p⁴. The oxide ion (O²⁻) gains two electrons: 1s²2s²2p⁶ - also isoelectronic with neon.
Cations (Positive Ions)
Cations are formed when atoms lose electrons. Here is where the critical rule comes in:
For transition metals, electrons are removed from the highest n value first - even if that is not the subshell that filled last.
This means for transition metals: s electrons are removed before d electrons, even though 4s fills before 3d.
Why the s Electrons Leave First
Although 4s fills before 3d (because 4s has a lower n + l value), once the d subshell begins to fill, the 3d electrons actually become lower in energy than the 4s electrons. This happens because d electrons penetrate more closely to the nucleus and become more tightly held. The 4s electrons, being in a higher shell, are farther from the nucleus and easier to remove.
Common Transition Metal Ions
Neutral Atom
Configuration
Ion
Ion Configuration
Fe (Z = 26)
[Ar] 4s²3d⁶
Fe²⁺
[Ar] 3d⁶
Fe (Z = 26)
[Ar] 4s²3d⁶
Fe³⁺
[Ar] 3d⁵
Cu (Z = 29)
[Ar] 4s¹3d¹⁰
Cu⁺
[Ar] 3d¹⁰
Cu (Z = 29)
[Ar] 4s¹3d¹⁰
Cu²⁺
[Ar] 3d⁹
Zn (Z = 30)
[Ar] 4s²3d¹⁰
Zn²⁺
[Ar] 3d¹⁰
Cr (Z = 24)
[Ar] 4s¹3d⁵
Cr³⁺
[Ar] 3d³
Isoelectronic Species
Isoelectronic species are atoms or ions that have the same number of electrons (and therefore the same electron configuration). For example:
N³⁻, O²⁻, F⁻, Ne, Na⁺, Mg²⁺, Al³⁺ - all have 10 electrons and the configuration 1s²2s²2p⁶.
Despite having the same electron configuration, these species have different sizes because they have different numbers of protons pulling on those 10 electrons. More protons = smaller radius. This concept is essential for understanding ionic radius trends (covered in Chapter 2).
What is the electron configuration of Fe³⁺? Is it paramagnetic or diamagnetic?
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[Ar] 3d⁵. Paramagnetic. Start with neutral Fe: [Ar] 4s²3d⁶. Remove 2 electrons from 4s first (Fe²⁺ = [Ar] 3d⁶), then remove 1 from 3d (Fe³⁺ = [Ar] 3d⁵). With 5 unpaired d electrons (all parallel spins by Hund's rule), Fe³⁺ is strongly paramagnetic.
List three ions that are isoelectronic with neon.
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F⁻, Na⁺, and Mg²⁺ (also O²⁻, N³⁻, Al³⁺). All have 10 electrons with the configuration 1s²2s²2p⁶, the same as neon. They differ in the number of protons (nuclear charge), which affects their size.
The photoelectric effect is one of the most important experiments in modern physics, and the MCAT tests it directly. It proved that light is not just a wave - it also behaves as particles (photons), each carrying a discrete packet of energy.
The Experiment
When light shines on a metal surface, electrons can be ejected from the metal. This is the photoelectric effect. But the results do not match what classical wave theory predicts:
Classical prediction: Brighter light (higher intensity) should give electrons more energy and eventually eject them, regardless of the light’s color (frequency).
Actual observation: Only light above a certain threshold frequency ejects electrons. Below this frequency, no electrons are ejected no matter how bright the light. Above the threshold, even dim light ejects electrons instantly.
Einstein’s Explanation
Einstein explained the photoelectric effect in 1905 by proposing that light consists of individual particles (photons), each with energy E = hf. His key insight:
Each photon interacts with one electron
If the photon’s energy (hf) is greater than or equal to the work function (phi) of the metal, the electron is ejected
If the photon’s energy is less than the work function, nothing happens - no matter how many photons hit the surface
Key Concepts
Threshold frequency (f₀): The minimum frequency of light needed to eject electrons. At the threshold: hf₀ = phi, so KE = 0 (the electron barely escapes).
