Problem 17 - Chemistry and Chemical Reactivity

Question

Verify that the atomic weight of lithium is \(6.94,\) given the following information: \(^{6} \mathrm{Li},\) mass \(=6.015121 \mathrm{u} ;\) percent abundance \(=7.50 \%\) \(^{7} \mathrm{Li},\) mass \(=7.016003 \mathrm{u} ;\) percent abundance \(=92.50 \%\)

Step-by-Step Solution

Answer

The calculated atomic weight of lithium is 6.94, confirming the information given.

Step 1: Understanding Isotopic Abundance

The atomic weight of an element is the weighted average of the masses of its isotopes, considering their natural abundances. In this case, lithium has two isotopes: \( ^{6}\text{Li} \) and \( ^{7}\text{Li} \), with given masses and percent abundances.

Step 2: Convert Percent Abundance to Decimal Form

Convert the given percent abundances of \( ^{6}\text{Li} \) and \( ^{7}\text{Li} \) to decimal form. For \( ^{6}\text{Li} \), it is \( 0.0750 \) and for \( ^{7}\text{Li} \), it is \( 0.9250 \). This makes the math easier when calculating the weighted average.

Step 3: Calculate Weighted Contributions

Find the contribution of each isotope to the atomic weight by multiplying their masses by their relative abundances. For \( ^{6}\text{Li} \), it is \( 6.015121 \times 0.0750 = 0.451134075 \). For \( ^{7}\text{Li} \), it is \( 7.016003 \times 0.9250 = 6.486802775 \).

Step 4: Calculate the Atomic Weight of Lithium

Add the contributions from both isotopes to find the atomic weight of lithium: \( 0.451134075 + 6.486802775 = 6.93793685 \), which rounds to \( 6.94 \) when considering significant figures.

Step 5: Concluding Verification

The calculated atomic weight is \( 6.94 \), which matches the given atomic weight for lithium. This verifies that the isotopic composition provided correctly calculates the known atomic weight.

Key Concepts

Isotopic Abundance

Isotopic abundance refers to the percentage of each isotope of an element naturally found on Earth. Elements can have different isotopes, which are forms of the same element with different numbers of neutrons in the nucleus. This means they have slightly different masses.

For lithium, there are two main isotopes:

• ⁶Li, which makes up 7.50% of naturally occurring lithium
• ⁷Li, making up 92.50% of lithium

These percentages help us calculate the atomic weight by showing how much each isotope contributes to the overall average.

Weighted Average

A weighted average is a way to calculate the average value of a set of numbers, where each number has a specific weight or importance. In the case of atomic weight, these weights are the isotopic abundances.

To find the weighted average for lithium's atomic weight:

• Convert the percentage abundances to decimal form:
  • For ⁶Li: 7.50% becomes 0.0750
  • For ⁷Li: 92.50% becomes 0.9250
• Multiply each isotope's mass by its decimal abundance
• Add these contributions together to find the atomic weight

This method ensures each isotope's contribution to the total atomic weight is accurately represented based on how common each isotope is.

Lithium Isotopes

Lithium is an element with two key isotopes, ⁶Li and ⁷Li. These isotopes have different masses but are chemically similar.

• ⁶Li has a mass of 6.015121 u
• ⁷Li has a mass of 7.016003 u

Lithium isotopes are important for scientific calculations and understanding the properties of lithium as a whole, including its atomic weight and chemical behavior. Each isotope has a distinct role and contribution to the overall atomic mass.

Significant Figures

Significant figures are essential in scientific calculations to convey the precision of a measured or calculated value. They indicate which digits in a number are meaningful and help ensure that calculations are accurate yet not over-precise.

When calculating the atomic weight of lithium from its isotopes, the number was rounded to 6.94:

• The calculation resulted in 6.93793685, rounded
• Considering precision and given data, 6.94 is accurate using two significant figures

Significant figures help communicate the reliability and accuracy of scientific results, making them crucial in reporting atomic weights.