Problem 102 - Chemistry Matter and Change

Question

Boron-10 and boron-11 are the naturally occurring isotopes of elemental boron. If boron has an atomic mass of 10.81 amu, which isotope occurs in greater abundance?

Step-by-Step Solution

Answer

Boron-11 occurs in greater abundance (81%).

Step 1: Identify Isotopes and Their Atomic Masses

Boron has two naturally occurring isotopes: Boron-10 with an atomic mass of 10 amu and Boron-11 with an atomic mass of 11 amu.

Step 2: Represent the Atomic Mass Using a Weighted Average Formula

The atomic mass of boron is a weighted average of its isotopes' masses, which can be represented as \( \text{{atomic mass}} = (p \times 10) + ((1-p) \times 11) \), where \( p \) is the fraction of Boron-10.

Step 3: Substitute the Known Atomic Mass into the Formula

We know the atomic mass is 10.81 amu, so we set up the equation: \( 10.81 = (p \times 10) + ((1-p) \times 11) \).

Step 4: Simplify and Solve the Equation for p

Simplify the equation: \( 10.81 = 10p + 11 - 11p \). Combine like terms to get \( 10.81 = 11 - p \). Solve for \( p \) by subtracting 11 from both sides: \( p = 11 - 10.81 \) which gives \( p = 0.19 \).

Step 5: Interpret the Results

The solution \( p = 0.19 \) indicates that 19% of the boron is Boron-10 and, therefore, the remaining 81% is Boron-11. This shows that Boron-11 is more abundant.

Key Concepts

Atomic Mass

Atomic mass is a crucial concept in chemistry that refers to the average mass of atoms of an element, measured in atomic mass units (amu). It takes into account the mass of each isotope of that element as well as their respective abundances.
You can think of atomic mass as an average, similar to how you would average grades on a test, but it’s "weighted" based on how common each isotope is. In nature, elements often exist as a mixture of isotopes, each contributing to the overall atomic mass depending on how much of each isotope is present.
For example, in boron, the two major isotopes are boron-10 and boron-11. Thus, the atomic mass is 10.81 amu because it accounts for both of these isotopes in their abundances.

Weighted Average

A weighted average is used when some entries in a set contribute more to the total than others. This is quite settled in the case of atomic mass, where different isotopes bear varying weights.
The formula for calculating a weighted average is:

• Multiply each isotope's mass by its fractional abundance (the percentage of each isotope in decimal form).
• Sum these values together.
• The result is the weighted average, or atomic mass.

In boron's case, we apply the idea of weighted averages by using the isotopic weights: (10 amu for Boron-10 and 11 amu for Boron-11) and their relative abundance to arrive at the average atomic mass of 10.81 amu.

Abundance

Abundance in chemistry refers to the relative amount of a particular isotope present in a sample. It is usually expressed as a percentage or fraction, denoting its proportion among all isotopes of that element.
For boron, using the given atomic mass of 10.81 amu, we determined the abundance of its isotopes by calculating how much each isotope contributes to the overall atomic mass.
By solving the formula: \[10.81 = (p \times 10) + ((1-p) \times 11)\]we found that boron-10 has an abundance of 19%, while boron-11 is more abundant with an 81% presence. Understanding abundance helps us deduce which isotope is more common in nature.

Boron Isotopes

Boron naturally occurs as two isotopes: boron-10 and boron-11. These two isotopes have different neutron numbers, giving them different atomic masses, yet they share the number of protons characteristic of boron.
Boron-10 has an atomic mass of 10 amu, and Boron-11 has an atomic mass of 11 amu. Together, they provide unique characteristics to boron's atomic structure, which affects its macroscopic properties like atomic mass.
In Earth's crust, these isotopes do not exist in equal proportions. By analyzing the isotopic data and using weighted averages, we find that boron-11 is more prevalent. With boron-11 making up 81% and boron-10 constituting 19% of the natural composition of boron, it becomes clear why boron's average atomic mass is closer to 11 than 10.