The atomic number is the number of protons in an atom’s nucleus. It defines the element: if two atoms have the same atomic number, they are the same element.

The atomic weight is the weighted average of the atomic masses of all naturally occurring isotopes of that element. Here, an atom’s atomic mass is essentially the mass of its nucleus (the sum of protons + neutrons), since electrons contribute very little to the total mass.

In standard atomic notation, the atomic number is written as a subscript (bottom), and the atomic weight is written as a superscript (top).

The mass of an electron is $9.11$ $\times 1 0^{- 31} kg .$

Neutrons, protons, isotopes

Neutrons are neutral particles found in the nucleus of an atom. Protons are positively charged particles also found in the nucleus. Together, protons and neutrons are called nucleons, and they account for nearly all of an atom’s mass.

Isotopes are atoms of the same element (same number of protons, so the same atomic number) that differ in their number of neutrons. Because the neutron count changes, isotopes have different atomic masses.

For reference:

A proton has a mass of about 1 atomic mass unit (amu) and a charge of +1.
A neutron has a mass of about 1 amu and a charge of 0.
An electron has a charge of −1 and negligible mass compared with nucleons.

Because isotopes have the same electron structure when neutral, they have nearly identical chemical properties. However, their nuclear stability can differ: some isotopes are stable, while others undergo radioactive decay.

Nuclear forces, binding energy

Inside the nucleus, two key forces matter:

The strong force binds nucleons together and is responsible for nuclear binding.
The electromagnetic force causes repulsion between positively charged protons.

A nucleus stays intact because, at very short distances, the strong force is much stronger than the electromagnetic repulsion.

Binding energy usually means nuclear binding energy: the energy that holds nucleons together in the nucleus, arising from the strong force. The binding energy per nucleon is often used as a measure of nuclear stability. For example, Iron-56 has one of the highest binding energies per nucleon (very stable), while Deuterium (a two-nucleon isotope of hydrogen) has one of the lowest.

Separately, electron binding energy (more commonly called ionization energy) is the energy required to remove an electron from an atom.

Radioactive decay

Radioactive decay is the process by which an unstable nucleus releases energy by emitting particles or radiation. There are three primary types:

Alpha decay, where an alpha particle (a helium nucleus) is emitted at low speed
Beta decay, which involves the ejection of a high-speed electron known as a beta particle
Gamma decay, where a high-energy gamma ray (an electromagnetic wave or photon) is released.

These decay processes follow conservation laws, so the total atomic weight and atomic number are conserved across the full decay process.

Half-life, exponential decay, semi-log plots, fission and fusion

Stability describes whether a substance resists decay. A stable substance does not decay, while an unstable one does. A shorter half-life means the substance decays more quickly.

The standard half-life equation relates the remaining amount of material (Nₜ) to the initial quantity (N₀) and the elapsed time (t). Alternative forms of this equation are sometimes used to make calculations more convenient.

Radioactive decay typically follows an exponential pattern: the amount decreases by a constant fraction over equal time intervals.

A common way to analyze exponential decay is with a semi-log plot, which graphs the logarithm of the remaining quantity versus time. On a semi-log plot, exponential decay appears as a straight line with a negative slope. The line intercepts the x-axis when the initial quantity equals one.

In nuclear reactions:

Fission occurs when a heavy nucleus splits into smaller nuclei after being struck by a free neutron. This releases additional neutrons that can trigger a chain reaction, as in uranium.
Fusion occurs when two nuclei combine. This powers the sun by fusing hydrogen atoms into helium.

Mass spectrometer

A mass spectrometer measures the mass-to-charge ratio of ionized particles.

In the ion source, the sample is ionized, turning molecules into charged particles.
The ions are accelerated by an electric field.
In the mass analyzer, a magnetic field deflects the ions based on their mass-to-charge ratio.
Lighter ions are deflected more than heavier ones, so the ions separate.
A detector records the ions, producing a mass spectrum.

The mass spectrum provides information about molecular structure, composition, and abundance, which is why mass spectrometry is widely used in chemistry, biochemistry, and environmental science.

Electronic structure

In the hydrogen atom, the electron’s arrangement is described by its orbital structure. The Bohr model pictured the electron moving in fixed paths around the nucleus, but quantum mechanics describes the electron as existing in a spherical probability cloud.

The principal quantum number ( $n$ ) identifies the shell (energy level). Values of $n$ start at 1 and increase for higher-energy shells. Each shell contains $n^{2}$ orbitals. Since each orbital can hold 2 electrons, the maximum number of electrons in a shell is $2 n^{2}$ .

