Radioactive Decay and Nuclear Equations for ESAT Physics

The Unstable Nucleus and Decay

Many nuclides occur naturally in our universe. While some are stable and exist indefinitely, others are unstable. An unstable nucleus will eventually undergo a process known as decay, during which it breaks down and emits radiation to become more stable. Nuclides that undergo this process are described as radioactive. An element can have many isotopes, but typically only one or two are stable, with the remainder being radioactive.

A nucleus may decay directly into a stable nuclide, or it may transition through a series of unstable states. This sequence is known as a decay chain, which only terminates when a stable nuclide is finally reached. These decays can involve various types of radiation, such as alpha, beta, and gamma.

The Random Nature of Emissions

Radioactive decay is fundamentally a random process. It is impossible to predict the exact moment a specific nucleus will decay. There is no known trigger or external cause for decay, making it spontaneous. Consequently, the rate of decay cannot be altered by chemical or physical changes, such as raising the temperature or changing the pressure.

Although individual decays are unpredictable, we can determine the probability that a specific type of nucleus will decay over a given period. This is similar to throwing a six sided dice: while you cannot predict which throw will result in a six, the probability for any single throw is always $1/6$. Because of this random nature, count rates measured from radioactive sources will always show some variation. To obtain an accurate reading, scientists take the mean value of the count rate over an extended period.

Worked Example: Understanding Decay Properties

Statement 1: Raising the temperature of a radioactive substance makes it decay faster. Answer: Incorrect. Nuclear decay is spontaneous and unaffected by external physical conditions like temperature.

Statement 2: An unstable nucleus can decay into another unstable nucleus. Answer: Correct. This is the basis of a decay chain.

Statement 3: The isotopes of an element are either all stable or all unstable. Answer: Incorrect. Most elements possess both stable and unstable isotopes.

The Nature of Alpha, Beta, and Gamma Radiation

When decay occurs, the nucleus emits ionising radiation. The three primary types are alpha particles, beta particles, and gamma rays. These processes release significantly more energy than chemical reactions.

Type of radiation	alpha ($\alpha$)	beta ($\beta^-$)	gamma ($\gamma$)
Nuclide Notation	$^4_2\alpha$ or $^4_2\text{He}$	$^0_{-1}\beta$ or $^0_{-1}e$	$^0_0\gamma$
Nature	2 protons and 2 neutrons	Fast moving electron	Electromagnetic radiation
Relative Mass	$4$	$1/2000$ (approx.)	$0$
Relative Charge	$+2$	$-1$	$0$
Speed	approx. $0.1c$	approx. $0.8c$	$c$ ($3 \times 10^8$ ms$^{-1}$)

An alpha particle is identical to a helium nucleus. It is the heaviest type of radiation and carries a double positive charge. A beta particle is a fast moving electron created when a neutron in the nucleus transforms into a proton and an electron. Gamma radiation is high energy electromagnetic waves with no mass or charge, travelling at the speed of light.

Nuclear Equations and the Effect of Decay

The emission of radiation changes the composition of the nucleus. Because alpha and beta decay change the number of protons, they transform the original atom into an atom of a different element.

Alpha Decay

In alpha decay, the nucleus ejects two protons and two neutrons. This causes the atomic number ($Z$) to decrease by $2$ and the mass number ($A$) to decrease by $4$.

General Equation: $^A_Z X \rightarrow ^{A-4}_{Z-2} Y + ^4_2\alpha$

Example: $^{226}_{88}\text{Ra} \rightarrow ^{222}_{86}\text{Rn} + ^4_2\alpha$

Beta Decay

In beta decay, a neutron changes into a proton (which remains in the nucleus) and an electron (which is emitted). The mass number ($A$) remains unchanged, but the atomic number ($Z$) increases by $1$.

General Equation: $^A_Z X \rightarrow ^{A}_{Z+1} Y + ^0_{-1}\beta$

Example: $^{14}_{6}\text{C} \rightarrow ^{14}_{7}\text{N} + ^0_{-1}\beta$

Gamma Decay

Gamma decay involves the release of excess energy. It does not change the number of protons or neutrons, so the atomic and mass numbers remain the same.

Worked Examples: Applying Nuclear Equations

Example 1: Thorium 232 undergoes alpha decay to become an isotope of radium. Complete the equation: $^{232}_{90}\text{Th} \rightarrow ^{A}_{88}\text{Ra} + ^4_2\alpha$ Solution: The mass number must balance: $232 = A + 4$, so $A = 228$. The final equation is $^{232}_{90}\text{Th} \rightarrow ^{228}_{88}\text{Ra} + ^4_2\alpha$.

Example 2: The radium isotope produced above ($^{228}_{88}\text{Ra}$) undergoes beta decay to become actinium (Ac). Write the equation. Solution: In beta decay, the mass number stays $228$. The atomic number increases by $1$, from $88$ to $89$. The equation is $^{228}_{88}\text{Ra} \rightarrow ^{228}_{89}\text{Ac} + ^0_{-1}\beta$.

Example 3: Identify the resulting nucleus of $^A_Z X$ after one alpha and two beta decays. Solution: One alpha decay reduces $A$ by $4$ and $Z$ by $2$, resulting in $^{A-4}_{Z-2} Y$. Two beta decays each increase $Z$ by $1$ and do not change $A$. The net change to $Z$ is $-2 + 1 + 1 = 0$. The resulting nucleus has the notation $^{A-4}_{Z} X$. The element remains the same as the original because the atomic number is unchanged.