Problem 54 - Chemistry: An Atoms-Focused Approach

Question

Sulfur in nature is a mixture of four isotopes: \(^{32} \mathrm{S}\) \((31.9721 \mathrm{amu}, 95.04 \%) ;^{33} \mathrm{S}(32.9715 \mathrm{amu}, 0.75 \%) ;^{34} \mathrm{S}\) \((33.9679 \mathrm{amu}, 4.20 \%) ;\) and \(^{36} \mathrm{S}(35.9671 \mathrm{amu}, 0.01 \%) .\) Use this information to calculate the average atomic mass of sulfur.

Step-by-Step Solution

Answer

Question: Using the given isotopes of sulfur, their masses, and their natural abundance percentages, calculate the average atomic mass of sulfur. Isotopes and Masses: 1. \(^{32}S\) (95.04%): 31.9721 amu 2. \(^{33}S\) (0.75%): 32.9715 amu 3. \(^{34}S\) (4.20%): 33.9679 amu 4. \(^{36}S\) (0.01%): 35.9671 amu Answer: The average atomic mass of sulfur is approximately 32.0334 amu.

Step 1: Convert the given percentages to decimals

To convert the percentages to decimals, divide each percentage by 100. \(95.04 \% = 0.9504\) \(0.75 \% = 0.0075\) \(4.20 \% = 0.0420\) \(0.01 \% = 0.0001\)

Step 2: Multiply the mass of each isotope by its corresponding decimal

Multiply the mass of each sulfur isotope by the corresponding decimal we found in step 1: \(^{32}S: 31.9721 \cdot 0.9504 = 30.3559\) \(^{33}S: 32.9715 \cdot 0.0075 = 0.2474\) \(^{34}S: 33.9679 \cdot 0.0420 = 1.4265\) \(^{36}S: 35.9671 \cdot 0.0001 = 0.0036\)

Step 3: Sum up the products

Add the products from step 2 to calculate the average atomic mass of sulfur: \(30.3559 + 0.2474 + 1.4265 + 0.0036 = 32.0334\) The average atomic mass of sulfur is approximately 32.0334 amu.

Key Concepts

Isotopes

Isotopes are fascinating variants of elements. They have the same number of protons but differ in the number of neutrons in their nuclei. This difference in neutron number leads to different atomic masses for each isotope. For a given element like sulfur, isotopes have unique identities and slightly different properties.

When discussing sulfur, it’s a great example because it naturally occurs as four major isotopes:

• ^{32}S - the most abundant isotope.
• ^{33}S
• ^{34}S
• ^{36}S - the rarest isotope among the four.

These isotopes are crucial in determining the average atomic mass of sulfur. The presence of multiple isotopes for an element is a natural phenomenon, contributing to the element's overall atomic mass. Understanding isotopes is essential for comprehending the broader concept of atomic mass.

Sulfur

Sulfur is an abundant nonmetal, essential for many biological functions and industrial applications. It appears in various forms, but the focus here is on its molecular structure as isotopes. Sulfur's four main isotopes have varying atomic masses, ranging from around 31.9721 amu to 35.9671 amu.

Due to these varying isotopes, sulfur does not have a singular atomic mass. Instead, its atomic mass is an average calculated from its isotopes. Along with its atomic masses, each isotope has a specific natural abundance represented in percentage terms. These values are pivotal in calculating sulfur's average atomic mass.

Sulfur’s isotopic diversity underpins a great deal of its chemical and physical properties, making it integral to understanding fundamental chemistry principles.

Atomic Mass Units (amu)

Atomic mass units, abbreviated as amu, serve as a small mass unit used to express atomic and molecular weights. The amu is incredibly useful because it helps scientists and students avoid cumbersome numbers when representing the mass of atoms. In most cases, the amu is defined using carbon's isotope, ^{12}C, with one amu being one-twelfth the mass of this carbon isotope.

When dealing with sulfur and its isotopes, atomic masses are recorded in amu, typically to four decimal places for precision. For example, consider ^{32}S with a mass of 31.9721 amu. It provides a standard, precise way to convey its mass in the realm of chemistry. Since isotopes of sulfur each have their specific atomic mass in amu, these values become indispensable when calculating the element’s average atomic mass.

Weighted Average

The concept of a weighted average is crucial when calculating the average atomic mass of elements like sulfur. Unlike a simple average, a weighted average accounts for the relative abundance of each isotope. This method ensures that isotopes occurring more frequently in nature have a larger impact on the overall average compared to rare isotopes. To compute sulfur's average atomic mass, we multiply each isotope's atomic mass by its relative abundance (converted from a percentage to a decimal).

These products are then summed up to yield the final average atomic mass. For example, the calculation involves:

• Multiplying the atomic mass of ^{32}S by 0.9504 (its decimal abundance),
• Then doing the same for the other isotopes,
• Before adding all the results to get 32.0334 amu.

This accurate measurement reflects how often each isotope appears in nature and is critical for precise scientific analysis and applications.