Question

Naturally occurring boron is 80.20% boron-11 (atomic mass = 11.01 \(\mathrm{u}\)) and 19.80% of some other isotopic form of boron. What must the atomic mass of this second isotope be in order to account for the 10.81 \(\mathrm{u}\) average atomic mass of boron? (Write the answer to two decimal places.)

Short answer

The atomic mass of the second isotope is 10.00 u.

Step-by-step solution

- Understand the Problem

We need to find the atomic mass of the second isotope of boron given the average atomic mass of boron and the percentage and atomic mass of boron-11.

- Set Up the Equation

Let’s denote the atomic mass of the second isotope as \(x\). The average atomic mass can be calculated using the formula for weighted average: \[ 0.8020 \times 11.01 + 0.1980 \times x = 10.81 \]

- Simplify the Equation

First, calculate the contribution of boron-11 to the average atomic mass: \[ 0.8020 \times 11.01 = 8.83102 \] Now the equation becomes: \[ 8.83102 + 0.1980 \times x = 10.81 \]

- Isolate the Variable

Subtract 8.83102 from both sides to isolate the term involving \(x\): \[ 0.1980 \times x = 10.81 - 8.83102 \] \[ 0.1980 \times x = 1.97898 \]

- Solve for x

Now divide both sides by 0.1980 to solve for \(x\): \[ x = \frac{1.97898}{0.1980} = 9.999 \] When rounded to two decimal places, \(x\) is approximately 10.00.

Key concepts

isotopes

When we talk about isotopes, we're referring to variants of a particular chemical element that have different numbers of neutrons. All isotopes of an element have the same number of protons but can differ in neutron count. This difference in neutrons does not affect the chemical properties of the element but changes its atomic mass.
For example, boron has two main isotopes: boron-10 and boron-11. Both have 5 protons, but boron-10 has 5 neutrons while boron-11 has 6 neutrons. These variations are crucial in calculating an element's average atomic mass.

weighted average

Understanding the concept of a weighted average is key to solving problems like the average atomic mass calculation. A weighted average is a type of mean that considers the relative importance or frequency of each component.
For instance, in calculating the average atomic mass of boron, we need to account for the percentage of each isotope present in nature. The formula looks like this: \[ \text{Weighted Average} = (\text{fraction of isotope 1} \times \text{atomic mass of isotope 1}) + (\text{fraction of isotope 2} \times \text{atomic mass of isotope 2}) \] This weighted average gives us a more accurate representation of the atomic mass based on natural abundance.

boron isotopes

Boron occurs in nature primarily as two isotopes: boron-10 and boron-11. Boron-11 is more abundant, making up about 80.20% of naturally occurring boron, while boron-10 makes up roughly 19.80%.
Let's delve deeper:

• Boron-11: has an atomic mass of approximately 11.01 u.
• Boron-10: we calculated in the exercise to have an atomic mass around 10.00 u.

By understanding the individual properties and abundances of these isotopes, we can accurately compute the average atomic mass of boron.

atomic masses

Atomic mass is the weighted average mass of an element's isotopes, measured in atomic mass units (u). It gives you an idea of how heavy an atom is relative to others.
In calculations, such as the one in our exercise, knowing the atomic masses of isotopes is essential. The atomic mass reflects both the mass and the relative abundance of each isotope, providing a precise average that aligns with natural occurrence.
For example, the atomic mass of boron is not a whole number like 11, but 10.81 u, because it’s a mix of the masses and proportions of boron-10 and boron-11.