Question

When vanadium- 51 captures a deuteron \(\left({ }_{1}^{2} \mathrm{H}\right),\) what compound nucleus forms? (Write its symbol.) This particle expels a proton \(\left({ }_{1}^{1} p\right) .\) Write the balanced nuclear equation for the overall change from vanadium- 51 .

Short answer

The compound nucleus formed is \({ }_{24}^{53} \text{Cr}\), and the balanced nuclear equation for the overall change is \({ }_{23}^{51} \text{V} + { }_{1}^{2} \text{H} \to { }_{23}^{52} \text{V} + { }_{1}^{1} \text{p}\).

Step-by-step solution

Identifying the Initial Nucleus and Incoming Particle

First, identify the initial nucleus and the incoming particle. The initial nucleus is vanadium-51, which has the symbol \({ }_{23}^{51} \text{V}\), and the incoming particle is a deuteron, represented by \({ }_{1}^{2} \text{H}\). The deuteron consists of one proton and one neutron.

Determining the Compound Nucleus

Combine the atomic number and mass number of vanadium-51 with those of the deuteron to form the new compound nucleus. Atomic numbers add up to \(23 + 1 = 24\), which corresponds to the element chromium (Cr). The mass numbers add up to \(51 + 2 = 53\). Thus, the compound nucleus formed is \({ }_{24}^{53} \text{Cr}\).

Accounting for the Ejected Particle

Following the formation of the compound nucleus, a proton is ejected. The proton has an atomic number of 1 and a mass number of 1. Subtracting this from the compound nucleus gives us the resulting nucleus: \(24 - 1 = 23\) for the atomic number, which is again vanadium (V), and \(53 - 1 = 52\) for the mass number. The resulting nucleus is \({ }_{23}^{52} \text{V}\).

Writing the Balanced Nuclear Equation

Combine the steps above to write the balanced nuclear equation for the process. The initial vanadium-51 nucleus and deuteron combine to form the intermediate chromium-53 nucleus, which then expels a proton to become vanadium-52. The balanced equation is: \({ }_{23}^{51} \text{V} + { }_{1}^{2} \text{H} \to { }_{24}^{53} \text{Cr} \to { }_{23}^{52} \text{V} + { }_{1}^{1} \text{p}\).

Key concepts

Nuclear Reactions

Nuclear reactions involve changes in an atom's nucleus and often result in the transformation of elements. Unlike chemical reactions that involve electrons and the formation of molecules, nuclear reactions deal with protons and neutrons, known as nucleons, and can lead to the release or absorption of enormous amounts of energy.

During a nuclear reaction, a nucleus might capture a particle, emit new particles, or even split into two or more smaller nuclei. The exercise provided showcases a type of nuclear reaction where a nucleus captures a deuteron. Deuteron is a name for a nucleus of deuterium, which consists of one proton and one neutron. After this capture, another particle, a proton in this case, is emitted. These sequential steps change the original nucleus into a different element or isotope, showing the transmutative power of nuclear reactions.

Nuclear reactions are guided by several conservation laws, such as the conservation of mass-energy and the conservation of charge. These principles ensure that, despite the changes within the nucleus, certain properties remain constant.

Chemical Symbols

In nuclear chemistry, it's crucial to understand chemical symbols, which succinctly convey a wealth of information about an atom or isotope. Each chemical symbol consists of one or two letters and often includes numbers that indicate the atomic number and mass number of the element or isotope. The atomic number, located as a subscript before the symbol, tells us the number of protons in the nucleus and defines the type of element. The mass number, shown as a superscript before the symbol, gives the total count of protons and neutrons.

For instance, the chemical symbol \( { }_{23}^{51} \text{V} \) represents an isotope of vanadium with an atomic number of 23 (indicating it has 23 protons) and a mass number of 51 (sum of its protons and neutrons). A correct interpretation of these symbols is fundamental for understanding nuclear reactions, as the symbols are used to represent the changes occurring within a nucleus.

Balanced Nuclear Equations

Balancing nuclear equations is akin to balancing chemical equations, but instead of dealing with atoms and molecules, we're focusing on isotopes and subatomic particles. A balanced nuclear equation respects the laws of conservation of mass number and atomic number, ensuring that both sides of the equation have equal totals for these quantities.

The exercise demonstrated the process of balancing a nuclear equation step by step. The initial capture of a deuteron by a vanadium-51 nucleus resulted in the formation of a new compound nucleus which subsequently emitted a proton. In the final balanced nuclear equation \( { }_{23}^{51} \text{V} + { }_{1}^{2} \text{H} \to { }_{24}^{53} \text{Cr} \to { }_{23}^{52} \text{V} + { }_{1}^{1} \text{p} \) you can verify the balance by adding up the atomic numbers and mass numbers on each side of the equation. This skill is crucial for anyone studying nuclear chemistry, as it enables them to predict the products of nuclear reactions and understand the subtleties involved in the transmutation of elements.