Problem 23 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Given that the abundances of isotopes \({ }^{54} \mathrm{Fe},{ }^{56} \mathrm{Fe}\) and \({ }^{\mathrm{s} 7} \mathrm{Fe}\) are \(5 \%, 90 \%\) and \(5 \%\), respectively, the atomic mass of \(\mathrm{Fe}\) is (a) \(55.85\) (b) \(55.95\) (c) \(55.75\) (d) \(56.05\)

Step-by-Step Solution

Answer

The atomic mass of Fe is 55.95, which is option (b).

Step 1: Write down the given data

We have three isotopes of iron with their respective abundance percentages given as follows: for \(^{54}\mathrm{Fe}\), the abundance is 5%; for \(^{56}\mathrm{Fe}\), the abundance is 90%; and for \(^{57}\mathrm{Fe}\), the abundance is 5%.

Step 2: Convert percentage abundances to decimal form

Convert each percentage to a decimal by dividing by 100: \(^{54}\mathrm{Fe}\) abundance is 0.05, \(^{56}\mathrm{Fe}\) abundance is 0.90, and \(^{57}\mathrm{Fe}\) abundance is 0.05.

Step 3: Calculate the contribution of each isotope to the atomic mass

The contribution of each isotope to the atomic mass is calculated by multiplying the isotope's mass number by its abundance in decimal form:1. Contribution of \(^{54}\mathrm{Fe}\) = 54 \( \times 0.05 = 2.7\).2. Contribution of \(^{56}\mathrm{Fe}\) = 56 \( \times 0.90 = 50.4\).3. Contribution of \(^{57}\mathrm{Fe}\) = 57 \( \times 0.05 = 2.85\).

Step 4: Sum the contributions to find the average atomic mass

Add up the contributions calculated for each isotope: \[2.7 + 50.4 + 2.85 = 55.95\]

Step 5: Compare the result to the given options

The calculated average atomic mass is 55.95, which matches option (b).

Key Concepts

Isotopic Abundance

Isotopic abundance is a crucial factor in understanding how elements naturally occur. It tells us the percentage or fraction of an isotope compared to all isotopes of that element found in nature. In the example of iron, we have three isotopes:

• For \(^ {54} \mathrm{Fe} \), the isotopic abundance is 5%.
• For \(^ {56} \mathrm{Fe} \), the isotopic abundance is 90%.
• For \(^ {57} \mathrm{Fe} \), the isotopic abundance is 5%.

These percentages help us determine the element's average atomic mass. When calculating, it's essential to convert these percentages into decimal form by dividing each by 100. This simplifies calculations and avoids errors. Understanding isotopic abundance is key to calculating other properties of an element and understanding its natural distribution.

Average Atomic Mass

The average atomic mass is a weighted average of the masses of an element's isotopes based on their abundances. It provides a single-value representation of the element's mass, facilitating easier use in calculations and comparisons. To find this average, we perform the following steps:

• Convert the isotopic abundances from percentages to decimals.
• Multiply each isotope's mass number by its abundance.
• Sum the results of these multiplications.

For instance, in our original exercise with iron isotopes, we multiply each isotope's mass by its respective decimal abundance. Calculated as:\[ (54 \times 0.05) + (56 \times 0.90) + (57 \times 0.05) = 55.95 \]This result shows us the average atomic mass of iron. Hence, the selected option is 55.95. The knowledge of average atomic mass aids in various scientific calculations, including stoichiometry and molecular chemistry.

Isotope Identification

Isotope identification involves recognizing that the same element can exist in different forms with varying numbers of neutrons, known as isotopes. For example, iron has several isotopes, such as \(^ {54} \mathrm{Fe} \), \(^ {56} \mathrm{Fe} \), and \(^ {57} \mathrm{Fe} \). By identifying these isotopes, scientists can explore an element's properties more deeply.An isotope is identified by its mass number, which is the sum of protons and neutrons. While atoms of a given element have the same number of protons, the difference in neutron numbers gives rise to various isotopes. This identification is crucial in applications such as radiometric dating, nuclear medicine, and studying environmental changes. Recognizing isotopes helps chemists and physicists to track substances and their transformations in experiments, making the concept of isotope identification a foundational pillar in scientific studies.