Question

An element has two naturally occurring isotopes. Isotope 1 has a mass of \(106.905 \mathrm{amu}\) and a relative abundance of \(51.8 \%\), and isotope 2 has a mass of \(108.904 \mathrm{amu}\) and a relative abundance of \(48.2 \%\). Find the atomic weight of this element and, by comparison to the periodic table, identify it.

Short answer

The atomic weight of the element is approximately 107.87 amu, which corresponds to Silver (Ag).

Step-by-step solution

Convert Percent Abundance to Decimal Form

Convert the percent abundance of each isotope to a decimal by dividing by 100. For isotope 1 with 51.8%, we have 0.518, and for isotope 2 with 48.2%, we have 0.482.

Calculate the Weighted Average of the Isotopes

Multiply the mass of each isotope by its corresponding abundance in decimal form and then add the results together to find the weighted average, which is the atomic weight of the element.

Perform the Calculations

For isotope 1: \(106.905 \mathrm{amu} \times 0.518 = 55.37769 \mathrm{amu}\), and for isotope 2: \(108.904 \mathrm{amu} \times 0.482 = 52.487928 \mathrm{amu}\). Adding these together gives the atomic weight: \(55.37769 \mathrm{amu} + 52.487928 \mathrm{amu} = 107.865618 \mathrm{amu}\).

Round the Atomic Weight

Round the calculated atomic weight to an appropriate number of significant figures. The atomic weight to two decimal places is 107.87 amu.

Identify the Element

Compare the calculated atomic weight to the atomic weights listed in the periodic table. The atomic weight of 107.87 amu corresponds to the element Silver (Ag).

Key concepts

Isotopic Abundance

Isotopic abundance refers to the proportion of each isotope of an element that exists in nature. Each isotope of an element has a different number of neutrons in the nucleus, leading to different atomic masses, even though they share the same number of protons. Understanding isotopic abundance is essential for determining the average atomic mass of an element, which is what we see on the periodic table.

For instance, if an element has two isotopes, their relative abundances must sum to 100%. These are typically given as percentages, as seen in the given exercise. To calculate anything related to these abundances, such as the average atomic weight, we first need to convert the percentages to decimal form.

Weighted Average of Isotopes

The weighted average of isotopes takes into account their individual masses and their abundances in nature. This average determines the atomic weight of an element. To find it, we multiply the mass of each isotope by its isotopic abundance (converted to decimal form) and add these products together.

To apply this concept, using the data from the original exercise, we multiply each isotope's mass by its relative abundance to see its contribution to the overall atomic weight. The sum of these products gives us the element's atomic weight. It is essential to perform these calculations with precision to ensure the final atomic weight is accurate. This value is critical as it must match the atomic weights found on the periodic table to identify the element correctly.

Significant Figures

Significant figures are the digits in a number that are known with some degree of confidence. They are important in scientific calculations because they communicate how precise a measurement or calculation is. When we calculate the atomic weight, we must be mindful of the number of significant figures from the isotope masses and their abundances.

In the solution, after finding the contribution of each isotope to the atomic weight, the sum is rounded to match the significant figures of the original measurements. The rule of thumb is to round the final answer to the least number of significant figures present in the input data. For example, if the least number of significant figures in the data is three, the final atomic weight will be rounded to three significant figures as well.

Periodic Table

The periodic table is an organized chart of elements arranged by increasing atomic number, which is the number of protons in an atom's nucleus. Each element on the periodic table also has an associated atomic weight, which is the weighted average of the masses of all naturally occurring isotopes of that element.

To identify an element based on atomic weight, one must compare the calculated average atomic mass to those listed in the periodic table. The exercise demonstrated this by calculating an atomic weight of 107.87 amu and then identifying the element as Silver (Ag) when matched with the periodic table. Understanding the periodic table and its arrangement allows for the identification of elements and an understanding of their properties.