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Question

Which one of the following elements will be an isobar of calcium if the atomic number of calcium is 20 and its mass number is 40?

This question was previously asked in
CDS I 2018 Elementary Mathematics Previous Year Paper (04-Feb-2018)
The correct answer is

Element with 22 neutrons and 18 protons

Understanding Isobars and Calcium Properties

The question asks us to identify an element that is an isobar of calcium. Let's first understand what an isobar is and what information we are given about calcium.

What is an Isobar?

Isobars are atoms of different elements that have the same mass number but different atomic numbers.

  • They have a different number of protons (atomic number \(\text{Z}\)).
  • They have a different number of neutrons.
  • They have the same total number of nucleons (protons + neutrons), which is the mass number \(\text{A}\).

The relationship between mass number (\(\text{A}\)), atomic number (\(\text{Z}\)), and the number of neutrons (\(\text{N}\)) is given by:

\(\text{A} = \text{Z} + \text{N}\)

Properties of Calcium:

We are given that calcium has an atomic number of 20 and a mass number of 40.

  • Atomic Number (\(\text{Z}\) of Calcium) = 20. This means Calcium has 20 protons.
  • Mass Number (\(\text{A}\) of Calcium) = 40.
  • Number of neutrons (\(\text{N}\) of Calcium) = \(\text{A} - \text{Z} = 40 - 20 = 20\) neutrons.

So, Calcium can be represented as \(^{40}_{20}\text{Ca}\).

Identifying the Isobar of Calcium

For an element to be an isobar of calcium (\(^{40}_{20}\text{Ca}\)), it must satisfy two conditions:

  • Its mass number (\(\text{A}\)) must be 40.
  • Its atomic number (\(\text{Z}\)) must be different from 20.

Let's examine each option provided. For each option, we will calculate the atomic number (number of protons) and the mass number (number of protons + number of neutrons).

Option Number of Neutrons (\(\text{N}\)) Number of Protons (\(\text{Z}\)) Mass Number (\(\text{A} = \text{Z} + \text{N}\)) Is it an Isobar of \(^{40}_{20}\text{Ca}\)? (A=40, Z\(\neq\)20)
1 18 20 20 + 18 = 38 No (\(\text{A}=38 \neq 40\))
2 19 18 18 + 19 = 37 No (\(\text{A}=37 \neq 40\))
3 19 20 20 + 19 = 39 No (\(\text{A}=39 \neq 40\))
4 22 18 18 + 22 = 40 Yes (\(\text{A}=40\) and \(\text{Z}=18 \neq 20\))

Analyzing Each Option for Isobar Property

  • Option 1: Element with 18 neutrons and 20 protons
    • Protons (\(\text{Z}\)) = 20
    • Neutrons (\(\text{N}\)) = 18
    • Mass Number (\(\text{A}\)) = \(\text{Z} + \text{N} = 20 + 18 = 38\)
    • This element has \(\text{Z}=20\) and \(\text{A}=38\). Since its mass number (38) is not equal to the mass number of calcium (40), it is not an isobar of calcium. It is actually an isotope of calcium (\(^{38}_{20}\text{Ca}\)).
  • Option 2: Element with 19 neutrons and 18 protons
    • Protons (\(\text{Z}\)) = 18
    • Neutrons (\(\text{N}\)) = 19
    • Mass Number (\(\text{A}\)) = \(\text{Z} + \text{N} = 18 + 19 = 37\)
    • This element has \(\text{Z}=18\) and \(\text{A}=37\). Since its mass number (37) is not equal to the mass number of calcium (40), it is not an isobar of calcium. This element is Argon (\(^{37}_{18}\text{Ar}\)).
  • Option 3: Element with 19 neutrons and 20 protons
    • Protons (\(\text{Z}\)) = 20
    • Neutrons (\(\text{N}\)) = 19
    • Mass Number (\(\text{A}\)) = \(\text{Z} + \text{N} = 20 + 19 = 39\)
    • This element has \(\text{Z}=20\) and \(\text{A}=39\). Since its mass number (39) is not equal to the mass number of calcium (40), it is not an isobar of calcium. It is another isotope of calcium (\(^{39}_{20}\text{Ca}\)).
  • Option 4: Element with 22 neutrons and 18 protons
    • Protons (\(\text{Z}\)) = 18
    • Neutrons (\(\text{N}\)) = 22
    • Mass Number (\(\text{A}\)) = \(\text{Z} + \text{N} = 18 + 22 = 40\)
    • This element has \(\text{Z}=18\) and \(\text{A}=40\). Its atomic number (18) is different from calcium's atomic number (20), but its mass number (40) is the same as calcium's mass number (40). Therefore, this element is an isobar of calcium. This element is Argon (\(^{40}_{18}\text{Ar}\)).

Conclusion on Calcium Isobar

Based on the analysis, the element described in option 4 has the same mass number (40) as calcium but a different atomic number (18). This fits the definition of an isobar.

Revision Table: Key Nuclear Terms

Term Definition Same Property Different Property
Isotopes Atoms of the same element Atomic number (\(\text{Z}\), protons) Mass number (\(\text{A}\), protons + neutrons), Neutrons (\(\text{N}\))
Isobars Atoms of different elements Mass number (\(\text{A}\), protons + neutrons) Atomic number (\(\text{Z}\), protons), Neutrons (\(\text{N}\))
Isotones Atoms of different elements Neutrons (\(\text{N}\)) Atomic number (\(\text{Z}\), protons), Mass number (\(\text{A}\), protons + neutrons)

Additional Information: Atomic Structure and Isobars

Atoms are made up of protons, neutrons, and electrons. Protons and neutrons are located in the nucleus and are collectively called nucleons. The number of protons determines the element (atomic number), while the total number of protons and neutrons determines the mass number. Isobars highlight that different elements can have nuclei with the same total number of particles, even though the mix of protons and neutrons is different. For example, \(^{40}_{20}\text{Ca}\) has 20 protons and 20 neutrons, while its isobar \(^{40}_{18}\text{Ar}\) has 18 protons and 22 neutrons. Both have a mass number of 40.

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