Question

Complete and balance the nuclear equations for the following fission reactions: (a) \({ }_{99}^{239} \mathrm{Pu}+{ }_{0} \mathrm{n} \longrightarrow{ }_{52}^{137} \mathrm{Te}+{ }_{42}^{100} \mathrm{Mo}+\) (b) \({ }_{100}^{256} \mathrm{Fm}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{46}^{113} \mathrm{Pd}+{ }_{-}+4{ }_{0}^{1} \mathrm{n}\)

Short answer

The short answer is as follows: (a) \({ }_{99}^{239} \mathrm{Pu}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{52}^{137} \mathrm{Te}+{ }_{42}^{100} \mathrm{Mo}+{ }_{5}^3 \mathrm{B}\) (b) \({ }_{100}^{256} \mathrm{Fm}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{46}^{113} \mathrm{Pd}+{ }_{54}^{140} \mathrm{Xe}+4{ }_{0}^{1} \mathrm{n}\)

Step-by-step solution

Calculate atomic number imbalance

Find the difference between the total atomic numbers on both sides of the equation : Left side atomic numbers: 99 Right side atomic numbers: 52 + 42 = 94 Imbalance: 99 - 94 = 5

Calculate mass number imbalance

Find the difference between the total mass numbers on both sides of the equation : Left side mass numbers: 239 + 1 = 240 Right side mass numbers: 137 + 100 = 237 Imbalance: 240 - 237 = 3

Add the missing particle

We need to add a particle with atomic number 5 and mass number 3 to balance the equation. This particle is an isotope of boron, represented as \({ }_{5}^3 \mathrm{B}\): \({ }_{99}^{239} \mathrm{Pu}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{52}^{137} \mathrm{Te}+{ }_{42}^{100} \mathrm{Mo}+{ }_{5}^3 \mathrm{B}\) #b)# For the fission reaction: \({ }_{100}^{256} \mathrm{Fm}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{46}^{113} \mathrm{Pd}+{ }_{-}+4{ }_{0}^{1} \mathrm{n}\)

Calculate atomic number imbalance

Find the difference between the total atomic numbers on both sides of the equation : Left side atomic numbers: 100 Right side atomic numbers: 46 + (-) Imbalance: 100 - 46 = 54 (not counting the unknown element)

Calculate mass number imbalance

Find the difference between the total mass numbers on both sides of the equation : Left side mass numbers: 256 + 1 = 257 Right side mass numbers: 113 + 4 * 1 = 117 (not counting the unknown element) Imbalance: 257 - 117 = 140

Add the missing particle

We need to add a particle with atomic number 54 and mass number 140 to balance the equation. This particle is an isotope of xenon, represented as \({ }_{54}^{140} \mathrm{Xe}\): \({ }_{100}^{256} \mathrm{Fm}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{46}^{113} \mathrm{Pd}+{ }_{54}^{140} \mathrm{Xe}+4{ }_{0}^{1} \mathrm{n}\)

Key concepts

Fission Reactions

Nuclear fission reactions are fascinating processes where a large nucleus splits into smaller ones. This process is essential for nuclear power generation and atomic weapons. In a fission reaction, a heavy nucleus, like uranium or plutonium, absorbs a neutron. This causes the nucleus to become unstable and eventually break apart into two or more smaller nuclei, along with additional free neutrons and a large amount of energy.
Each newly formed neutron can go on to induce further fission reactions, creating a chain reaction.

• Fission significantly releases energy due to the conversion of mass into energy according to Einstein's famous equation, \( E = mc^2\).
• The nuclei produced from fission are typically more stable but have different mass numbers and atomic numbers compared to the original nucleus.

Fission reactions need to be carefully controlled; otherwise, they can lead to catastrophic outcomes. In practical applications, moderating materials like water are often used to control the speed of the reaction by managing neutron activity.

Atomic Number Imbalance

When balancing nuclear equations for fission, it’s crucial to ensure that atomic numbers are balanced on both sides. The atomic number of an element tells us how many protons are in the nucleus, defining what element it is.
The atomic number must remain consistent to maintain the identity of the involved elements. Here's an easy way to think about it:

• The left and right sides of the equation should have the same total atomic number.
• In the given exercise, you compare the starting atomic numbers with those of the products.

For example, in equation (a): - Left side has an atomic number of 99 (\(_{99}^{239} \mathrm{Pu}\))- The known products on the right sum to an atomic number of 94 (\(_{52}^{137} \mathrm{Te} + { }_{42}^{100} \mathrm{Mo}\))This shows an imbalance, indicating the need for additional particles, such as isotopes, to restore balance. Adding the correct isotope ensures the sum of atomic numbers on both sides matches.

Mass Number Imbalance

Alongside atomic number balance, mass number balance is crucial in nuclear equations. The mass number is the sum of protons and neutrons in the nucleus, giving the overall mass of an atomic nucleus. It's crucial in maintaining conservation of mass within the nuclear reaction.
Let's break this down for clarity:

• The mass on the left (reactants) and the right (products) of the equation must sum equally, including the any emitted neutrons or missing isotopes.
• Check the mass numbers for both sides in the given fission reactions.

For instance, in reaction (a) from the exercise:- The left side mass number is 240 (accounts for \(_{99}^{239} \mathrm{Pu}\) and one neutron \(_{0}^{1} \mathrm{n}\))- The right side totals up to only 237 with the identified products (\(_{52}^{137} \mathrm{Te} + { }_{42}^{100} \mathrm{Mo}\))To correct this imbalance, an isotope is added with a mass number of 3 (like \({_5}^{3} \mathrm{B}\)) ensuring both sides of the equation reflect the same mass. Proper balancing of mass numbers provides accurate reflection of the conservation laws in nuclear physics.