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

The incorrect notation is (b).

Step-by-step solution

Understand the Given Notations

Each notation represents a nuclear reaction. It follows the pattern: Target (Incoming Particle, Outgoing Particle) Product. The target and product are isotopes, written in the form \( _{Z}^{A}X \), where \( Z \) is the atomic number, \( A \) the mass number, and \( X \) the element symbol. Incoming \( \alpha \), \( \mathrm{n} \), etc., represent particles, and outgoing particles are similar.

Check Notation (a)

Verify if notation \( _{5} \mathrm{B}^{10}(\alpha, \mathrm{n})_{7} \mathrm{N}^{13} \) is correct. The reaction: Boron-10 absorbs an alpha particle and emits a neutron, producing a nitrogen isotope: \[ ^{10}_5 \mathrm{B} + ^{4}_2 \alpha \rightarrow ^{13}_7 \mathrm{N} + ^{1}_0 \mathrm{n}. \] Ensure mass numbers and atomic numbers balance. Here, on the left: mass is 14 (10 + 4), and atomic number is 7 (5 + 2). On the right: mass is 14 (13 + 1), and atomic number is 7 (7 + 0). This notation is correct.

Check Notation (b)

Verify \( _{96} \mathrm{Cm}^{242}(\alpha, 2\mathrm{n})_{97}\mathrm{Bk}^{243} \). The reaction: Curium-242 absorbs an alpha particle, emits two neutrons, and forms berkelium: \[ ^{242}_{96} \mathrm{Cm} + ^{4}_{2} \alpha \rightarrow ^{243}_{97} \mathrm{Bk} + 2 imes ^{1}_{0} \mathrm{n}. \] Check numbers: Mass on the left is 246 (242 + 4), atomic number is 98 (96 + 2); right: mass 245 (243 + 2), atomic number 97. Balance is incorrect in mass. This notation is incorrect.

Check Notation (c)

Verify \( _{7} \mathrm{N}^{14}(\mathrm{n}, \mathrm{p})_{6} \mathrm{C}^{14} \). The nitrogen-14 absorbs a neutron and emits a proton, forming a carbon isotope: \[ ^{14}_{7} \mathrm{N} + ^{1}_{0} \mathrm{n} \rightarrow ^{14}_{6} \mathrm{C} + ^{1}_{1} \mathrm{p}. \] Check: Mass on the left is 15 (14 + 1), atomic number is 7 (7 + 0); right: mass 15 (14 + 1), atomic number 7 (6 + 1). Numbers balance, so this notation is correct.

Conclusion Based on Check

After evaluating each notation, it is evident that option (b) is incorrectly balanced as per the nuclear reaction representation.

Key concepts

Isotope Notation

Nuclear reactions often use a special notation called isotope notation to represent isotopes involved in these reactions. An isotope is a variant of an element with the same number of protons but different numbers of neutrons. This notation follows a specific format:

• A: Mass number, which is the sum of protons and neutrons in the atom's nucleus.
• Z: Atomic number, representing the number of protons in the atom.
• X: The chemical symbol of the element.

For example, in the notation \(_{5}^{10}\mathrm{B}\), Boron (\(\mathrm{B}\)) has a mass number of 10 and an atomic number of 5. This means it has 5 protons and 5 neutrons. Recognizing these components helps understand how nuclear reactions are balanced.

Mass Number Balance

In any nuclear reaction, ensuring that the total mass number on the left side of the equation equals the total mass number on the right is essential. The mass number represents the sum of protons and neutrons, and this rule is fundamental in terms of conserving the overall mass during the reaction. For example, in the reaction:\[ ^{10}_{5}\mathrm{B} + ^{4}_{2} \alpha \rightarrow ^{13}_{7} \mathrm{N} + ^{1}_{0} \mathrm{n} \]The sums of the mass numbers on both sides are 14, demonstrating mass number balance. Here, understanding balance helps quickly identify any imbalances, such as in case (b), where the mass number did not balance correctly, alerting us to an error in the reaction.

Atomic Number Balance

Atomic number balance is crucial in nuclear reactions involving isotopes. The atomic number (or the number of protons) should be conserved on both sides of the nuclear equation to ensure the same element identities before and after the reaction. For example, consider the reaction:\[^{14}_{7}\mathrm{N} + ^{1}_{0}\mathrm{n} \rightarrow ^{14}_{6}\mathrm{C} + ^{1}_{1}\mathrm{p} \]Here, the atomic number is 7 on both sides (7 from nitrogen equals 6 from carbon plus 1 from the proton). If the atomic numbers don't balance, it's indicative of an error, as seen in the incorrect notation (b) where the atomic number did not match.

Alpha Particle

An alpha particle is one of the common components involved in nuclear reactions. It is essentially a helium nucleus, consisting of 2 protons and 2 neutrons, and is represented as \(^4_2\alpha\). In nuclear reactions, the absorption of an alpha particle can significantly change the composition of the nucleus.For example, adding an alpha particle to Boron-10 in the reaction:\[ ^{10}_{5}\mathrm{B} + ^{4}_{2} \alpha \rightarrow ^{13}_{7} \mathrm{N} + ^{1}_{0} \mathrm{n} \]Increases the mass number by 4 and the atomic number by 2. Understanding how alpha particles affect nuclear reactions is crucial for balancing equations and predicting reaction products.

Neutron Emission

Neutron emission occurs when a neutron is ejected from the nucleus as part of a nuclear reaction. This can affect the mass number of the resulting isotope, as neutrons contribute to the total mass number but do not change the atomic number. In reactions like:\[^{242}_{96}\mathrm{Cm} + ^{4}_{2} \alpha \rightarrow ^{243}_{97}\mathrm{Bk} + 2\times^{1}_{0}\mathrm{n} \]Two neutrons are emitted, decreasing the mass number by 2. Understanding neutron emission helps in examining and constructing balanced nuclear reactions effectively. By knowing this concept, students can gauge the importance of ensuring mass number conservation and accurately predict potential isotope products.