Nuclear Reactions

At the core of an atom lies the nucleus, made of protons (positively charged) and neutrons (neutral) — collectively called nucleons. Surrounding the nucleus are electrons, which are negatively charged and contribute very little to the atom's overall mass. The atomic number (Z) is the number of protons in the nucleus, and the mass number (A) is the total number of protons and neutrons. Together, these values let us represent an atom using standard nuclear notation, where the mass number is written as a superscript and the atomic number as a subscript next to the elemental symbol: ᴬZX. For example, oxygen is written as ¹⁶₈O, where 16 is the mass number and 8 is the atomic number.

Key Takeaways

• Nuclear notation writes mass number as a superscript and atomic number as a subscript: ᴬZX.

• Fusion combines small nuclei into a larger one; fission splits a large nucleus into smaller ones — both release energy.

• Radioactive decay conserves total nucleons and total charge across parent nucleus → daughter nucleus + emitted particle.

• Alpha decay: Z −2, A −4. Beta-minus: Z +1, A same. Beta-plus: Z −1, A same. Gamma decay: Z and A unchanged (energy only). Electron capture: Z −1, A same.

• Half-life (T₁/₂) and the decay constant (λ) describe exponential decay: n = n₀e^(−λt), with T₁/₂ = ln2/λ ≈ 0.693/λ.

Fusion and Fission

Nuclear reactions involve changes in the composition of the nucleus, usually accompanied by the release of energy. There are two main types:

• Fusion: two smaller nuclei combine to form a larger, more stable nucleus, releasing a tremendous amount of energy. This is the reaction that powers the Sun, where hydrogen nuclei fuse to form helium.

• Fission: a large, unstable nucleus splits into smaller nuclei, often releasing neutrons and energy. Fission is the basis of nuclear reactors and atomic bombs.

Radioactive Decay and Conservation Rules

Radioactive decay is a naturally occurring process in which unstable nuclei spontaneously emit particles or radiation to become more stable. On the MCAT, you should be prepared for three general types of radioactive decay problems:

1. The integer arithmetic of particles and isotopes.

2. Radioactive half-life problems.

3. The use of exponential decay curves and decay constants.

Answering these requires understanding isotope decay arithmetic and nucleon conservation. Radioactive decay transforms one isotope (the parent nucleus) into another (the daughter nucleus), along with the emission of a decay particle. These reactions follow strict conservation rules: the total number of nucleons and the total charge must remain constant across the reaction.

Alpha Decay

Alpha decay occurs when an unstable nucleus emits an alpha particle — essentially a helium-4 nucleus (2 protons + 2 neutrons, no electrons) — to become more stable:

ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂He

The atomic number decreases by 2 and the mass number decreases by 4. Alpha decay is common in heavy nuclei (such as uranium and radium), where the strong nuclear force can't fully counteract the repulsive electrostatic forces between so many protons. By emitting an alpha particle, the nucleus reduces its size and charge, increasing stability.

Beta Decay (Beta-Minus and Beta-Plus)

A beta particle is either an electron (β⁻) or a positron (β⁺) — a particle with an electron's mass but a positive charge. Neither is normally found in the nucleus; both are produced during beta decay as nucleons transform.

Beta-minus (β⁻) decay: a neutron converts into a proton, emitting an electron (and, technically, an antineutrino — not MCAT-testable detail):

ᴬZX → ᴬZ₊₁Y + ⁰₋₁e

The atomic number increases by 1 (a neutron became a proton); the mass number is unchanged.

Beta-plus (β⁺) decay, or positron emission: a proton converts into a neutron, releasing a positron (and a neutrino):

ᴬZX → ᴬZ₋₁Y + ⁰₊₁e

The atomic number decreases by 1 (a proton became a neutron); the mass number is unchanged.

Gamma Decay

Gamma decay releases energy as a high-energy photon (a gamma ray, γ) without emitting any particle. It occurs as a nucleus drops from an excited state to a lower-energy or ground state — similar to how electrons emit light falling to lower energy levels:

ᴬZX* → ᴬZX + γ

Because gamma rays have no charge or mass, gamma decay changes neither the atomic number nor the mass number — it's purely a release of energy.

Electron Capture

Electron capture is best thought of as the reverse of beta-minus decay. A nucleus absorbs an inner orbital electron, which combines with a proton to form a neutron:

ᴬZX + ⁰₋₁e → ᴬZ₋₁Y

The atomic number decreases by 1; the mass number is unchanged, since no nucleons are gained or lost.

