Problem 95 - Chemistry: The Central Science

Question

The element lead ( \(\mathrm{Pb}\) ) consists of four naturally occurring isotopes with atomic masses \(203.97302,205.97444,206.97587\), and \(207.97663\) amu. The relative abundances of these four isotopes are \(1.4,24.1,22.1\), and \(52.4 \%\) respectively. From these data, calculate the atomic weight of lead.

Step-by-Step Solution

Answer

The atomic weight of lead can be calculated using the formula: Atomic weight = Σ (atomic mass * relative abundance). After converting the relative abundances to decimals and multiplying them by their respective atomic masses, we sum the products to get the atomic weight of lead as approximately \(207.217607\) amu.

Step 1: Convert relative abundances to decimal form

To convert the percentages to decimals, we will divide each percentage by 100. Isotope 1: 1.4% = 0.014 Isotope 2: 24.1% = 0.241 Isotope 3: 22.1% = 0.221 Isotope 4: 52.4% = 0.524

Step 2: Multiply atomic masses by their relative abundances

Now, we will multiply the atomic masses of the isotopes by their corresponding decimal abundance values. Isotope 1: 203.97302 amu * 0.014 = 2.855622 amu Isotope 2: 205.97444 amu * 0.241 = 49.639668 amu Isotope 3: 206.97587 amu * 0.221 = 45.741197 amu Isotope 4: 207.97663 amu * 0.524 = 108.981120 amu

Step 3: Sum the products

Now, we will add the products we obtained in step 2. Atomic weight of lead = 2.855622 amu + 49.639668 amu + 45.741197 amu + 108.981120 amu Atomic weight of lead ≈ 207.217607 amu So, the atomic weight of lead is approximately 207.217607 amu.

Key Concepts

Isotopes

Isotopes are variations of elements that have the same number of protons but different numbers of neutrons. This means they have the same atomic number but different atomic masses. In the case of lead, the four isotopes mentioned have distinct atomic masses: 203.97302, 205.97444, 206.97587, and 207.97663 amu (atomic mass units). These isotopes all belong to the element lead because they share the same number of protons in their atomic structure.

Despite having different masses, isotopes of an element often behave similarly in chemical reactions. However, these varying masses can affect their physical properties, such as density. Understanding isotopes is crucial in calculating the atomic weight of an element based on its natural abundance.

Relative Abundance

Relative abundance is a way of expressing the proportion of each isotope in a sample of an element. It is usually given as a percentage to show how much of each isotope is present compared to the others. In the exercise, the isotopes of lead have the following relative abundances: 1.4%, 24.1%, 22.1%, and 52.4%.

To use these percentages in calculations, you first convert them into decimal form by dividing them by 100. For example:

• 1.4% becomes 0.014
• 24.1% becomes 0.241
• 22.1% becomes 0.221
• 52.4% becomes 0.524

These decimal values are then used to find the weighted average or atomic weight of an element.

Atomic Mass

Atomic mass is the mass of an individual atom, usually measured in atomic mass units (amu). It is essentially the sum of protons and neutrons in the nucleus of an isotope. In our exercise, each isotope of lead has a different atomic mass because they have varied numbers of neutrons.

To calculate the atomic weight of an element, you use the atomic masses of all its isotopes alongside their relative abundances. By multiplying each isotope's atomic mass by its decimal abundance, you find the contribution of each isotope to the overall atomic weight. The final atomic weight is determined by summing all these contributions together, as shown in the exercise.

Lead Element

Lead is a chemical element with the symbol Pb and atomic number 82. It is a versatile metal that is soft and malleable at room temperature. Though it has mischievous historical uses, particularly in plumbing and paints, lead is now known to be toxic to humans and animals.

The atomic weight of lead is calculated by considering all its naturally occurring isotopes and their relative abundances. In the given exercise, we established that the atomic weight of lead is approximately 207.217607 amu. This value represents a weighted average of the atomic masses of all lead isotopes based on their abundance in nature.

Despite its known toxicity, lead continues to be valuable in certain industrial applications, including batteries, radiation shielding, and weights due to its dense, heavy nature.