Ground state, excited states, absorption and emission line spectra

Electrons in an atom usually occupy the ground state, the lowest possible energy level. If an electron absorbs energy, it can move to a higher-energy excited state. Because excited states are less energetically favorable, the electron tends to return to the ground state, releasing energy as light.

This absorption and release of energy produces characteristic absorption spectra and emission spectra:

An absorption spectrum appears as a series of dark lines on a continuous rainbow background. The dark lines mark wavelengths absorbed by the atom.
An emission spectrum appears as bright, colored lines on a dark background. The bright lines mark wavelengths emitted as electrons drop to lower energy levels.

The line patterns in absorption and emission spectra closely match, although emitted wavelengths are often slightly longer.

Absorption and emission spectra

Quantum numbers and quantum states

Quantum numbers describe the allowed states for electrons in an atom.

The angular momentum quantum number (l) ranges from $0$ to $n^{- 1}$ . It determines the orbital shape and corresponds to subshell labels:

$s$ for $l = 0$
$p$ for $l = 1$
$d$ for $l = 2$
$f$ for $l = 3$

Each subshell contains a specific number of orbitals - one in the s-subshell, three in the p-subshell, five in the d-subshell, and seven in the f-subshell. Since each orbital holds up to two electrons, a subshell can hold up to $4 l + 2$ electrons.

The magnetic quantum number ( $m$ ) takes integer values from $- l$ to $+ l$ and specifies an orbital’s orientation in space.

The spin quantum number ( $s$ ) describes the electron’s intrinsic angular momentum and can be $+ 1/2$ or $- 1/2$ .

Together, these quantum numbers specify an electron’s complete quantum state.

Paramagnetism and diamagnetism

Paramagnetism occurs in materials with unpaired electrons, which create a net magnetic moment. In an external magnetic field, these materials are attracted because their magnetic moments tend to align with the field.

Diamagnetism occurs when all electrons are paired, so there is no permanent magnetic moment. In a magnetic field, these materials develop an induced field that opposes the applied field, producing a weak repulsion.

Conventional notation for electronic structure

Conventional notation for electronic structure (often shown in an orbital diagram) follows these rules:

The Aufbau principle says electrons fill shells and subshells in order of increasing energy. For example, $1 s$ fills before $2 s$ , and $2 p$ fills next; $d$ subshells fill after the corresponding $s$ orbital.
Hund’s rule says that within a subshell with multiple orbitals (such as $p, d$ , or $f$ ), electrons occupy orbitals one at a time with the same spin until each orbital is half-filled. This minimizes electron-electron repulsion.
The Pauli exclusion principle says no two electrons in the same orbital can have identical spin; they must have opposite spins. Special cases occur in elements with d subshells, such as d⁴ and d⁹ configurations, where half-full or fully filled d subshells (for example, $s^{1} d^{5}$ or $s^{1} d^{10}$ ) are favored for increased stability.

Bohr atom and effective nuclear charge

The Bohr atom model shows electrons moving in fixed circular orbits around the nucleus. As the principal quantum number ( $n$ ) increases, the orbit size increases. This model is a simplified way to visualize electron positions.

The effective nuclear charge is the net positive charge felt by an electron. You find it by subtracting the shielding effect of shielding electrons (inner, lower-energy electrons) from the total nuclear charge. A higher effective nuclear charge holds electrons more tightly, which increases ionization energy and contributes to atomic stability. Moving left to right across a period of the periodic table, the effective nuclear charge on outer electrons generally increases.

Heisenberg uncertainty principle

The Heisenberg uncertainty principle sets a fundamental limit on how precisely you can know certain pairs of properties at the same time, such as position and momentum. The more precisely one is known, the less precisely the other can be known.

This idea also applies to other pairs of canonically conjugate variables, such as energy and time. For example, if you measure a particle’s position very accurately, the uncertainty in its momentum increases.

This limit is expressed with an inequality involving the standard deviations of the variables, with a lower bound set by Planck’s constant ( $6.62607015 \times 1 0^{- 34}$ joule second).

Photoelectric effect

The photoelectric effect (also called photoemission) occurs when light hits a metal surface and ejects electrons called photoelectrons.

A photon’s energy is given by Planck’s equation:

$E_{(} p h o t o n_{)}$ = $h ν$

where $h$ is Planck’s constant and $ν$ is the light frequency. Photoemission occurs when the photon energy is sufficient to overcome the metal’s work function ( $Φ$ ), the minimum energy needed to free an electron.

Any extra energy becomes the electron’s kinetic energy:

$K E_{e l ec t ro n} = h ν - Φ$

This shows that the kinetic energy increases linearly with frequency, as long as the photon energy exceeds the work function.

