## Learning Outcomes

- Interpret atomic notation to determine protons, neutrons, electrons, isotopes, and ion charge.
- Distinguish atomic mass from atomic weight and use weighted averages for isotope abundance questions.
- Predict changes in atomic number and mass number during alpha, beta, gamma decay, and electron capture.
- Apply half-life and exponential decay equations to radioactive samples.
- Relate photon energy, wavelength, frequency, and electronic transitions in atoms.
- Use quantum numbers, orbital filling rules, and common exceptions to write electron configurations.
- Explain periodic trends using valence electrons, shielding, and effective nuclear charge.

## Medical College Admission Test (MCAT) Syllabus

This subtopic belongs to the Chemical and Physical Foundations of Biological Systems section, which contains 59 questions in 95 minutes, including both passage-based and discrete questions. Atomic structure is mainly general chemistry, but nuclear decay and photon interactions connect strongly to physics. Expect MCAT questions to test quick recognition, equation use, and data interpretation rather than long calculations.

- Atomic number, mass number, isotopes, ions, atomic mass, and atomic weight
- Nuclear binding energy, mass defect, fission, fusion, and radioactivity
- Alpha, beta-negative, beta-positive, gamma decay, and electron capture
- Half-life, decay constant, exponential decay, and remaining sample calculations
- Planck’s equation, Bohr model, absorption, emission, and spectra
- Quantum numbers, orbitals, Pauli exclusion principle, Hund’s rule, and Aufbau principle
- Electron configurations of atoms and ions, including transition metal exceptions
- Periodic trends and chemical behavior based on valence electrons and effective nuclear charge

## Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, look more closely at that area during your revision.

1. How many protons, neutrons, and electrons are present in $$^{133}\mathrm{Cs}^{+}$$?
2. In beta-negative decay, what happens to atomic number and mass number?
3. A radioactive isotope has a half-life of 6 hours. What fraction remains after 18 hours?
4. Which transition in hydrogen absorbs more energy: $$n=2$$ to $$n=3$$ or $$n=2$$ to $$n=6$$?
5. Why does chromium have the electron configuration $$[\mathrm{Ar}]4s^13d^5$$ rather than $$[\mathrm{Ar}]4s^23d^4$$?

## Introduction

Atoms explain both the chemical identity of matter and many physical signals measured in medicine and research. The number of protons defines an element; the arrangement of electrons explains bonding, reactivity, ionization, and spectroscopy; the nucleus explains radioactivity, isotopes, and nuclear medicine. MCAT passages may place these ideas in contexts such as PET imaging, radioactive tracers, mass spectrometry, atomic absorption, or biological ion chemistry.

For the MCAT, 4E is not just memorization. You must move between symbols, equations, diagrams, and words. A question might describe an isotope in prose, show a decay equation, or ask which electron transition gives the greatest photon energy. The safest approach is to tie each fact to a physical rule: conservation of nucleons, conservation of charge, quantized energy levels, and electron filling principles.

> **Key Term: Isotope**
> Atoms of the same element with the same number of protons but different numbers of neutrons.

### Nuclear composition, isotopes, and atomic mass

Atomic notation is written as:

$$
{}^{A}_{Z}X
$$

Here, $$X$$ is the chemical symbol, $$Z$$ is the atomic number, and $$A$$ is the mass number. The atomic number equals the number of protons. The mass number equals protons plus neutrons. Therefore:

$$
\text{neutrons}=A-Z
$$

For a neutral atom, the number of electrons equals the number of protons. For an ion, adjust the electron count using the charge: cations have lost electrons, and anions have gained electrons. For example, cesium has atomic number 55. Thus $$^{133}\mathrm{Cs}^{+}$$ has 55 protons, $$133-55=78$$ neutrons, and 54 electrons because the $$+1$$ charge means one electron has been removed.

This distinction is frequently tested because answer choices may include protons, neutrons, electrons, or total subatomic particles. Read the charge and isotope notation carefully.

