Question

Write a balanced nuclear equation for the reaction of argon- 40 with a proton: $$ { }_{18}^{40} \mathrm{Ar}+{ }_{1}^{1} \mathrm{H} \longrightarrow \mathrm{P}+{ }_{0}^{1} \mathrm{n} $$

Short answer

The balanced nuclear equation is \({ }_{18}^{40} \mathrm{Ar} + { }_{1}^{1} \mathrm{H} \rightarrow { }_{19}^{40} \mathrm{K} + { }_{0}^{1} \mathrm{n}\).

Step-by-step solution

Identify Initial Reactants

The reactants are a nucleus of Argon-40, represented as \({ }_{18}^{40} \mathrm{Ar}\), and a proton, represented as \({ }_{1}^{1} \mathrm{H}\).

Analyze Given Reaction and Identify Unknown Particle

The reaction given is \({ }_{18}^{40} \mathrm{Ar} + { }_{1}^{1} \mathrm{H} \rightarrow \mathrm{P} + {}_{0}^{1} \mathrm{n}\). We need to determine the identity of \(\mathrm{P}\). The possible reaction suggests a neutron \({}_{0}^{1}\mathrm{n}\) is produced alongside the unknown particle \(\mathrm{P}\).

Apply Conservation of Mass Number and Atomic Number

According to the law of conservation, the total mass number and atomic number before and after the reaction must be equal. The initial mass number is \(40 + 1 = 41\), and the atomic number is \(18 + 1 = 19\). After emitting a neutron \({ }_{0}^{1} \mathrm{n}\), mass number changes to \(41 - 1 =40\), and the atomic number remains \(19\) as \(\mathrm{n}\) doesn't affect it.

Determine Identity of Particle P by Balancing Equation

The particle \(\mathrm{P}\) should have a mass number of 40 and an atomic number of 19 to balance both mass number and atomic number after reaction. This corresponds to an isotope of potassium, expressed as \({ }_{19}^{40} \mathrm{K}\).

Write the Balanced Nuclear Equation

The balanced nuclear reaction considering the identities found is \({ }_{18}^{40} \mathrm{Ar} + { }_{1}^{1} \mathrm{H} \rightarrow { }_{19}^{40} \mathrm{K} + { }_{0}^{1} \mathrm{n}\).

Key concepts

Balanced Reactions

A balanced reaction ensures that the number of each type of atom and total charge remains constant before and after a reaction. In nuclear equations, this means the sum of the mass numbers and the atomic numbers should be equal on both sides of the equation.
For example, in the reaction \({ }^{40}_{18} \text{Ar} + { }^{1}_{1} \text{H} \to { }^{40}_{19} \text{K} + { }^{1}_{0} \text{n}\):

• The sum of mass numbers on the left is \(40 + 1 = 41\).
• This should match the sum of mass numbers on the right \((40 + 1 = 41)\).
• The atomic numbers also balance: \(18 + 1 = 19\) on the left and \(19 + 0 = 19\) on the right.

It's crucial in nuclear chemistry that every reaction remains balanced to follow the laws of physics, just as we balance chemical reactions to ensure mass conservation.

Conservation of Mass Number

The conservation of mass number is a principle stating that the total mass number in a nuclear equation must remain the same before and after the reaction. This is a cornerstone of nuclear reactions.
In the given nuclear equation, we begin with a total mass number of 41 (from Argon-40 and a proton).
After the reaction, we have Potassium-40 and a neutron, which together also add up to 41. To verify:

• The mass number for Argon is 40.
• The proton contributes a mass number of 1.
• This totals to \(40 + 1 = 41\).
• The resulting Potassium has a mass number of 40, and the neutron has a mass number of 1, summing to \(40 + 1 = 41\).

In nuclear reactions, mass number conservation verifies that no "mass" is lost or gained, simply transformed.

Atomic Number

The atomic number is a fundamental concept in chemistry and nuclear reactions, representing the number of protons in an atom's nucleus. This number defines the element's identity and dictates its position in the periodic table.
When working with nuclear equations, maintaining atomic number conservation is key to identifying elements before and after reactions.
For example, in the reaction:

• Argon starts with an atomic number of 18.
• The proton introduced has an atomic number of 1.
• Total atomic number before the reaction is \(18 + 1 = 19\).
• After the reaction occurs, Potassium has an atomic number of 19, matching the sum of the reactants.

The neutron, having an atomic number of zero, does not affect this count. By ensuring atomic numbers are balanced, we can correctly identify the resulting products.

Isotopes

Isotopes are variations of a chemical element with the same atomic number but different mass numbers due to differing numbers of neutrons. Understanding isotopes is critical when studying nuclear reactions.
In the balanced reaction we examined, Argon-40 and Potassium-40 are isotopes of different elements because they have different atomic numbers but the same mass number (40).
To identify an isotope:

• Look at the atomic number to determine the element.
• Compare the mass numbers to see how many neutrons differ.
• Potassium-40 (\({ }^{40}_{19} \text{K}\)) has one more proton than Argon (\({ }^{40}_{18} \text{Ar}\)), indicating the change from one isotope to another due to reactions.

This change highlights the dynamic nature of isotopes and their roles in nuclear chemistry, as they transform through various reactions while still conserving atomic identities.