You can also relate kinetic energy to the electron’s speed (v):

$K E_{e l ec t ro n} = \frac{1}{2} m_{e} v^{2}$

where $m_{e}$ is the electron’s mass.

Atomic number, atomic weight

Atomic number = number of protons; defines element identity
Atomic weight = weighted average of isotopic atomic masses
Atomic mass ≈ protons + neutrons (nucleus mass)

Neutrons, protons, isotopes

Protons: +1 charge, ~1 amu; Neutrons: 0 charge, ~1 amu; Electrons: −1 charge, negligible mass
Isotopes: same protons (atomic number), different neutrons (atomic mass)
Isotopes: similar chemical properties, differing nuclear stability

Nuclear forces, binding energy

Strong force: binds nucleons; electromagnetic force: proton repulsion
Binding energy: energy holding nucleus together (from strong force)
Binding energy per nucleon = measure of nuclear stability

Radioactive decay

Alpha decay: emits alpha particle (He nucleus)
Beta decay: emits high-speed electron (beta particle)
Gamma decay: emits gamma ray (photon); atomic number and weight conserved

Half-life, exponential decay, semi-log plots, fission and fusion

Half-life: time for half of substance to decay; shorter half-life = faster decay
Radioactive decay follows exponential law; semi-log plot yields straight line
Fission: heavy nucleus splits, releases neutrons (chain reaction); Fusion: light nuclei combine (e.g., hydrogen to helium in sun)

Mass spectrometer

Measures mass-to-charge ratio of ions
Ion source (ionization), acceleration (electric field), mass analyzer (magnetic field), detector
Produces mass spectrum: reveals structure, composition, abundance

Electronic structure

Bohr model: electrons in fixed orbits; quantum mechanics: probability clouds
Principal quantum number ( $n$ ): shell/energy level; max electrons per shell = $2 n^{2}$
Each shell: $n^{2}$ orbitals; each orbital holds 2 electrons

Ground state, excited states, absorption and emission line spectra

Ground state: lowest energy; excited state: higher energy (after absorption)
Absorption spectrum: dark lines (absorbed wavelengths) on rainbow
Emission spectrum: bright lines (emitted wavelengths) on dark background

Quantum numbers and quantum states

Principal ( $n$ ): shell; Angular momentum ( $l$ ): subshell/orbital shape ( $s$ , $p$ , $d$ , $f$ )
Magnetic ( $m$ ): orbital orientation ( $- l$ to $+ l$ ); Spin ( $s$ ): $+ 1/2$ or $- 1/2$
Subshell holds $4 l + 2$ electrons; quantum numbers specify electron’s state

Paramagnetism and diamagnetism

Paramagnetism: unpaired electrons, attracted to magnetic field
Diamagnetism: all electrons paired, weakly repelled by magnetic field

Conventional notation for electronic structure

Aufbau principle: fill orbitals in order of increasing energy
Hund’s rule: one electron per orbital (same spin) before pairing
Pauli exclusion principle: no two electrons in same orbital with same spin; d⁴/d⁹ exceptions for stability

Bohr atom and effective nuclear charge

Bohr atom: electrons in fixed orbits; higher $n$ = larger orbit
Effective nuclear charge = nuclear charge − shielding electrons
Higher effective nuclear charge increases ionization energy, atomic stability

Heisenberg uncertainty principle

Cannot precisely know position and momentum simultaneously
Applies to other variable pairs (e.g., energy and time)
Uncertainty limit set by Planck’s constant ( $h$ )

Photoelectric effect

Light ejects electrons from metal if photon energy > work function ( $Φ$ )
Photon energy: $E = h ν$ ; Kinetic energy: $K E = h ν - Φ$
$K E = \frac{1}{2} m_{e} v^{2}$ ; kinetic energy increases with frequency above threshold

Atomic nucleus and electronic structure

Atomic number, atomic weight

The atomic number is the number of protons in an atom’s nucleus. It defines the element: if two atoms have the same atomic number, they are the same element.

In standard atomic notation, the atomic number is written as a subscript (bottom), and the atomic weight is written as a superscript (top).

The mass of an electron is $9.11$ $\times 1 0^{- 31} kg .$

Neutrons, protons, isotopes

For reference:

A proton has a mass of about 1 atomic mass unit (amu) and a charge of +1.
A neutron has a mass of about 1 amu and a charge of 0.
An electron has a charge of −1 and negligible mass compared with nucleons.

Nuclear forces, binding energy

Inside the nucleus, two key forces matter:

The strong force binds nucleons together and is responsible for nuclear binding.
The electromagnetic force causes repulsion between positively charged protons.

A nucleus stays intact because, at very short distances, the strong force is much stronger than the electromagnetic repulsion.