> **Test Tip:**
> For ion electron counts, do not use mass number. Use atomic number, then adjust for charge. A $$+2$$ ion has two fewer electrons than the neutral atom; a $$-1$$ ion has one extra electron.

Atomic mass and atomic weight are related but not identical. The atomic mass of a specific isotope is the mass of one atom of that isotope, usually reported in atomic mass units. The atomic weight shown on the periodic table is a weighted average of naturally occurring isotopes.

> **Key Term: Atomic weight**
> The average atomic mass of an element based on the relative abundance of its naturally occurring isotopes.

A common MCAT style question gives the atomic weight and asks for isotope abundance. Hydrogen’s atomic weight is approximately 1.008 amu. If the only isotopes considered are $$^{1}\mathrm{H}$$ and $$^{2}\mathrm{H}$$, the average is very close to 1, meaning almost all atoms are $$^{1}\mathrm{H}$$. The extra 0.008 above 1 corresponds to about 0.8% deuterium in this simplified two-isotope model.

Nuclei also have binding energy. The measured mass of a nucleus is slightly less than the sum of the masses of its individual protons and neutrons. This missing mass is the mass defect, converted into binding energy according to Einstein’s equation:

$$
E=mc^2
$$

> **Key Term: Mass defect**
> The difference between the mass of a nucleus and the sum of the masses of its separated protons and neutrons.

Greater binding energy generally means a more stable nucleus. Nuclear reactions such as fission and fusion can release energy when products are more stable than reactants. In both cases, some mass is converted to energy.

### Nuclear decay, half-life, and radiation changes

Radioactive decay changes unstable nuclei into more stable forms. On the MCAT, you should be able to predict how atomic number $$Z$$ and mass number $$A$$ change for each major decay type.

Alpha decay emits an alpha particle, which is a helium nucleus:

$$
{}^{4}_{2}\mathrm{He}
$$

The parent nucleus loses 2 protons and 2 neutrons, so atomic number decreases by 2 and mass number decreases by 4.

Beta-negative decay converts a neutron into a proton and emits an electron plus an antineutrino. Since a neutron becomes a proton, atomic number increases by 1, while mass number stays the same.

Beta-positive decay, also called positron emission, converts a proton into a neutron and emits a positron plus a neutrino. Atomic number decreases by 1, while mass number stays the same. Positron emission is especially relevant to PET imaging, where emitted positrons annihilate with electrons and produce gamma photons detected by the scanner.

Electron capture occurs when the nucleus captures an inner-shell electron. A proton combines with that electron to form a neutron, so atomic number decreases by 1 and mass number stays the same. Because an inner electron is removed, outer electrons may fall into the vacancy and emit characteristic radiation.

Gamma decay emits a high-energy photon from an excited nucleus. It does not change atomic number or mass number.

> **Key Term: Half-life**
> The time required for half of the radioactive nuclei in a sample to decay.

Radioactive decay follows first-order kinetics. If $$n_0$$ is the number of undecayed nuclei at time zero, then the number remaining at time $$t$$ is:

$$
n=n_0e^{-\lambda t}
$$

The decay constant $$\lambda$$ is related to half-life by:

$$
\lambda=\frac{\ln 2}{T_{1/2}}=\frac{0.693}{T_{1/2}}
$$

For many MCAT questions, using half-lives is faster than using the exponential form:

$$
n=n_0\left(\frac{1}{2}\right)^k
$$

where $$k$$ is the number of half-lives elapsed.

> **Question:**
> A 64 mg sample has a half-life of 4 hours. How much remains after 12 hours?
> Twelve hours is three half-lives:
>
> $$
> k=\frac{12}{4}=3
> $$
>
> The sample halves three times:
>
> $$
> 64\to32\to16\to8\ \mathrm{mg}
> $$
>
> So 8 mg remains.