Half-Life and Exponential Decay

Half-life (T₁/₂) is the time required for half of the atoms in a radioactive sample to decay. It's unique to each radioactive isotope and independent of the initial amount of material — a predictable timescale that follows an exponential decay model.

Let n be the number of radioactive nuclei in a sample that haven't yet decayed. The rate of decay, Δn/Δt, is directly proportional to the number of undecayed nuclei remaining:

Δn/Δt = −λn

Here, λ is the decay constant — a value unique to each isotope quantifying how quickly it decays. The negative sign shows the undecayed nuclei count decreasing over time. Solving this relationship gives the exponential decay formula:

n = n₀e^(−λt)

where n₀ is the initial number of undecayed nuclei at t = 0, n is the number remaining at time t, and e is Euler's number (≈ 2.718). The decay constant is related to the half-life by:

T₁/₂ = ln2/λ ≈ 0.693/λ

Worked example. Suppose a sample starts with n₀ = 800 g of a radioactive isotope with a half-life of 6 hours. How much remains after 18 hours?

Method 1 — counting half-lives directly: 18 hours ÷ 6 hours = 3 half-lives.

n = n₀(½)³ = 800 g × ⅛ = 100 g

Method 2 — using the decay constant: λ = 0.693 / 6 h ≈ 0.1155 h⁻¹

n = n₀e^(−λt) = 800 g × e^(−(0.1155)(18)) = 800 g × e^(−2.079) ≈ 800 g × 0.125 = 100 g

Both methods agree: 100 g of the original 800 g sample remains after 18 hours.

Common MCAT Mistakes

• Confusing alpha decay's effect on Z and A. Alpha decay decreases the atomic number by 2 and the mass number by 4 (an alpha particle carries away 2 protons and 2 neutrons) — not by 1 and 2, and not the reverse.

• Mixing up beta-minus and beta-plus direction. Beta-minus decay increases Z by 1 (neutron → proton, electron emitted); beta-plus decay decreases Z by 1 (proton → neutron, positron emitted). The mass number never changes in either case.

• Thinking gamma decay changes the identity of the nucleus. Gamma decay releases only energy (a photon) — Z and A are both unchanged, so the parent and "daughter" are the same isotope, just at a lower energy state.

• Assuming half-life means the whole sample decays after two half-lives. Each half-life only removes half of what's currently present — after 2 half-lives, 1/4 remains (not zero); after 3 half-lives, 1/8 remains, and so on.

MCAT-Style Concept Check

Question: A radioactive nucleus undergoes beta-plus (β⁺) decay. What happens to its atomic number (Z) and mass number (A)?

• A) Z increases by 1, A stays the same

• B) Z decreases by 1, A stays the same

• C) Z decreases by 2, A decreases by 4

• D) Z stays the same, A decreases by 1

Answer: B

Explanation: In beta-plus decay, a proton converts into a neutron and the nucleus emits a positron. Since a proton became a neutron, the number of protons (atomic number, Z) decreases by 1, while the total number of nucleons (mass number, A) is unchanged because a proton was simply converted into a neutron rather than lost. Option A describes beta-minus decay, and option C describes alpha decay.

FAQ

What is the difference between nuclear fusion and nuclear fission?

Fusion combines two smaller nuclei into one larger, more stable nucleus, releasing energy — this is how the Sun generates energy. Fission splits one large, unstable nucleus into smaller nuclei, also releasing energy — this is the basis of nuclear reactors and atomic bombs.

How do you know how much Z and A change in each type of radioactive decay?

Alpha decay decreases Z by 2 and A by 4 (loses a helium-4 nucleus). Beta-minus decay increases Z by 1 with A unchanged. Beta-plus decay decreases Z by 1 with A unchanged. Gamma decay changes neither Z nor A. Electron capture decreases Z by 1 with A unchanged.

What is the relationship between half-life and the decay constant?

Half-life (T₁/₂) and the decay constant (λ) are inversely related: T₁/₂ = ln2/λ ≈ 0.693/λ. A larger decay constant means faster decay and therefore a shorter half-life.

Does the amount of a substance affect how long its half-life is?

No. Half-life is a fixed property of a given radioactive isotope, independent of how much material you start with. A 10 g sample and a 10,000 g sample of the same isotope both take the same amount of time for half of the material to decay.