Work function (phi): The minimum energy required to remove an electron from the metal surface. Different metals have different work functions. Metals with low work functions (like cesium) eject electrons more easily.
Above threshold: Any photon with f > f₀ will eject an electron. The excess energy (hf - phi) becomes the kinetic energy of the ejected electron.
Intensity vs. Frequency
This distinction is the core of what the MCAT tests:
| Factor | What it affects | What it does NOT affect |
|--------|----------------|----------------------|
| Frequency (color) | Whether electrons are ejected; KE of each ejected electron | Number of electrons ejected |
| Intensity (brightness) | Number of electrons ejected (if above threshold) | Whether electrons are ejected; KE of ejected electrons |
Increasing frequency (above threshold): Each ejected electron has more kinetic energy
Increasing intensity (above threshold): More electrons are ejected per second, but each has the same KE
Increasing intensity (below threshold): Nothing happens. Zero electrons ejected. No exceptions.
Graphical Representation
The photoelectric effect: what frequency changes and what brightness changes
Quantised light
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Why every line is parallelThe slope of each line is Planck's constant. It does not depend on which metal you shine light at, because it is a fact about the photon, not the surface. Only the intercept moves.
What the intercept meansWhere a line crosses the axis is that metal's threshold frequency, f₀ = φ/h. Below it the graph does not go negative, it simply stops: no electrons come off at all, and turning up the intensity does not change that.
The result that broke classical physicsA wave should let you accumulate energy, so a dim blue lamp ought to eventually free an electron and a bright red one certainly should. Neither happens. Energy arrives in single photons of E = hf, and one photon either carries enough to pay the work function or it does not.
Brightness and colour do completely different jobs. Raising the intensity sends more photons, so more electrons come off, but each one leaves with exactly the same kinetic energy. Raising the frequency raises the energy of every electron. That split is the whole experiment.
MCAT passages may show a graph of kinetic energy vs. frequency for the photoelectric effect:
The graph is a straight line with slope = h (Planck’s constant)
The x-intercept is the threshold frequency (f₀)
The y-intercept (extrapolated) gives -phi (negative work function)
Below f₀, the line does not exist - KE = 0, no electrons ejected
Light with frequency 8.0×1014 Hz hits a metal with work function 3.0×10−19 J. Are electrons ejected? If so, what is their kinetic energy? (h=6.626×10−34 J·s)
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Yes, electrons are ejected. KE = 2.3×10−19 J. Photon energy = hf=(6.626×10−34)(8.0×1014)=5.3×10−19 J. This exceeds the work function (3.0×10−19 J), so electrons are ejected. KE=hf−ϕ=5.3×10−19−3.0×10−19=2.3×10−19 J.
If you double the intensity of light that is below the threshold frequency, what happens?
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Absolutely nothing. No electrons are ejected. Doubling the intensity means doubling the number of photons, but each photon still lacks sufficient energy (hf < phi) to eject an electron. The photoelectric effect requires each individual photon to have enough energy - you cannot “add up” the energies of multiple low-energy photons.
Every element has a unique set of energy levels. When electrons jump between these levels, they absorb or emit photons with very specific energies - and therefore very specific wavelengths. This creates a spectral “fingerprint” that is unique to each element.
Emission Spectra
When atoms are heated or electrically excited, electrons jump to higher energy levels. These excited states are unstable, so electrons quickly fall back to lower levels, releasing photons in the process. Each photon has a wavelength determined by the energy difference between the two levels:
The result is a line spectrum - a set of discrete, bright lines on a dark background. Each line corresponds to a specific electron transition. Because each element has a unique set of energy levels, each element produces a unique line spectrum.
