Problem 116 - Chemistry: An Atoms-Focused Approach

Question

Suppose we know the atomic mass of each of the three stable isotopes of an element to six significant figures, and we know the natural abundances of the isotopes to the nearest \(0.01 \% .\) How well can we know the average atomic mass - that is, how many significant figures should be used to express its value?

Step-by-Step Solution

Answer

Question: Given the atomic masses of three stable isotopes of an element and their natural abundances, find the average atomic mass with the correct number of significant figures. Answer: The average atomic mass should be expressed with three significant figures.

Step 1: Calculate the average atomic mass

To find the average atomic mass, we multiply the mass of each isotope by its natural abundance and sum up the results. Let \(m_1\), \(m_2\), and \(m_3\) be the atomic masses of the three isotopes, and let \(A_1\), \(A_2\), and \(A_3\) be their respective natural abundances (in decimal form). The average atomic mass can be calculated as: \(M = m_1 \cdot A_1 + m_2 \cdot A_2 + m_3 \cdot A_3\)

Step 2: Determine the significant figures of abundances

We are given the natural abundances of the isotopes to the nearest 0.01%. This information suggests that the abundances (In decimal form) should have three significant figures: \(A_1: 0.XXX\) \(A_2: 0.YYY\) \(A_3: 0.ZZZ\)

Step 3: Analyze the significant figures in the calculation

In the calculation for average atomic mass, we have: \(M = m_1 \cdot A_1 + m_2 \cdot A_2 + m_3 \cdot A_3\) Note that atomic masses are given to six significant figures, and the abundances are given to three significant figures. For multiplication, the number of significant figures in the result should be equal to the least number of significant figures in either the multiplicand or the multiplier. Therefore, the result of each multiplication (the terms \(m_1 \cdot A_1,\) \(m_2 \cdot A_2,\) and \(m_3 \cdot A_3\)) should have three significant figures.

Step 4: Find the number of significant figures for the final result

Since the terms to be added in step 1 are in three significant figures, the final result (average atomic mass) after summing up the terms should also be expressed in three significant figures. This is because, in addition, we must maintain the number of decimal places consistent. Finally, we can conclude that the average atomic mass should be expressed with three significant figures as per the given data.

Key Concepts

Significant Figures

In any scientific calculation, the concept of significant figures is crucial for conveying precision. Significant figures represent the digits in a number that contribute to its accuracy. Knowing how many significant figures to use is essential for maintaining the integrity of numerical data.

For example, if an atomic mass is measured as 12.3456 g/mol, it has six significant figures. This indicates a high level of precision in the measurement. When performing calculations, such as finding an average atomic mass using isotopic abundances, the number of significant figures dictates the precision of the final result.

• Multiplication and Division: The result should have the same number of significant figures as the factor with the fewest significant figures.
• Addition and Subtraction: The result should maintain the precision of the least precise decimal place.

Handling significant figures correctly ensures that your calculations are as precise as the measured values allow.

Isotopes

Isotopes are variants of a chemical element that have the same number of protons but a different number of neutrons in their nucleus. This means isotopes of an element have the same atomic number but different mass numbers.

For example, carbon-12 and carbon-14 are isotopes of carbon. Both have six protons, but carbon-12 has six neutrons while carbon-14 has eight neutrons. Because of this difference in neutron count, isotopes exhibit different atomic masses even though they behave similarly in chemical reactions.

• Each isotope of an element contributes differently to the average atomic mass based on its mass and natural abundance.
• Despite varying masses, isotopes belong to the same element, ensuring consistency in chemical reactions.

Understanding isotopes is essential for accurately calculating atomic masses and predicting elemental properties.

Natural Abundances

Natural abundance refers to the relative proportion of different isotopes of an element found in nature. This percentage provides a statistical insight into how common an isotope is compared to others.

For instance, the isotopes of magnesium might have natural abundances expressed as: 79% for magnesium-24, 10% for magnesium-25, and 11% for magnesium-26. These percentages are crucial for calculating the average atomic mass of an element.

• The natural abundance of each isotope is typically represented as a percentage.
• To use in calculations, these percentages are converted to decimal form (e.g., 79% becomes 0.79).

Natural abundances dictate the weight each isotope has in computing the average atomic mass, affecting the final precision based on the given data.

Atomic Masses

Atomic mass or atomic weight is the weighted average of the masses of an element's isotopes, accounting for their natural abundances. This property is essential in understanding an element's mass on the atomic scale.

To illustrate, consider the atomic masses of various isotopes used in calculating an element's average atomic mass. Utilizing a precise formula, each isotope's mass is multiplied by its natural abundance to determine its contribution to the total atomic mass.

The formula for calculating the average atomic mass is:\[ M = m_1 \cdot A_1 + m_2 \cdot A_2 + m_3 \cdot A_3 \]

• Here \(m_1, m_2,\) and \(m_3\) represent the masses of the isotopes.
• \(A_1, A_2,\) and \(A_3\) stand for their respective natural abundances (in decimal form).

In the final result, ensure to express the average atomic mass with the correct number of significant figures, balancing both precision and accuracy of the measurements used.