Problem 21 - Chemistry and Chemical Reactivity

Question

Verify that the atomic weight of lithium is 6.94 given the following information: \(^{6} \mathrm{Li},\) mass \(=6.015121 \mathrm{u} ;\) percent abundance\(=7.50 \%\) \(^{7} \mathrm{Li},\) mass \(=7.016003 \mathrm{u} ;\) percent abundance \(=92.50 \%\)

Step-by-Step Solution

Answer

The calculated atomic weight is 6.94, verifying the given atomic weight of lithium.

Step 1: Understanding the Concept

To verify the atomic weight, we need to calculate the weighted average atomic mass using the isotopic masses and their percent abundances.

Step 2: Calculate Weighted Mass for Lithium Isotopes

We calculate the contribution of each isotope to the atomic weight by multiplying its mass with its abundance (as a decimal). For \(^{6}\mathrm{Li}\), the contribution is \(6.015121 \, \text{u} \times \frac{7.50}{100} = 0.451134075 \, \text{u}\). For \(^{7}\mathrm{Li}\), the contribution is \(7.016003 \, \text{u} \times \frac{92.50}{100} = 6.493802775 \, \text{u}\).

Step 3: Sum the Contributions

Add the weighted contributions from each isotope: \(0.451134075 \, \text{u} + 6.493802775 \, \text{u} = 6.94493685 \, \text{u}\).

Step 4: Round to Significant Figures

The calculated atomic weight must be rounded to reflect the appropriate number of significant figures based on the data provided. Here, the data used for calculation suggests results should be rounded to 3 significant figures, yielding an atomic weight of \(6.94 \, \text{u}\).

Key Concepts

Weighted Average Atomic Mass

The atomic weight of an element such as lithium is essentially its weighted average atomic mass. This average takes into account the different isotopes of the element, along with their relative abundances. To achieve this, we use a formula that combines the isotopic masses with their percent abundances. Each isotope's mass is multiplied by its abundance (expressed as a decimal), and these products are summed to find the average. This weighted average reflects the atomic weight that we observe on the periodic table. It's crucial in understanding how elements behave during chemical reactions and in physical processes.

Isotopic Masses

Isotopic mass is the mass of a specific isotope of an element and it is expressed in atomic mass units (u). Each isotope of an element has a different number of neutrons, which results in varying masses. For lithium, the isotopes are

• \(^{6}\text{Li}\): with a mass of 6.015121 u.
• \(^{7}\text{Li}\): with a mass of 7.016003 u.

These differences in isotopic masses affect how the element is distributed naturally and its overall atomic weight. Understanding isotopic masses is important in fields such as nuclear physics and geochemistry, where isotopic ratios can tell us much about a sample's origin and history.

Percent Abundance

Percent abundance refers to how much of each isotope is naturally found in a sample of the element. It is expressed as a percentage of the total amount. For lithium, the given isotopic abundances are:

• \(^{6}\text{Li}\): 7.50% abundance.
• \(^{7}\text{Li}\): 92.50% abundance.

These percentages reflect the natural occurrence of each isotope and are essential for calculating the weighted average atomic mass of an element. They help in determining the atomic weight comprehensively since they indicate the proportion of each isotope present in a naturally occurring sample.

Significant Figures

Significant figures are used in scientific calculations to express precision. When dealing with measurements and resulting calculations like atomic weight, it's important to consider the number of significant figures to ensure that the answer reflects the precision of the input values. In the lithium example, our calculations should align with the least number of significant figures presented in the data, which is typically three. This helps maintain consistency and accuracy in reporting results. Rounding the atomic weight to 6.94 from an unrounded calculation ensures the measurement appropriately reflects the quality of the input data, enhancing clarity and precision in scientific communication.