Separately, electron binding energy (more commonly called ionization energy) is the energy required to remove an electron from an atom.

Radioactive decay

Radioactive decay is the process by which an unstable nucleus releases energy by emitting particles or radiation. There are three primary types:

Alpha decay, where an alpha particle (a helium nucleus) is emitted at low speed
Beta decay, which involves the ejection of a high-speed electron known as a beta particle
Gamma decay, where a high-energy gamma ray (an electromagnetic wave or photon) is released.

These decay processes follow conservation laws, so the total atomic weight and atomic number are conserved across the full decay process.

Half-life, exponential decay, semi-log plots, fission and fusion

Stability describes whether a substance resists decay. A stable substance does not decay, while an unstable one does. A shorter half-life means the substance decays more quickly.

Radioactive decay typically follows an exponential pattern: the amount decreases by a constant fraction over equal time intervals.

In nuclear reactions:

Fission occurs when a heavy nucleus splits into smaller nuclei after being struck by a free neutron. This releases additional neutrons that can trigger a chain reaction, as in uranium.
Fusion occurs when two nuclei combine. This powers the sun by fusing hydrogen atoms into helium.

Mass spectrometer

A mass spectrometer measures the mass-to-charge ratio of ionized particles.

In the ion source, the sample is ionized, turning molecules into charged particles.
The ions are accelerated by an electric field.
In the mass analyzer, a magnetic field deflects the ions based on their mass-to-charge ratio.
Lighter ions are deflected more than heavier ones, so the ions separate.
A detector records the ions, producing a mass spectrum.

The mass spectrum provides information about molecular structure, composition, and abundance, which is why mass spectrometry is widely used in chemistry, biochemistry, and environmental science.

Electronic structure

Ground state, excited states, absorption and emission line spectra

This absorption and release of energy produces characteristic absorption spectra and emission spectra:

An absorption spectrum appears as a series of dark lines on a continuous rainbow background. The dark lines mark wavelengths absorbed by the atom.
An emission spectrum appears as bright, colored lines on a dark background. The bright lines mark wavelengths emitted as electrons drop to lower energy levels.

The line patterns in absorption and emission spectra closely match, although emitted wavelengths are often slightly longer.

Absorption and emission spectra

Quantum numbers and quantum states

Quantum numbers describe the allowed states for electrons in an atom.

The angular momentum quantum number (l) ranges from $0$ to $n^{- 1}$ . It determines the orbital shape and corresponds to subshell labels:

$s$ for $l = 0$
$p$ for $l = 1$
$d$ for $l = 2$
$f$ for $l = 3$

The magnetic quantum number ( $m$ ) takes integer values from $- l$ to $+ l$ and specifies an orbital’s orientation in space.

The spin quantum number ( $s$ ) describes the electron’s intrinsic angular momentum and can be $+ 1/2$ or $- 1/2$ .

Together, these quantum numbers specify an electron’s complete quantum state.

Paramagnetism and diamagnetism

Conventional notation for electronic structure

Conventional notation for electronic structure (often shown in an orbital diagram) follows these rules:

The Aufbau principle says electrons fill shells and subshells in order of increasing energy. For example, $1 s$ fills before $2 s$ , and $2 p$ fills next; $d$ subshells fill after the corresponding $s$ orbital.
Hund’s rule says that within a subshell with multiple orbitals (such as $p, d$ , or $f$ ), electrons occupy orbitals one at a time with the same spin until each orbital is half-filled. This minimizes electron-electron repulsion.
The Pauli exclusion principle says no two electrons in the same orbital can have identical spin; they must have opposite spins. Special cases occur in elements with d subshells, such as d⁴ and d⁹ configurations, where half-full or fully filled d subshells (for example, $s^{1} d^{5}$ or $s^{1} d^{10}$ ) are favored for increased stability.

Bohr atom and effective nuclear charge

Heisenberg uncertainty principle

This limit is expressed with an inequality involving the standard deviations of the variables, with a lower bound set by Planck’s constant ( $6.62607015 \times 1 0^{- 34}$ joule second).

Photoelectric effect

The photoelectric effect (also called photoemission) occurs when light hits a metal surface and ejects electrons called photoelectrons.

A photon’s energy is given by Planck’s equation:

$E_{(} p h o t o n_{)}$ = $h ν$

Any extra energy becomes the electron’s kinetic energy:

$K E_{e l ec t ro n} = h ν - Φ$

This shows that the kinetic energy increases linearly with frequency, as long as the photon energy exceeds the work function.

You can also relate kinetic energy to the electron’s speed (v):

$K E_{e l ec t ro n} = \frac{1}{2} m_{e} v^{2}$

where $m_{e}$ is the electron’s mass.

