Problem 26 - Chemistry The Molecular Science

Question

Verify that the average atomic mass of lithium is 6.941 , given this information: \({ }^{6} \mathrm{Li},\) exact mass \(=6.015121 \mathrm{u}\) percent abundance \(=7.500 \%\) \({ }^{7} \mathrm{Li},\) exact mass \(=7.016003 \mathrm{u}\) percent abundance \(=92.50 \%\)

Step-by-Step Solution

Answer

The average atomic mass of lithium, 6.941 u, is correct based on isotope data.

Step 1: Understanding Percent Abundance

The percentage abundance of each isotope indicates how much of that isotope is present in a typical sample of lithium. For example, \(^{6} Li\) has an abundance of 7.500%, which means that out of 100 lithium atoms, approximately 7.5 atoms will be \(^{6} Li\). Similarly, \(^{7} Li\) has an abundance of 92.50%.

Step 2: Convert Percent to Decimal

Convert the percent abundance into a decimal for calculation. For \(^{6} Li\), the abundance is \(7.500\%\) or \(0.0750\) in decimal form. For \(^{7} Li\), the abundance is \(92.50%\) or \(0.9250\) in decimal form.

Step 3: Calculate Weighted Mass Contribution

Multiply the exact mass of each isotope by its decimal abundance to find its contribution to the average atomic mass. For \(^{6} Li\):\[6.015121 \, \text{u} \times 0.0750 = 0.4511341 \\]For \(^{7} Li\):\[7.016003 \, \text{u} \times 0.9250 = 6.4888028\\]

Step 4: Calculate Average Atomic Mass

Add up the contributions of each isotope to get the average atomic mass of lithium:\[0.4511341 \, \text{u} + 6.4888028 \, \text{u} = 6.9399369 \, \text{u}\\]Rounding this to three decimal places gives 6.940 u.

Step 5: Verification

Compare the calculated average atomic mass (6.940 u) with the given average ( 6.941 u). The calculated mass is very close to the provided data, verifying that the average atomic mass provided is accurate given the isotope data.

Key Concepts

Isotope Abundance

Isotope abundance, often expressed in percentages, tells us the distribution of different isotopes of an element found in nature. For example, if we look at lithium, it primarily exists as two isotopes:

• ^{6}Li with an abundance of 7.500%
• ^{7}Li with an abundance of 92.50%

This percent abundance helps us understand how common each isotope is within a given sample.
Every sample of lithium will have about 7.5% of ^{6}Li and 92.5% of ^{7}Li, highlighting the dominance of ^{7}Li in nature. This distribution is crucial when calculating the average atomic mass of lithium or any other element.

Weighted Average

The weighted average is essential when calculating the average atomic mass. This concept ensures that each isotope's mass is properly proportioned based on its abundance.
To compute the weighted average, convert percent abundances into their decimal forms by dividing by 100. This allows precise calculations:

• ^{6}Li abundance as decimal: 0.0750
• ^{7}Li abundance as decimal: 0.9250

Next, multiply the mass of each isotope by its decimal abundance:

• ^{6}Li: 6.015121 u × 0.0750 = 0.4511341 u
• ^{7}Li: 7.016003 u × 0.9250 = 6.4888028 u

Summing these products provides the average atomic mass:
6.9399369 u, rounded to 6.940 u. This demonstrates how significant isotope abundances are in calculating weighted averages.

Lithium Isotopes

Lithium, one of the lightest metals, exists naturally in two stable isotopes:

• ^{6}Li
• ^{7}Li

These isotopes vary by the number of neutrons. ^{6}Li contains three protons and three neutrons, while ^{7}Li contains three protons and four neutrons.
Despite the small difference in their mass numbers, ^{7}Li is significantly more abundant. It contributes more heavily to the average atomic mass of lithium.
This heavier isotope's predominance impacts physical properties and is crucial for applications such as nuclear fusion research and battery technology.

Mass Spectrometry

Mass spectrometry is an analytical technique that helps us determine isotope abundance and natural atomic masses. It separates isotopes based on their mass-to-charge ratio, allowing us to quantify and identify isotopes present in a sample.
For elements like lithium, this method can accurately measure the relative abundance of different isotopes such as ^{6}Li and ^{7}Li.
During mass spectrometry, the element's ions are generated, sorted, and detected, resulting in a spectrum that displays isotope abundance. This technology is integral in verifying isotope distribution and calculating precise average atomic masses of elements.