Problem 120 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following pairs are isodiapheric pairs? (a) \({ }_{29} \mathrm{Cu}^{65}\) and \({ }_{24} \mathrm{Cr}^{55}\) (b) \({ }_{29} \mathrm{Cu}^{65}\) and \({ }_{24} \mathrm{Cr}^{52}\) (c) \({ }_{92} \mathrm{U}^{235}\) and \({ }_{90}^{20} \mathrm{Th}^{231}\) (d) \({ }_{92} \mathrm{U}^{238}\) and \({ }_{90} \mathrm{Th}^{231}\)

Step-by-Step Solution

Answer

Pairs (a) and (c) are isodiapheric.

Step 1: Understand Isodiapheric Pairs

Isodiapheric pairs of nuclides have the same difference between their neutron number and atomic number. In other words, for two nuclides to be isodiapheric, the difference \( (A - 2Z) \) should be equal for both nuclides, where \( A \) is the mass number and \( Z \) is the atomic number.

Step 2: Calculate for Pair (a)

For \({ }_{29} \mathrm{Cu}^{65} \, A - 2Z = 65 - 2 \times 29 = 7.\)For \({ }_{24} \mathrm{Cr}^{55} \, A - 2Z = 55 - 2 \times 24 = 7.\)Since both calculations give 7, the pair \( (a) \) is isodiapheric.

Step 3: Calculate for Pair (b)

For \({ }_{29} \mathrm{Cu}^{65} \, A - 2Z = 65 - 2 \times 29 = 7.\)For \({ }_{24} \mathrm{Cr}^{52} \, A - 2Z = 52 - 2 \times 24 = 4.\)Different results (7 and 4) indicate that the pair \( (b) \) is not isodiapheric.

Step 4: Calculate for Pair (c)

For \({ }_{92} \mathrm{U}^{235} \, A - 2Z = 235 - 2 \times 92 = 51.\)For \({ }_{90} \mathrm{Th}^{231} \, A - 2Z = 231 - 2 \times 90 = 51.\)Since both calculations give 51, the pair \( (c) \) is isodiapheric.

Step 5: Calculate for Pair (d)

For \({ }_{92} \mathrm{U}^{238} \, A - 2Z = 238 - 2 \times 92 = 54.\)For \({ }_{90} \mathrm{Th}^{231} \, A - 2Z = 231 - 2 \times 90 = 51.\)Different results (54 and 51) indicate that the pair \( (d) \) is not isodiapheric.

Key Concepts

Nuclides

Nuclides are the building blocks of atomic nuclei. In simple terms, a nuclide is a specific type of atom or nucleus characterized by a certain number of protons and neutrons.
Each nuclide is unique in its combination of these subatomic particles. The total number of protons in a nuclide is what defines its chemical element.
It's useful to know that:

• A change in the number of neutrons results in a different isotope of the same element.
• A change in the number of protons transforms the nuclide into a different element altogether.

The representation of a nuclide uses the format \({}_Z^{A}\text{X}\), where \(Z\) is the atomic number, \(A\) is the mass number, and \((X)\) is the chemical symbol of the element.
For example, \({}_{29}^{65}\text{Cu}\) represents a copper atom with 29 protons and a total mass number of 65, encompassing both protons and neutrons.

Atomic Number

The atomic number is a fundamental concept that characterizes an element. It is a count of the protons found in the nucleus of a single atom of an element.
Not only does the atomic number determine the identity of the element, but it also establishes its position in the periodic table.
Consider these points:

• As the atomic number increases, the number of electrons typically increases in a neutral atom, maintaining an equal balance of protons to electrons.
• Differences in atomic numbers between isotopes result in different elements entirely.

In equations discussing nuclides, the atomic number \(Z\) is crucial as it dictates the elemental symbol in the nuclear notation, like in \({}_Z^{A}\text{X}\). An astronaut can discern the type of element just by this atomic number. For instance, an atomic number of 8 indicates the element oxygen.

Mass Number

Mass number speaks to the sum total of protons and neutrons in an atomic nucleus. It gives a rough idea of the atom's total mass and is vital for distinguishing between different isotopes of the same element.
While the protons and neutrons both contribute largely to the mass, electrons are negligible in weight.
Here's what you should remember:

• Each proton and each neutron roughly equate to 1 atomic mass unit (u), making the mass number a near match to the actual atomic mass.
• Nuclides with the same atomic number can have different mass numbers, revealing a variation in the number of neutrons.

In nuclide notation \({}_Z^{A}\text{X}\), \(A\) stands for the mass number—telling us "how heavy" the atom is relative to others.
For example, in \({}_{92}^{235}\text{U}\), uranium has a mass number of 235, indicating 235 nucleons (protons and neutrons).
Understanding mass number is essential in calculating isodiapheric pairs, where comparing \(A - 2Z\) values helps determine if nuclides share a unique characteristic relationship.