Achievable MCAT

4. Chem/phys

4.5. Atoms, nuclear decay, electronic structure, and atomic chemical behavior

Atomic nucleus and electronic structure

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Atomic number, atomic weight

The atomic number is the number of protons in an atom’s nucleus. It defines the element: if two atoms have the same atomic number, they are the same element.

In standard atomic notation, the atomic number is written as a subscript (bottom), and the atomic weight is written as a superscript (top).

The mass of an electron is $9.11$ $\times 1 0^{- 31} kg .$

Neutrons, protons, isotopes

For reference:

A proton has a mass of about 1 atomic mass unit (amu) and a charge of +1.
A neutron has a mass of about 1 amu and a charge of 0.
An electron has a charge of −1 and negligible mass compared with nucleons.

Nuclear forces, binding energy

Inside the nucleus, two key forces matter:

The strong force binds nucleons together and is responsible for nuclear binding.
The electromagnetic force causes repulsion between positively charged protons.

A nucleus stays intact because, at very short distances, the strong force is much stronger than the electromagnetic repulsion.

Separately, electron binding energy (more commonly called ionization energy) is the energy required to remove an electron from an atom.

Radioactive decay

Radioactive decay is the process by which an unstable nucleus releases energy by emitting particles or radiation. There are three primary types:

Alpha decay, where an alpha particle (a helium nucleus) is emitted at low speed
Beta decay, which involves the ejection of a high-speed electron known as a beta particle
Gamma decay, where a high-energy gamma ray (an electromagnetic wave or photon) is released.

These decay processes follow conservation laws, so the total atomic weight and atomic number are conserved across the full decay process.

Half-life, exponential decay, semi-log plots, fission and fusion

Stability describes whether a substance resists decay. A stable substance does not decay, while an unstable one does. A shorter half-life means the substance decays more quickly.

Radioactive decay typically follows an exponential pattern: the amount decreases by a constant fraction over equal time intervals.

In nuclear reactions:

Fission occurs when a heavy nucleus splits into smaller nuclei after being struck by a free neutron. This releases additional neutrons that can trigger a chain reaction, as in uranium.
Fusion occurs when two nuclei combine. This powers the sun by fusing hydrogen atoms into helium.

Mass spectrometer

A mass spectrometer measures the mass-to-charge ratio of ionized particles.

In the ion source, the sample is ionized, turning molecules into charged particles.
The ions are accelerated by an electric field.
In the mass analyzer, a magnetic field deflects the ions based on their mass-to-charge ratio.
Lighter ions are deflected more than heavier ones, so the ions separate.
A detector records the ions, producing a mass spectrum.

The mass spectrum provides information about molecular structure, composition, and abundance, which is why mass spectrometry is widely used in chemistry, biochemistry, and environmental science.

Electronic structure

Ground state, excited states, absorption and emission line spectra

This absorption and release of energy produces characteristic absorption spectra and emission spectra:

An absorption spectrum appears as a series of dark lines on a continuous rainbow background. The dark lines mark wavelengths absorbed by the atom.
An emission spectrum appears as bright, colored lines on a dark background. The bright lines mark wavelengths emitted as electrons drop to lower energy levels.

The line patterns in absorption and emission spectra closely match, although emitted wavelengths are often slightly longer.

Absorption and emission spectra

Quantum numbers and quantum states

Quantum numbers describe the allowed states for electrons in an atom.

The angular momentum quantum number (l) ranges from $0$ to $n^{- 1}$ . It determines the orbital shape and corresponds to subshell labels:

$s$ for $l = 0$
$p$ for $l = 1$
$d$ for $l = 2$
$f$ for $l = 3$

The magnetic quantum number ( $m$ ) takes integer values from $- l$ to $+ l$ and specifies an orbital’s orientation in space.

The spin quantum number ( $s$ ) describes the electron’s intrinsic angular momentum and can be $+ 1/2$ or $- 1/2$ .

Together, these quantum numbers specify an electron’s complete quantum state.

Paramagnetism and diamagnetism

Conventional notation for electronic structure

Conventional notation for electronic structure (often shown in an orbital diagram) follows these rules:

The Aufbau principle says electrons fill shells and subshells in order of increasing energy. For example, $1 s$ fills before $2 s$ , and $2 p$ fills next; $d$ subshells fill after the corresponding $s$ orbital.
Hund’s rule says that within a subshell with multiple orbitals (such as $p, d$ , or $f$ ), electrons occupy orbitals one at a time with the same spin until each orbital is half-filled. This minimizes electron-electron repulsion.
The Pauli exclusion principle says no two electrons in the same orbital can have identical spin; they must have opposite spins. Special cases occur in elements with d subshells, such as d⁴ and d⁹ configurations, where half-full or fully filled d subshells (for example, $s^{1} d^{5}$ or $s^{1} d^{10}$ ) are favored for increased stability.

