Question

The following equations are for nuclear reactions that are known to occur in the explosion of an atomic bomb. Identify X. (a) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{56}^{140} \mathrm{Ba}+3{ }_{0}^{1} \mathrm{n}+\mathrm{X}\) (b) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{55}^{144} \mathrm{Cs}+{ }_{37}^{90} \mathrm{Rb}+2 \mathrm{X}\) (c) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{35}^{87} \mathrm{Br}+3{ }_{0}^{1} \mathrm{n}+\mathrm{X}\) (d) \({ }_{92}^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{6}^{160} \mathrm{Sm}+{ }_{30}^{72} \mathrm{Zn}+4 \mathrm{X}\)

Short answer

(a) \(^{92}_{36}\text{Kr}\), (b) 2 neutrons, (c) \(^{146}_{57}\text{La}\), (d) 4 neutrons.

Step-by-step solution

Set Up the Nuclear Reaction for Equation (a)

For equation (a), the initial nuclide is Uranium-235, which captures a neutron to form the final products.Initial mass and charge equations are:\[^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{140}_{56}\text{Ba} + 3 \, ^{1}_{0}\text{n} + X\]By conserving atomic number (Z):\[92 = 56 + 0 + Z_X\]\[Z_X = 36\]By conserving mass number (A):\[236 = 140 + 3(1) + A_X\]\[A_X = 92\]This results in Krypton-92, \(^{92}_{36}\text{Kr}\), for X.

Determine X for Equation (b)

For equation (b), balance the same components as done previously:Initial mass and charge equations are:\[^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{144}_{55}\text{Cs} + ^{90}_{37}\text{Rb} + 2X\]Conserve atomic number (Z):\[92 = 55 + 37 + 2Z_X\]\[2Z_X = 0\]By conserving mass number (A):\[236 = 144 + 90 + 2A_X\]\[2A_X = 2\]\( Z_X = 0 \) and \( A_X = 1 \), so X are two neutrons, \(2 \, ^{1}_{0}\text{n}\).

Handle the Reaction for Equation (c)

For equation (c), apply similar balance rules:Initial mass and charge equations are:\[^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{87}_{35}\text{Br} + 3 \, ^{1}_{0}\text{n} + X\]Conserve atomic number (Z):\[92 = 35 + 0 + Z_X\]\[Z_X = 57\]Conserve mass number (A):\[236 = 87 + 3(1) + A_X\]\[A_X = 146\]The X in this reaction is Lanthanum-146, \(^{146}_{57}\text{La}\).

Balance Equation (d)

Now for equation (d):Initial mass and charge equations are:\[^{235}_{92}\text{U} + ^{1}_{0}\text{n} \rightarrow ^{160}_{62}\text{Sm} + ^{72}_{30}\text{Zn} + 4X\]Conserve atomic number (Z):\[92 = 62 + 30 + 4Z_X\]\[Z_X = 0\] (Each X must be a neutron)Conserve mass number (A):\[236 = 160 + 72 + 4A_X\]\[4A_X = 4\]All X are again found to be neutrons, \(4 \, ^{1}_{0}\text{n}\).

Key concepts

Atomic Number Conservation

Atomic number conservation is a key principle in nuclear reactions. It states that the total atomic number, which is the number of protons in an atom, remains constant during the reaction. This means the sum of atomic numbers of the reactants should equal the sum of atomic numbers of the products.
When dealing with nuclear reactions, such as in the given equations involving Uranium-235, ensuring atomic number conservation is crucial. For example, if you start with Uranium-235, which has an atomic number of 92, you must end with products whose total atomic numbers also sum up to 92.

• In equation (a), Uranium (92) plus a neutron (0) results in products with a combined atomic number equal to 92.
• This balancing allows identification of unknown products or missing particles.

Additionally, maintaining this balance is vital in nuclear fission and other nuclear processes that involve splitting atoms, ensuring no protons 'disappear' from the reaction.

Mass Number Conservation

Mass number conservation is another essential principle in nuclear chemistry. It asserts that the total mass number (a combination of protons and neutrons) is conserved throughout the nuclear reaction. Therefore, the sum of mass numbers of all reactant particles must match the sum of mass numbers of the product particles.
Consider the reaction equations with Uranium-235 as an example. Uranium-235 absorbs a neutron, increasing its mass number to 236, and this new total is then distributed among the resulting products.

• Equation (a) demonstrates mass number conservation, where Uranium-235 plus a neutron balances the mass numbers of the resultant isotopes, including unknown element X.
• Each reaction step involves careful calculation to ensure all products align with the initial summed mass number.

Maintaining mass number conservation helps confirm the identity of resultant particles, such as new isotopes or neutrons, in nuclear reactions.

Nuclear Fission

Nuclear fission is a reaction in which a heavy nucleus, like Uranium-235, splits into two or more smaller nuclei, along with the release of energy and additional neutrons. This process is fundamental in nuclear power generation and atomic bombs.
During fission, Uranium-235 absorbs a neutron, becomes unstable, and then divides into lighter elements and releases a significant amount of energy. This is accompanied by the emission of additional neutrons, which can further induce fission in nearby nuclei, potentially leading to a chain reaction.

• In the given nuclear reactions, Uranium-235 undergoes fission, resulting in various isotopes of elements like Barium, Cesium, and Bromine, depending on the initial arrangement.
• These products vary because different fission paths and energy levels can alter end results.

Understanding nuclear fission allows for control in reactors and calculating materials in weaponry.

Uranium-235

Uranium-235 is a crucial element in nuclear science due to its capability to sustain chain reactions. It is a fissile isotope of uranium, meaning it can easily split and cause further splitting (fission) of other uranium nuclei when it absorbs a neutron.
Uranium-235's abundance in nature is low, around 0.7%, but it is key for nuclear energy and weapons because of its remarkable fission capabilities.

• When Uranium-235 undergoes fission, as in the presented equations, it leads to the formation of various products from its decay.
• Each fission event releases about 200 MeV of energy, which is harnessed in nuclear power plants.

The ability of Uranium-235 to undergo fission and generate energy makes it not only a valuable resource but also requires careful handling and regulation.