The Hydrogen Emission Series
Hydrogen’s emission spectrum is the simplest and most important for the MCAT. The transitions are grouped into named series based on the final energy level:
Hydrogen energy levels and the lines they emit
Bohr model
1
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Why the rungs crowdEnergy goes as 1/n², so the n = 1 to n = 2 gap is 10.2 eV while n = 5 to n = 6 is under 0.2 eV. Every question that asks which transition emits the shortest wavelength is really asking which gap is biggest, and the biggest gaps are always at the bottom.
Reading a series off the ladderA series is named for where the electron lands, not where it starts. Landing on n = 1 means falling the whole way, so Lyman lines are ultraviolet. Landing on n = 2 gives the four visible Balmer lines. Landing on n = 3 gives infrared.
The sign conventionEnergies are negative because the electron is bound: zero is the free electron at n = ∞. Absorbing a photon moves the electron up toward zero, and it takes the full 13.6 eV to ionise hydrogen from the ground state.
The ladder and the spectrum are the same fact. Each coloured line on the right is drawn at the wavelength its arrow on the left produces, through λ = hc/ΔE. A bigger drop makes a bluer photon, which is why H-α from n = 3 is red and H-δ from n = 6 is violet.
| Series | Final level (nf) | Transitions from | Region of spectrum |
|--------|-------------------|------------------|--------------------|
| Lyman | 1 | n = 2, 3, 4, … to 1 | Ultraviolet (UV) |
| Balmer | 2 | n = 3, 4, 5, … to 2 | Visible light |
| Paschen | 3 | n = 4, 5, 6, … to 3 | Infrared (IR) |
Key Energy Relationships
The largest energy transition in any series is from n = infinity to nf (the series limit). The smallest is from nf + 1 to nf.
Within the Balmer series (visible light):
n = 3 to n = 2: red light (656 nm) - lowest energy visible transition
n = 4 to n = 2: blue-green (486 nm)
n = 5 to n = 2: blue-violet (434 nm)
n = 6 to n = 2: violet (410 nm) - highest energy visible transition
Remember: shorter wavelength = higher frequency = higher energy.
Absorption Spectra
An absorption spectrum is the reverse of an emission spectrum. When white light (containing all wavelengths) passes through a gas of atoms, the atoms absorb photons that match specific energy transitions. The result is a continuous spectrum with dark lines where specific wavelengths have been absorbed.
The wavelengths of the dark lines in an absorption spectrum match exactly the wavelengths of the bright lines in the emission spectrum for the same element. This makes sense - the same energy gaps are involved, whether the electron is jumping up (absorption) or falling down (emission).
| Feature | Emission Spectrum | Absorption Spectrum |
|---------|------------------|-------------------|
| Appearance | Bright lines on dark background | Dark lines on continuous background |
| Process | Electrons fall to lower levels | Electrons jump to higher levels |
| Energy | Released as photons | Absorbed from photons |
| Same wavelengths? | Yes - same element, same lines |
Continuous vs. Line Spectra
A continuous spectrum contains all wavelengths of light (like a rainbow) and is produced by hot, dense objects like the sun or an incandescent light bulb.
A line spectrum contains only specific wavelengths and is produced by excited atoms in the gas phase. The discrete lines reflect the quantized energy levels of the atoms.
Which hydrogen emission series produces visible light? What is the final energy level for transitions in this series?
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The Balmer series produces visible light. All transitions in the Balmer series end at n = 2 (nf = 2). Transitions from n = 3, 4, 5, and 6 down to n = 2 produce red, blue-green, blue-violet, and violet light respectively.
How do emission and absorption spectra of the same element compare?
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They are complementary. The bright lines in the emission spectrum appear at exactly the same wavelengths as the dark lines in the absorption spectrum. Both correspond to the same electron transitions between the same energy levels - emission lines result from electrons falling down, absorption lines result from electrons jumping up.
We have established that electrons absorb photons to jump to higher energy levels and emit photons to fall back down. Fluorescence and phosphorescence are two specific types of light emission that the MCAT expects you to distinguish.
Fluorescence
Fluorescence occurs when a substance absorbs light at one wavelength and immediately re-emits it at a longer wavelength (lower energy). The emission stops almost instantly when the light source is removed.