A radioactive sample never reaches complete decay in a finite number of half-lives. After each half-life, half of the remaining sample decays, so the amount approaches zero asymptotically.

> **Exam Warning:**
> Do not say that 100% of a radioactive sample is gone after two half-lives. After two half-lives, 25% remains; after three, 12.5% remains; after ten, about 0.1% remains.

### Quantized electrons, photons, and spectra

Classical physics could not fully explain atomic spectra, the photoelectric effect, or why electrons do not collapse into the nucleus. Quantum theory solved this by treating energy as quantized. Planck’s equation relates photon energy to frequency:

$$
E=hf
$$

Because electromagnetic waves also obey:

$$
c=f\lambda
$$

photon energy can be written as:

$$
E=\frac{hc}{\lambda}
$$

Higher frequency means higher energy. Shorter wavelength also means higher energy. This is often tested in spectroscopy questions.

The Bohr model describes the hydrogen atom and hydrogen-like ions by placing electrons in allowed energy levels. Although modern quantum mechanics is more accurate, the Bohr model remains useful for MCAT energy-transition questions. Electron levels are labeled by the principal quantum number $$n$$. For hydrogen, energy is more negative at lower $$n$$ values, meaning lower levels are more stable.

When an electron absorbs a photon, it moves to a higher energy level. When it falls to a lower level, it emits a photon. The photon energy equals the difference between the two levels:

$$
\Delta E=E_{\text{final}}-E_{\text{initial}}
$$

For absorption, $$\Delta E$$ is positive. For emission, the atom loses energy and the photon carries away the magnitude of that energy.

Energy levels get closer together as $$n$$ increases. Therefore, transitions involving low $$n$$ levels often have larger energy differences than transitions between high levels. For example, moving from $$n=2$$ to $$n=6$$ requires more energy than moving from $$n=3$$ to $$n=4$$. A downward transition from $$n=6$$ to $$n=2$$ emits the same energy that the upward $$n=2$$ to $$n=6$$ transition absorbs.

Absorption and emission spectra are central experimental links between electronic structure and measurable data. In an absorption spectrum, atoms absorb specific wavelengths that match allowed transitions. In an emission spectrum, excited atoms release photons at specific wavelengths as electrons return to lower levels. The lines are discrete because electron energies are quantized.

> **Key Term: Quantum number**
> A value used to describe an electron’s energy level, subshell, orbital orientation, or spin.

### Quantum numbers, configurations, and periodic behavior

The quantum mechanical model replaces fixed circular orbits with orbitals: regions of high probability for finding electrons. Four quantum numbers describe each electron.

The principal quantum number $$n$$ describes the shell and overall energy level. It can be any positive integer: 1, 2, 3, and so on.

The azimuthal quantum number $$l$$ describes the subshell shape. It ranges from 0 to $$n-1$$. The common labels are:

- $$l=0$$: s
- $$l=1$$: p
- $$l=2$$: d
- $$l=3$$: f

The magnetic quantum number $$m_l$$ describes orbital orientation and ranges from $$-l$$ to $$+l$$. For example, a p subshell has $$l=1$$, so $$m_l=-1,0,+1$$, meaning there are three p orbitals.

The spin quantum number $$m_s$$ is either $$+\frac{1}{2}$$ or $$-\frac{1}{2}$$. Two electrons in the same orbital must have opposite spins.

The Pauli exclusion principle states that no two electrons in the same atom can have the same four quantum numbers. Hund’s rule states that electrons fill degenerate orbitals singly with parallel spins before pairing. The Aufbau principle states that electrons fill lower-energy orbitals before higher-energy orbitals.

> **Key Term: Electron configuration**
> A notation showing how electrons occupy atomic orbitals and subshells.

For main-group atoms, configurations are usually straightforward. Oxygen has eight electrons:

$$
1s^22s^22p^4
$$

The valence electrons are in the highest principal energy level, here $$2s^22p^4$$. These six valence electrons explain oxygen’s common chemical behavior, including its tendency to gain or share two electrons to complete an octet.

