Balanced Nuclear Equations
The Building Blocks of an Atom
Before we can balance nuclear equations, we need to understand the particles involved. An atom is made up of a tiny, dense nucleus surrounded by a cloud of electrons. The nucleus itself is composed of two types of particles:
- Protons (p): Positively charged particles. The number of protons defines the element. For example, any atom with 6 protons is a carbon atom.
- Neutrons (n): Neutral particles with no electric charge. They help hold the nucleus together.
Together, protons and neutrons are called nucleons. The total number of nucleons is the mass number, represented by the symbol $ A $. The number of protons is the atomic number, represented by $ Z $.
We represent an element $ X $ with its atomic and mass numbers as: $ ^{A}_{Z}X $.
For instance, a common form of carbon, Carbon-12, is written as $ ^{12}_{6}C $. This tells us it has 6 protons (atomic number 6) and 6 neutrons (because 12 total nucleons - 6 protons = 6 neutrons).
What is a Nuclear Reaction?
In a chemical reaction, atoms rearrange their electrons to form new molecules, but the atoms themselves remain the same. A nuclear reaction is different. It involves changes within the nucleus of an atom, transforming one element into another! This process is called transmutation.
These reactions release or absorb enormous amounts of energy, much more than chemical reactions. They are the power source for stars like our Sun and are used in nuclear power plants.
The Golden Rule: Conservation of Nucleon Number
The most important rule for balancing nuclear equations is the conservation of nucleon number. This law states that the total number of nucleons (protons + neutrons) cannot change in a nuclear reaction. They can be rearranged, but not created or destroyed.
Think of it like building blocks. If you start with 10 blocks, you must end with 10 blocks. You can build a different shape (a new element), but you can't have 9 or 11 blocks at the end.
Common Particles in Nuclear Reactions
To write and balance equations, you need to know the symbols for common particles. Here is a handy reference table:
| Particle Name | Symbol | Mass Number (A) | Atomic Number (Z) | Description |
|---|---|---|---|---|
| Alpha Particle | $ ^{4}_{2}\alpha $ or $ ^{4}_{2}He $ | 4 | 2 | A helium nucleus (2 protons, 2 neutrons) |
| Beta Particle | $ ^{0}_{-1}\beta $ or $ ^{0}_{-1}e $ | 0 | -1 | A high-speed electron emitted from the nucleus |
| Positron | $ ^{0}_{+1}\beta $ or $ ^{0}_{+1}e $ | 0 | +1 | The antimatter counterpart of an electron |
| Gamma Ray | $ ^{0}_{0}\gamma $ | 0 | 0 | High-energy electromagnetic radiation (a photon) |
| Neutron | $ ^{1}_{0}n $ | 1 | 0 | A neutral nucleon |
| Proton | $ ^{1}_{1}p $ or $ ^{1}_{1}H $ | 1 | 1 | A positively charged nucleon |
Step-by-Step Guide to Balancing Nuclear Equations
Let's learn how to balance a nuclear equation using a step-by-step approach. We'll use the alpha decay of Radium-226 as our example.
Step 1: Write the known part of the equation.
We start with Radium-226 on the left. It emits an alpha particle, so we put that on the right. We don't know the identity of the other product yet, so we call it $ ^{A}_{Z}X $.
$ ^{226}_{88}Ra \rightarrow ^{A}_{Z}X + ^{4}_{2}\alpha $
Step 2: Balance the Mass Numbers (A).
The sum of mass numbers on the left must equal the sum on the right.
$ 226 = A + 4 $
Solving for A: $ A = 226 - 4 = 222 $
Step 3: Balance the Atomic Numbers (Z).
The sum of atomic numbers on the left must equal the sum on the right.
$ 88 = Z + 2 $
Solving for Z: $ Z = 88 - 2 = 86 $
Step 4: Identify the Unknown Element.
We now know the new nucleus is $ ^{222}_{86}X $. We use the periodic table to find which element has an atomic number of 86. That element is Radon (Rn).
Step 5: Write the Complete Balanced Equation.
$ ^{226}_{88}Ra \rightarrow ^{222}_{86}Rn + ^{4}_{2}\alpha $
Let's verify: Left side: A=226, Z=88. Right side: A=222+4=226, Z=86+2=88. It's balanced!
A Closer Look at Different Decay Types
Now that we know the method, let's apply it to other common types of radioactive decay.
Beta Decay (β⁻)
In beta decay, a neutron inside the nucleus transforms into a proton and emits an electron (the beta particle). This means the mass number stays the same, but the atomic number increases by 1.