Bohr atom and effective nuclear charge

Heisenberg uncertainty principle

This limit is expressed with an inequality involving the standard deviations of the variables, with a lower bound set by Planck’s constant ( $6.62607015 \times 1 0^{- 34}$ joule second).

Photoelectric effect

The photoelectric effect (also called photoemission) occurs when light hits a metal surface and ejects electrons called photoelectrons.

A photon’s energy is given by Planck’s equation:

$E_{(} p h o t o n_{)}$ = $h ν$

Any extra energy becomes the electron’s kinetic energy:

$K E_{e l ec t ro n} = h ν - Φ$

This shows that the kinetic energy increases linearly with frequency, as long as the photon energy exceeds the work function.

You can also relate kinetic energy to the electron’s speed (v):

$K E_{e l ec t ro n} = \frac{1}{2} m_{e} v^{2}$

where $m_{e}$ is the electron’s mass.

Atomic number, atomic weight

Atomic number = number of protons; defines element identity
Atomic weight = weighted average of isotopic atomic masses
Atomic mass ≈ protons + neutrons (nucleus mass)

Neutrons, protons, isotopes

Protons: +1 charge, ~1 amu; Neutrons: 0 charge, ~1 amu; Electrons: −1 charge, negligible mass
Isotopes: same protons (atomic number), different neutrons (atomic mass)
Isotopes: similar chemical properties, differing nuclear stability

Nuclear forces, binding energy

Strong force: binds nucleons; electromagnetic force: proton repulsion
Binding energy: energy holding nucleus together (from strong force)
Binding energy per nucleon = measure of nuclear stability

Radioactive decay

Alpha decay: emits alpha particle (He nucleus)
Beta decay: emits high-speed electron (beta particle)
Gamma decay: emits gamma ray (photon); atomic number and weight conserved

Half-life, exponential decay, semi-log plots, fission and fusion

Half-life: time for half of substance to decay; shorter half-life = faster decay
Radioactive decay follows exponential law; semi-log plot yields straight line
Fission: heavy nucleus splits, releases neutrons (chain reaction); Fusion: light nuclei combine (e.g., hydrogen to helium in sun)

Mass spectrometer

Measures mass-to-charge ratio of ions
Ion source (ionization), acceleration (electric field), mass analyzer (magnetic field), detector
Produces mass spectrum: reveals structure, composition, abundance

Electronic structure

Bohr model: electrons in fixed orbits; quantum mechanics: probability clouds
Principal quantum number ( $n$ ): shell/energy level; max electrons per shell = $2 n^{2}$
Each shell: $n^{2}$ orbitals; each orbital holds 2 electrons

Ground state, excited states, absorption and emission line spectra

Ground state: lowest energy; excited state: higher energy (after absorption)
Absorption spectrum: dark lines (absorbed wavelengths) on rainbow
Emission spectrum: bright lines (emitted wavelengths) on dark background

Quantum numbers and quantum states

Principal ( $n$ ): shell; Angular momentum ( $l$ ): subshell/orbital shape ( $s$ , $p$ , $d$ , $f$ )
Magnetic ( $m$ ): orbital orientation ( $- l$ to $+ l$ ); Spin ( $s$ ): $+ 1/2$ or $- 1/2$
Subshell holds $4 l + 2$ electrons; quantum numbers specify electron’s state

Paramagnetism and diamagnetism

Paramagnetism: unpaired electrons, attracted to magnetic field
Diamagnetism: all electrons paired, weakly repelled by magnetic field

Conventional notation for electronic structure

Aufbau principle: fill orbitals in order of increasing energy
Hund’s rule: one electron per orbital (same spin) before pairing
Pauli exclusion principle: no two electrons in same orbital with same spin; d⁴/d⁹ exceptions for stability

Bohr atom and effective nuclear charge

Bohr atom: electrons in fixed orbits; higher $n$ = larger orbit
Effective nuclear charge = nuclear charge − shielding electrons
Higher effective nuclear charge increases ionization energy, atomic stability

Heisenberg uncertainty principle

Cannot precisely know position and momentum simultaneously
Applies to other variable pairs (e.g., energy and time)
Uncertainty limit set by Planck’s constant ( $h$ )

Photoelectric effect

Light ejects electrons from metal if photon energy > work function ( $Φ$ )
Photon energy: $E = h ν$ ; Kinetic energy: $K E = h ν - Φ$
$K E = \frac{1}{2} m_{e} v^{2}$ ; kinetic energy increases with frequency above threshold

Atomic nucleus and electronic structure

Atomic number, atomic weight

The atomic number is the number of protons in an atom’s nucleus. It defines the element: if two atoms have the same atomic number, they are the same element.