Here is the process:
A photon is absorbed, exciting an electron to a higher energy level
The electron loses some energy through non-radiative relaxation (vibrations, heat) - dropping to a slightly lower excited state
The electron then falls back to the ground state, emitting a photon
Because some energy was lost to heat in step 2, the emitted photon has less energy (longer wavelength) than the absorbed photon
This energy difference between absorbed and emitted light is called the Stokes shift.
Phosphorescence
Phosphorescence is similar to fluorescence but with a critical difference: the emission continues for a period of time after the excitation source is removed. Glow-in-the-dark materials are phosphorescent.
Fluorescence and phosphorescence on one diagram
Electron transitions
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Why emitted light is always redderThe molecule absorbs into a high vibrational level of the excited state, then sheds that vibrational energy as heat in about a picosecond before it emits. The photon that comes out is missing that heat, so it is lower in energy and longer in wavelength than the one that went in. That shift is why a fluorescent highlighter looks brighter than its surroundings under UV.
What makes phosphorescence slowIntersystem crossing flips the electron's spin, putting the molecule in a triplet state. Getting back to the singlet ground state means flipping the spin again, and that transition is formally forbidden. Forbidden does not mean impossible, it means rare, and rare means slow.
The one-line testTurn the lamp off. Fluorescence stops immediately because its lifetime is nanoseconds. Phosphorescence keeps glowing for milliseconds to minutes, which is exactly what a glow-in-the-dark star does.
One spin flip separates the two. Fluorescence returns to the ground state without changing spin, so it is fast and stops with the lamp. Phosphorescence has to cross into a triplet state and back, and because that spin flip is forbidden, the molecule is stuck holding the energy long enough to glow in a dark room.
The mechanism involves an additional step:
A photon is absorbed, exciting an electron to a higher singlet state
The electron undergoes intersystem crossing - a spin flip that converts it from a singlet state to a triplet state
The transition from the triplet state back to the ground state (singlet) is “spin-forbidden” - it violates quantum mechanical selection rules
Because it is forbidden, the transition is slow - the electron is “trapped” in the triplet state, releasing its energy gradually over seconds, minutes, or even hours
Comparison Table
| Feature | Fluorescence | Phosphorescence |
|---------|-------------|-----------------|
| Emission timing | Immediate (nanoseconds) | Delayed (seconds to hours) |
| After light removed | Stops instantly | Continues glowing |
| Excited state | Singlet (spins paired) | Triplet (spins parallel) |
| Transition type | Allowed | Spin-forbidden (slow) |
| Emitted wavelength | Longer than absorbed | Longer than absorbed |
| Example | Fluorescent lights, highlighters | Glow-in-the-dark stars, watch dials |
Practical Applications
Connecting Back to Atomic Structure
Fluorescence and phosphorescence are fundamentally about electron transitions - the same transitions that produce emission spectra. The key conceptual framework remains the same:
Electrons can only exist at specific energy levels
Moving to a higher level requires absorbing exactly the right amount of energy
Falling to a lower level releases energy as a photon
The energy of the photon equals the gap between levels: E = hf = hc/lambda
Whether we call it emission, fluorescence, or phosphorescence, the underlying physics is quantized electron transitions.
Why is the light emitted during fluorescence always a longer wavelength than the light absorbed?
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Because some absorbed energy is lost as heat during non-radiative relaxation. The electron drops to a slightly lower excited state before emitting a photon. Since some energy was lost to vibrations/heat, the emitted photon has less energy than the absorbed photon. Less energy means lower frequency and longer wavelength (E = hc/lambda). This energy difference is called the Stokes shift.
What is the key difference between fluorescence and phosphorescence?
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Timing. Fluorescence emission is immediate (nanoseconds) and stops when the light source is removed. Phosphorescence emission is delayed (seconds to hours) because the electron gets trapped in a triplet state via intersystem crossing, and the return transition to the ground state is spin-forbidden and therefore slow.
