Question

Which of the following notations shows the product incorrectly? (a) \({ }_{5} \mathrm{~B}^{10}(\alpha, \mathrm{n}){ }_{7} \mathrm{~N}^{13}\) (b) \({ }_{96} \mathrm{Cm}^{242}(\alpha, 2 \mathrm{n}){ }_{97} \mathrm{BK}^{243}\) (c) \({ }_{7} \mathrm{~N}^{14}(\mathrm{n}, \mathrm{p}){ }_{6} \mathrm{C}^{14}\) (d) none of these

Short answer

Option (a) shows the product incorrectly.

Step-by-step solution

Understanding the Notations

The given notations represent nuclear reactions where the reactant, particle involved in the reaction, and the final product are shown in a particular format. The format follows: \[ _{Z_1}^{A_1}E_1(x,y) _{Z_2}^{A_2}E_2 \]where \(E_1\) and \(E_2\) are elements with atomic number \(Z\) and mass number \(A\), \(x\) is the incident particle, and \(y\) is the emitted particle.

Calculating Each Reaction's Product

We will check if the provided notation respects the conservation of both atomic number and mass number in nuclear reactions.- (a)\: \( _{5}B^{10}(\alpha, n)_{7}N^{13} \): - \(\alpha\) (alpha particle) is \(_{2}^{4}He\) - Reaction: \( _5^{10}B + _2^4He \rightarrow _7^{13}N + _0^1n \) - Mass: \(10 + 4 eq 13 + 1\); Atomic: \(5 + 2 eq 7 + 0\) - Both balances are incorrect.- (b)\: \( _{96}Cm^{242}(\alpha, 2n)_{97}Bk^{243} \): - Reaction: \( _{96}^{242}Cm + _2^4He \rightarrow _{97}^{243}Bk + 2 _0^1n \) - Mass: \(242 + 4 = 243 + 1\); Atomic: \(96 + 2 = 97\) - Both balances are correct.- (c)\: \( _{7}N^{14}(n, p)_{6}C^{14} \): - \(n\) (neutron) is \(_{0}^{1}n\) - \(p\) (proton) is \(_{1}^{1}p\) - Reaction: \(_7^{14}N + _0^1n \rightarrow _6^{14}C + _1^1p\) - Mass: \(14 + 1 = 14 + 1\); Atomic: \(7 = 6 + 1\) - Both balances are correct.

Identifying the Incorrect Product Notation

From the calculations: - Option (a) does not conserve the atomic number and mass number, therefore the product notation is incorrect. - Options (b) and (c) follow the rules of nuclear reaction as they conserve atomic and mass numbers.

Conclusion

The incorrect notation is shown in option (a). Therefore, the product is incorrectly depicted in this notation.

Key concepts

Conservation of Atomic Number

In nuclear reactions, one of the key principles is the conservation of atomic number. This rule states that the sum of atomic numbers on the reactant side must equal the sum on the product side.
This means that during any nuclear reaction, the number of protons before and after the reaction remains unchanged. The atomic number, denoted by the letter Z, indicates the total number of protons in an atom's nucleus.
To understand this better, let's consider the notation for nuclear reactions:

• If the notation is \(_{Z_1}^{A_1}E_1(x,y)_{Z_2}^{A_2}E_2\), then \(Z_1 + Z_x = Z_2 + Z_y\)
• Here, \(E_1\) and \(E_2\) are elements with atomic numbers \(Z_1\) and \(Z_2\), \(x\) is the incident particle with atomic number \(Z_x\), and \(y\) is the emitted particle with atomic number \(Z_y\).

This principle is essential for verifying the correctness of a nuclear reaction notation. For example, in option (a) from the exercise, the atomic number does not balance because the sum of atomic numbers on the left (5 + 2) does not equal that on the right (7 + 0).

Conservation of Mass Number

Another fundamental principle in nuclear reactions is the conservation of mass number. This principle asserts that the total mass number on the reactant side must equal the total on the product side.
The mass number, denoted by the letter A, is the sum of protons and neutrons in an atom's nucleus. It's a crucial part of ensuring that nuclear reactions are balanced and scientifically accurate.
For example, in nuclear reaction notations, the following should hold true:

• If expressed as \(_{Z_1}^{A_1}E_1(x,y)_{Z_2}^{A_2}E_2\), then \(A_1 + A_x = A_2 + A_y\)
• Here, \(E_1\) and \(E_2\) are elements with mass numbers \(A_1\) and \(A_2\), \(x\) is the incident particle with mass number \(A_x\), and \(y\) is the emitted particle with mass number \(A_y\).

In the given exercise, option (a) shows an incorrect mass number balance: \(10 + 4\) does not equal \(13 + 1\). This illustrates the importance of checking both atomic and mass numbers to verify the correctness of nuclear reaction notations.

Nuclear Reaction Notations

Understanding nuclear reaction notations is vital when working with nuclear reactions. These notations are a way to represent the changes happening at a nuclear level when particles are exchanged in reactions.
A standard notation can be defined as follows: \(_{Z_1}^{A_1}E_1(x,y)_{Z_2}^{A_2}E_2\). This format tells us:

• \(E_1\) is the initial element, having \(Z_1\) as its atomic number and \(A_1\) as its mass number.
• \(x\) is the incident particle involved in the reaction, like an alpha particle \(\alpha\), a neutron \(n\), etc.
• \(y\) is the particle emitted as a result of the reaction.
• \(E_2\) is the resulting element, with \(Z_2\) as its atomic number and \(A_2\) as its mass number.

Correctly reading and writing these notations ensures accurate representation of nuclear changes. In the provided exercise, ensuring the notation follows the rules for atomic and mass number conservation highlights errors. In option (a), both are not conserved, which helps us identify its incorrectness.