In standard atomic notation, the atomic number is written as a subscript (bottom), and the atomic weight is written as a superscript (top).

The mass of an electron is $9.11$ $\times 1 0^{- 31} kg .$

Neutrons, protons, isotopes

For reference:

A proton has a mass of about 1 atomic mass unit (amu) and a charge of +1.
A neutron has a mass of about 1 amu and a charge of 0.
An electron has a charge of −1 and negligible mass compared with nucleons.

Nuclear forces, binding energy

Inside the nucleus, two key forces matter:

The strong force binds nucleons together and is responsible for nuclear binding.
The electromagnetic force causes repulsion between positively charged protons.

A nucleus stays intact because, at very short distances, the strong force is much stronger than the electromagnetic repulsion.

Separately, electron binding energy (more commonly called ionization energy) is the energy required to remove an electron from an atom.

Radioactive decay

Radioactive decay is the process by which an unstable nucleus releases energy by emitting particles or radiation. There are three primary types:

Alpha decay, where an alpha particle (a helium nucleus) is emitted at low speed
Beta decay, which involves the ejection of a high-speed electron known as a beta particle
Gamma decay, where a high-energy gamma ray (an electromagnetic wave or photon) is released.

These decay processes follow conservation laws, so the total atomic weight and atomic number are conserved across the full decay process.

Half-life, exponential decay, semi-log plots, fission and fusion

Stability describes whether a substance resists decay. A stable substance does not decay, while an unstable one does. A shorter half-life means the substance decays more quickly.

Radioactive decay typically follows an exponential pattern: the amount decreases by a constant fraction over equal time intervals.

In nuclear reactions:

Fission occurs when a heavy nucleus splits into smaller nuclei after being struck by a free neutron. This releases additional neutrons that can trigger a chain reaction, as in uranium.
Fusion occurs when two nuclei combine. This powers the sun by fusing hydrogen atoms into helium.

Mass spectrometer

A mass spectrometer measures the mass-to-charge ratio of ionized particles.

In the ion source, the sample is ionized, turning molecules into charged particles.
The ions are accelerated by an electric field.
In the mass analyzer, a magnetic field deflects the ions based on their mass-to-charge ratio.
Lighter ions are deflected more than heavier ones, so the ions separate.
A detector records the ions, producing a mass spectrum.

The mass spectrum provides information about molecular structure, composition, and abundance, which is why mass spectrometry is widely used in chemistry, biochemistry, and environmental science.

Electronic structure

Ground state, excited states, absorption and emission line spectra

This absorption and release of energy produces characteristic absorption spectra and emission spectra:

An absorption spectrum appears as a series of dark lines on a continuous rainbow background. The dark lines mark wavelengths absorbed by the atom.
An emission spectrum appears as bright, colored lines on a dark background. The bright lines mark wavelengths emitted as electrons drop to lower energy levels.

The line patterns in absorption and emission spectra closely match, although emitted wavelengths are often slightly longer.

Absorption and emission spectra

Quantum numbers and quantum states

Quantum numbers describe the allowed states for electrons in an atom.

The angular momentum quantum number (l) ranges from $0$ to $n^{- 1}$ . It determines the orbital shape and corresponds to subshell labels:

$s$ for $l = 0$
$p$ for $l = 1$
$d$ for $l = 2$
$f$ for $l = 3$

The magnetic quantum number ( $m$ ) takes integer values from $- l$ to $+ l$ and specifies an orbital’s orientation in space.

The spin quantum number ( $s$ ) describes the electron’s intrinsic angular momentum and can be $+ 1/2$ or $- 1/2$ .

Together, these quantum numbers specify an electron’s complete quantum state.

Paramagnetism and diamagnetism

Conventional notation for electronic structure

Conventional notation for electronic structure (often shown in an orbital diagram) follows these rules:

The Aufbau principle says electrons fill shells and subshells in order of increasing energy. For example, $1 s$ fills before $2 s$ , and $2 p$ fills next; $d$ subshells fill after the corresponding $s$ orbital.
Hund’s rule says that within a subshell with multiple orbitals (such as $p, d$ , or $f$ ), electrons occupy orbitals one at a time with the same spin until each orbital is half-filled. This minimizes electron-electron repulsion.
The Pauli exclusion principle says no two electrons in the same orbital can have identical spin; they must have opposite spins. Special cases occur in elements with d subshells, such as d⁴ and d⁹ configurations, where half-full or fully filled d subshells (for example, $s^{1} d^{5}$ or $s^{1} d^{10}$ ) are favored for increased stability.

