Question

Element 99 was first synthesized by bombarding uranium- 238 with neutrons. The reaction yielded Es- 253 and seven beta particles. How many neutrons are necessary to balance the equation?

Short answer

15 neutrons are necessary.

Step-by-step solution

Understanding the Reactants and Products

In this nuclear reaction, uranium-238 (\(\text{U}_{92}^{238}\)) is bombarded with neutrons (\(n\)\(\_{}^1\)) to produce einsteinium-253 (\(\text{Es}_{99}^{253}\)) and beta particles (\(\beta\) where \(\beta_{-1}^{0}\)). We need to determine the number of neutrons involved in this reaction that satisfy both mass number and atomic number conservation.

Setting Up the Conservation Equations

Using the law of conservation of mass and atomic numbers, we have two equations to satisfy for balancing a nuclear reaction. For mass numbers: \(238 + x = 253 + 0 \times 7\) (where \(x\) is the number of neutrons). For atomic numbers: \(92 + 0 = 99 + (-1) \times 7\).

Solving Mass Number Equation

The mass number equation is \(238 + x = 253\). Solving for \(x\) gives \(x = 253 - 238 = 15\). This means 15 neutrons are needed.

Verifying Atomic Number Conservation

For atomic numbers: \(92 = 99 + (-1) \times 7\) simplifies to \(92 = 99 - 7\) or \(92 = 92\), confirming the balance. This verifies that the initial number of atomic numbers is conserved.

Key concepts

Mass Number Conservation

In nuclear reactions, mass number conservation is a crucial principle. It states that the total mass number—the sum of protons and neutrons—in an atomic nucleus must be balanced between the reactants and the products.
When we consider a nuclear reaction like the one where uranium-238 is bombarded with neutrons, it results in the formation of a new element, einsteinium-253, among other products. To ensure the reaction is balanced, we use the equation:

• For reactants: Mass number of Uranium + Neutrons = 238 + x
• For products: Mass number of Einsteinium + Beta Particles = 253 + 0

Solving the equation:

• \[ 238 + x = 253 \]
• Therefore, \( x = 253 - 238 = 15 \).

This tells us 15 neutrons are necessary to balance the mass number on both sides of the equation. Mass number conservation ensures that the reactants' and products' total nuclei match, thus maintaining the same total before and after the reaction.

Atomic Number Conservation

Apart from mass number conservation, atomic number conservation is another fundamental law in nuclear reactions. This principle ensures that the total number of protons is the same before and after the reaction.
In the given example with uranium-238 transforming to einsteinium-253 and beta particles, the atomic numbers must also balance. The atomic number for each element is essentially the number of protons, which critically defines the element:

• Uranium, \( ext{U}_{92}^{238} \), starts with 92 protons.
• Einsteinium, \( ext{Es}_{99}^{253} \), ends with 99 protons.

Let's analyze the reaction:

• For reactants: 92 protons from Uranium + 0 from Neutrons = 92
• For products: 99 protons from Einsteinium + 7 changes due to beta particles

This leads to the equation:

• \[ 92 = 99 + (-1 imes 7) \]
• Simplifying the equation \( 92 = 99 - 7 \)
• This confirms
  \( 92 = 92 \).

Atomic number conservation assures us that despite the exchange and transformation among particles, the sum of atomic numbers remains unchanged throughout the reaction.

Neutron Bombardment

Neutron bombardment is a common technique to initiate nuclear reactions. It involves shooting neutrons at a target nucleus to create new elements or isotopes. Neutrons are particularly effective because they are neutral, allowing them to penetrate nuclei more easily than charged particles.
In our scenario, uranium-238 is bombarded with neutrons to produce einsteinium-253. Here's why neutrons are crucial:

• Neutrality: Neutrons aren't repelled by the positive charge of the nucleus, so they can easily enter and instigate reactions.
• Effective Impact: The bombardment increases the nucleus size, making it unstable and more likely to transform into a new element.

When uranium-238 is bombarded with enough neutrons—15 in this case— the nucleus becomes unstable, leading to the eventual formation of Einsteinium and release of beta particles. This process underlines the significance of neutron bombardment in synthesizing new elements and understanding nuclear reactions.