> **Key Term: Valence electron**
> An electron in the outermost occupied shell of an atom that participates most directly in bonding and chemical reactivity.

Transition metals require extra care. Chromium and copper are common exceptions because half-filled and fully filled d subshells have extra stability:

$$
\mathrm{Cr}:[\mathrm{Ar}]4s^13d^5
\mathrm{Cu}:[\mathrm{Ar}]4s^13d^{10}
$$

For transition metal cations, electrons are removed from the highest principal quantum number first. That means $$4s$$ electrons are removed before $$3d$$ electrons. For example, neutral iron is:

$$
\mathrm{Fe}:[\mathrm{Ar}]4s^23d^6
$$

Iron(II) is:

$$
\mathrm{Fe}^{2+}:[\mathrm{Ar}]3d^6
$$

> **Test Tip:**
> When writing transition metal ions, fill using the usual order, but remove electrons from the orbital with the highest $$n$$ first. For first-row transition metals, remove $$4s$$ before $$3d$$.

Periodic trends follow from shielding, distance from the nucleus, and effective nuclear charge.

> **Key Term: Effective nuclear charge**
> The net positive charge experienced by an electron after accounting for attraction to the nucleus and repulsion from other electrons.

Across a period from left to right, protons are added to the nucleus while electrons are added to the same general shell. Shielding does not increase enough to cancel the added nuclear charge, so effective nuclear charge increases. As a result, atomic radius generally decreases, while ionization energy and electronegativity generally increase.

Down a group, atoms gain new electron shells. The outer electrons are farther from the nucleus and more shielded by inner electrons. Atomic radius increases down a group, while ionization energy and electronegativity generally decrease.

Chemical behavior is strongly tied to valence electrons. Alkali metals have one valence electron and commonly form $$+1$$ ions. Alkaline earth metals often form $$+2$$ ions. Halogens have seven valence electrons and commonly form $$-1$$ ions. Noble gases have filled valence shells and are relatively unreactive.

Electron affinity generally becomes more favorable across a period, especially for halogens, because adding an electron helps complete the valence shell. There are exceptions due to electron repulsion and subshell stability, so MCAT questions usually focus on broad trends rather than obscure anomalies.

Mass spectrometry connects atomic mass, isotopes, and abundance. A mass spectrum can show peaks corresponding to isotopes; the relative height or area of each peak reflects abundance. If chlorine appears with major isotope peaks near 35 and 37 amu in about a 3:1 ratio, its weighted average atomic mass is near 35.5 amu.

## Key Point Checklist

This article has covered the following key knowledge points:

- Atomic number equals protons; mass number equals protons plus neutrons.
- Isotopes have the same atomic number but different neutron counts.
- Ion electron count is found from atomic number and charge, not mass number.
- Atomic weight is a weighted average of naturally occurring isotopes.
- Mass defect is converted to nuclear binding energy by $$E=mc^2$$.
- Alpha decay decreases $$A$$ by 4 and $$Z$$ by 2.
- Beta-negative decay increases $$Z$$ by 1; beta-positive decay and electron capture decrease $$Z$$ by 1.
- Gamma decay changes nuclear energy but not atomic number or mass number.
- Radioactive decay follows $$n=n_0e^{-\lambda t}$$ and $$\lambda=0.693/T_{1/2}$$.
- Photon energy is proportional to frequency and inversely proportional to wavelength.
- Quantum numbers describe shell, subshell, orbital orientation, and spin.
- Periodic trends are explained by valence electrons, shielding, and effective nuclear charge.

## Key Terms and Concepts

- Isotope
- Atomic weight
- Mass defect
- Half-life
- Quantum number
- Electron configuration
- Valence electron
- Effective nuclear charge


Foundational concept 4: Physical principles - 4E: Atoms, nuclear decay, electronic structure, atomic chemical behavior