Bohr atom and effective nuclear charge

Heisenberg uncertainty principle

This limit is expressed with an inequality involving the standard deviations of the variables, with a lower bound set by Planck’s constant ( $6.62607015 \times 1 0^{- 34}$ joule second).

Photoelectric effect

The photoelectric effect (also called photoemission) occurs when light hits a metal surface and ejects electrons called photoelectrons.

A photon’s energy is given by Planck’s equation:

$E_{(} p h o t o n_{)}$ = $h ν$

Any extra energy becomes the electron’s kinetic energy:

$K E_{e l ec t ro n} = h ν - Φ$

This shows that the kinetic energy increases linearly with frequency, as long as the photon energy exceeds the work function.

You can also relate kinetic energy to the electron’s speed (v):

$K E_{e l ec t ro n} = \frac{1}{2} m_{e} v^{2}$

where $m_{e}$ is the electron’s mass.

Key points

Atomic number, atomic weight

Atomic number = number of protons; defines element identity
Atomic weight = weighted average of isotopic atomic masses
Atomic mass ≈ protons + neutrons (nucleus mass)

Neutrons, protons, isotopes

Protons: +1 charge, ~1 amu; Neutrons: 0 charge, ~1 amu; Electrons: −1 charge, negligible mass
Isotopes: same protons (atomic number), different neutrons (atomic mass)
Isotopes: similar chemical properties, differing nuclear stability

Nuclear forces, binding energy

Strong force: binds nucleons; electromagnetic force: proton repulsion
Binding energy: energy holding nucleus together (from strong force)
Binding energy per nucleon = measure of nuclear stability

Radioactive decay

Alpha decay: emits alpha particle (He nucleus)
Beta decay: emits high-speed electron (beta particle)
Gamma decay: emits gamma ray (photon); atomic number and weight conserved

Half-life, exponential decay, semi-log plots, fission and fusion

Half-life: time for half of substance to decay; shorter half-life = faster decay
Radioactive decay follows exponential law; semi-log plot yields straight line
Fission: heavy nucleus splits, releases neutrons (chain reaction); Fusion: light nuclei combine (e.g., hydrogen to helium in sun)

Mass spectrometer

Measures mass-to-charge ratio of ions
Ion source (ionization), acceleration (electric field), mass analyzer (magnetic field), detector
Produces mass spectrum: reveals structure, composition, abundance

Electronic structure

Bohr model: electrons in fixed orbits; quantum mechanics: probability clouds
Principal quantum number ( $n$ ): shell/energy level; max electrons per shell = $2 n^{2}$
Each shell: $n^{2}$ orbitals; each orbital holds 2 electrons

Ground state, excited states, absorption and emission line spectra

Ground state: lowest energy; excited state: higher energy (after absorption)
Absorption spectrum: dark lines (absorbed wavelengths) on rainbow
Emission spectrum: bright lines (emitted wavelengths) on dark background

Quantum numbers and quantum states

Principal ( $n$ ): shell; Angular momentum ( $l$ ): subshell/orbital shape ( $s$ , $p$ , $d$ , $f$ )
Magnetic ( $m$ ): orbital orientation ( $- l$ to $+ l$ ); Spin ( $s$ ): $+ 1/2$ or $- 1/2$
Subshell holds $4 l + 2$ electrons; quantum numbers specify electron’s state

Paramagnetism and diamagnetism

Paramagnetism: unpaired electrons, attracted to magnetic field
Diamagnetism: all electrons paired, weakly repelled by magnetic field

Conventional notation for electronic structure

Aufbau principle: fill orbitals in order of increasing energy
Hund’s rule: one electron per orbital (same spin) before pairing
Pauli exclusion principle: no two electrons in same orbital with same spin; d⁴/d⁹ exceptions for stability

Bohr atom and effective nuclear charge

Bohr atom: electrons in fixed orbits; higher $n$ = larger orbit
Effective nuclear charge = nuclear charge − shielding electrons
Higher effective nuclear charge increases ionization energy, atomic stability

Heisenberg uncertainty principle

Cannot precisely know position and momentum simultaneously
Applies to other variable pairs (e.g., energy and time)
Uncertainty limit set by Planck’s constant ( $h$ )

Photoelectric effect

Light ejects electrons from metal if photon energy > work function ( $Φ$ )
Photon energy: $E = h ν$ ; Kinetic energy: $K E = h ν - Φ$
$K E = \frac{1}{2} m_{e} v^{2}$ ; kinetic energy increases with frequency above threshold