Example: The beta decay of Carbon-14 into Nitrogen-14.
$ ^{14}_{6}C \rightarrow ^{14}_{7}N + ^{0}_{-1}\beta $
Check: Left: A=14, Z=6. Right: A=14+0=14, Z=7+(-1)=6. Balanced!
Positron Emission (β⁺)
This is the opposite of beta decay. A proton transforms into a neutron and emits a positron. The mass number stays the same, but the atomic number decreases by 1.
Example: A proton-rich isotope of carbon decaying to Boron-11.
$ ^{11}_{6}C \rightarrow ^{11}_{5}B + ^{0}_{+1}\beta $
Check: Left: A=11, Z=6. Right: A=11+0=11, Z=5+1=6. Balanced!
Gamma Emission (γ)
Gamma decay is different. It doesn't change the composition of the nucleus (no change in A or Z). It only releases excess energy. Think of it as an excited nucleus "calming down" by emitting a high-energy photon.
Example: Technetium-99m is a metastable (excited) isotope used in medical imaging. It relaxes to its ground state by emitting a gamma ray.
$ ^{99m}_{43}Tc \rightarrow ^{99}_{43}Tc + ^{0}_{0}\gamma $
Check: Left: A=99, Z=43. Right: A=99+0=99, Z=43+0=43. Balanced! (The 'm' just denotes the excited state).
Nuclear Reactions in Action: Fission and Fusion
Balanced equations are also essential for understanding the two most famous nuclear processes: fission and fusion.
Nuclear Fission
Fission is the splitting of a heavy nucleus into lighter fragments. This is the process used in nuclear power plants. A common example is the fission of Uranium-235 when it absorbs a neutron.
Example Equation:
$ ^{235}_{92}U + ^{1}_{0}n \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3^{1}_{0}n $
Check Balance:
Mass Numbers (A): 235 + 1 = 236 on left. 141 + 92 + (3×1) = 236 on right.
Atomic Numbers (Z): 92 + 0 = 92 on left. 56 + 36 + (3×0) = 92 on right.
Notice that the single neutron on the left causes the fission and releases three neutrons on the right. These can go on to cause more fissions, creating a chain reaction.
Nuclear Fusion
Fusion is the combining of light nuclei to form a heavier nucleus. This process powers the Sun and other stars.
Example Equation (Proton-Proton Chain in the Sun):
One step in this chain is the fusion of two protons to form a deuterium nucleus.
$ ^{1}_{1}H + ^{1}_{1}H \rightarrow ^{2}_{1}H + ^{0}_{+1}\beta + \text{energy} $
Check Balance:
Mass Numbers (A): 1 + 1 = 2 on left. 2 + 0 + 0 = 2 on right.
Atomic Numbers (Z): 1 + 1 = 2 on left. 1 + 1 + 0 = 2 on right.
This reaction converts two protons into a deuterium nucleus, a positron, and releases a tremendous amount of energy.
Common Mistakes and Important Questions
Q: I balanced the mass number and atomic number, but my teacher said the equation is still wrong. Why?
A: A common mistake is forgetting to look up the element symbol based on the final atomic number (Z). For example, if you calculate Z=82, the product is Lead (Pb), not just "X". Also, ensure you are using the correct symbols for particles (e.g., $ \alpha $ for an alpha particle, not $ He $ unless specified).
Q: In beta decay, where does the electron come from? I thought electrons were only in the electron cloud, not the nucleus.
A: This is a great question! The electron (beta particle) is not just sitting in the nucleus waiting to be emitted. It is created at the moment of decay. Inside the nucleus, a neutron transforms into a proton and in the process creates and emits an electron and an antineutrino (a nearly massless particle we often don't write in basic equations). So, the electron is created from the energy and particles within the nucleus.
Q: Why is the mass number of a beta particle 0? Doesn't an electron have mass?
A: Yes, an electron does have a very small mass. However, in nuclear physics, the mass number (A) represents the number of protons and neutrons, which are about 1836 times more massive than an electron. The mass of an electron is so small compared to a nucleon that we round its mass number down to 0 to simplify the calculations and focus on the conservation of nucleons. This is a very good approximation for balancing equations.
Footnote
1 Nucleon: A collective term for a proton or a neutron, the particles found in an atomic nucleus.
2 Transmutation: The conversion of one chemical element into another through a nuclear reaction.
3 Radioactive Decay: The spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation.
4 Fission: A nuclear reaction in which a heavy nucleus splits into two or more lighter nuclei.
5 Fusion: A nuclear reaction in which two or more light atomic nuclei combine to form a heavier nucleus.