Problem 20 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The fractional abundance of \(\mathrm{Cl}^{35}\) in a sample of chlorine containing only \(\mathrm{Cl}^{35}\) (atomic weight \(=34.9\) ) and \(\mathrm{Cl}^{37}\) (atomic weight \(=36.9\) ) isotopes, is \(0.6\). The average mass number of chlorine is (a) \(35.7\) (b) \(35.8\) (c) \(18.8\) (d) \(35.77\)

Step-by-Step Solution

Answer

The average mass number of chlorine is (a) 35.7.

Step 1: Understand the concept of average atomic mass

The average atomic mass of an element with multiple isotopes is calculated by multiplying the mass number of each isotope by its fractional abundance and then adding the results together. The formula to find the average mass number is: \( \text{Average Mass} = (\text{Fractional abundance of isotope 1} \times \text{Mass of isotope 1}) + (\text{Fractional abundance of isotope 2} \times \text{Mass of isotope 2}) \).

Step 2: Calculate the fractional abundance of \(\mathrm{Cl}^{37}\)

Since the total fractional abundance must equal 1, we can subtract the fractional abundance of \(\mathrm{Cl}^{35}\) from 1 to find the fractional abundance of \(\mathrm{Cl}^{37}\). \[ \text{Fractional abundance of } \mathrm{Cl}^{37} = 1 - 0.6 = 0.4 \].

Step 3: Calculate the average atomic mass

Using the formula from Step 1 and the abundances from Step 2, we calculate the average atomic mass of chlorine. \[ \text{Average Mass} = (0.6 \times 34.9) + (0.4 \times 36.9) \].

Step 4: Perform the calculation

Now we perform the arithmetic operations to find the average mass. \[ \text{Average Mass} = (0.6 \times 34.9) + (0.4 \times 36.9) = 20.94 + 14.76 = 35.7 \]. Hence, the average mass number of chlorine in the sample is 35.7.

Key Concepts

Isotopic Abundance

Understanding isotopic abundance is essential to comprehend how elements exist in nature. Isotopes are atoms of the same element that have different numbers of neutrons, and consequently, different mass numbers.

Isotopic abundance refers to the relative amount in which each isotope of an element is found in a natural sample. It is usually expressed as a decimal or a percentage. For example, if an isotope has an abundance of 60%, this will be represented as 0.6 in calculations. To calculate the average atomic mass of an element, we weight each isotope's mass by its abundance.

Let's break it down with an example:

• When a chlorine sample contains isotopes Cl-35 and Cl-37, we first need to know their fractional abundancies, like 0.6 for Cl-35.
• If the total abundance must add up to 1 (or 100%), the abundance of Cl-37 would be 1 - 0.6 = 0.4.
• These abundances are crucial as they influence the calculated average atomic mass of the chlorine sample.

Atomic Weight

Atomic weight, often used interchangeably with atomic mass, is the weighted average mass of an atom of an element. It is measured in atomic mass units (amu) or Daltons. This value takes into account the different masses and abundances of the isotopes that compose the element.

The atomic weight reflects the average mass of all the atoms of an element as they occur in nature, not just one individual atom or one isotope. For example:

• Cl-35 has an atomic weight of approximately 34.9 amu.
• Cl-37 has an atomic weight around 36.9 amu.

These values are used along with the isotopic abundances to calculate the average atomic mass of an element. A misconception is that atomic weight is a constant for each element, but it slightly varies depending on the isotopic composition of the element in different samples.

Mass Number

Mass number is the total number of protons and neutrons in an atomic nucleus. It is denoted by the letter 'A' and is always a whole number. For isotopes, the mass number varies because while the number of protons (which determines the element) remains constant, the number of neutrons can differ.

For example, chlorine has two common isotopes:

• Cl-35 has 17 protons and 18 neutrons, so its mass number is 35.
• Cl-37 has 17 protons and 20 neutrons, making the mass number 37.

Understanding the mass number is crucial when performing average atomic mass calculations. It allows us to quantify the relative mass of each isotope (ignoring the very small mass of electrons), which when multiplied by the respective isotopic abundances, leads to the average atomic mass of the element.