Problem 21 - Chemistry and Chemical Reactivity

Question

Verify that the atomic mass 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 mass of lithium is \(6.94\,\text{u}\), confirming the given value.

Step 1: Understand the Concept of Atomic Mass

The atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes. It is calculated using the formula \( \text{Atomic Mass} = \sum (\text{Isotope Mass} \times \text{Fractional Abundance}) \).

Step 2: Convert Percent Abundance to Fractional Abundance

To use the formula, convert the given percent abundance of each isotope into fractional abundance by dividing by 100. For \(^{6}\text{Li}\), the fractional abundance is \(0.0750\), and for \(^{7}\text{Li}\), it is \(0.9250\).

Step 3: Calculate the Contribution of Each Isotope

Multiply the mass of each isotope by its fractional abundance to find its contribution to the atomic mass. For \(^{6}\text{Li}\), the contribution is \(6.015121 \times 0.0750 = 0.4511\,\text{u}\). For \(^{7}\text{Li}\), the contribution is \(7.016003 \times 0.9250 = 6.4898\,\text{u}\).

Step 4: Sum the Contributions

Add the contributions from all isotopes to get the atomic mass of lithium. This sum is \(0.4511 + 6.4898 = 6.9409\,\text{u}\).

Step 5: Compare with the Given Atomic Mass

The calculated atomic mass is \(6.9409\,\text{u}\), which is consistent with the expected atomic mass of lithium, \(6.94\,\text{u}\). Therefore, the calculation verifies that the atomic mass of lithium is \(6.94\).

Key Concepts

Isotopes

Atoms of the same element can have different numbers of neutrons in their nuclei. These variants are called isotopes. Isotopes of an element have the same number of protons but different numbers of neutrons.
This leads to differences in their atomic mass.
For instance, lithium has two naturally occurring isotopes: \( ^{6}\text{Li} \) and \( ^{7}\text{Li} \).
\( ^{6}\text{Li} \) has 3 protons and 3 neutrons, while \( ^{7}\text{Li} \) has 3 protons and 4 neutrons. These differences do not affect the chemical behavior, as the chemical properties are determined by the number of protons.

Isotopes matter when calculating the atomic mass of an element. Since different isotopes have different masses, their presence affects the overall atomic mass of the element.

Percent Abundance

When we talk about isotopes, we also need to understand what percent abundance means.
Percent abundance refers to the percentage of each isotope found in a naturally occurring sample of an element.
For lithium, \( ^{6}\text{Li} \) has a percent abundance of 7.50%, and \( ^{7}\text{Li} \) is much more common at 92.50%.

To calculate the atomic mass, percent abundance needs to be converted into fractional abundance:

• For \(^{6}\text{Li}\): \( 7.50\% = 0.0750 \)
• For \(^{7}\text{Li}\): \( 92.50\% = 0.9250 \)

This conversion is essential because calculations in chemistry often require these values as fractions.

Weighted Average

The atomic mass of an element is not just a simple average of its isotopes because each isotope does not contribute equally. Some are more abundant than others.
This is where the concept of a weighted average comes in.
In this context, a weighted average takes into account both the mass and the fractional abundance of each isotope.

The formula used is:\[\text{Atomic Mass} = \sum (\text{Isotope Mass} \times \text{Fractional Abundance})\]For lithium:

• \( ^{6}\text{Li} \): \( 6.015121 \times 0.0750 = 0.4511 \, \text{u} \)
• \( ^{7}\text{Li} \): \( 7.016003 \times 0.9250 = 6.4898 \, \text{u} \)

Adding these gives the atomic mass \( 0.4511 \, \text{u} + 6.4898 \, \text{u} = 6.9409 \, \text{u} \).
The result is very close to the accepted atomic mass for lithium, which is 6.94 u.

Lithium

Lithium is a soft, silvery-white metal that belongs to the alkali metal group.
It is the third element on the periodic table, with the chemical symbol Li.
Lithium has two naturally occurring isotopes: \(^{6}\text{Li} \) and \(^{7}\text{Li} \), with \(^{7}\text{Li} \) being the more common isotope.

This makes lithium unusual because most elements have more evenly distributed isotopes.
Its light weight and reactivity make lithium a valuable resource in various applications, from batteries to medication.

• Lithium batteries: Known for their high energy density and efficiency.
• Medication: Used to treat bipolar disorder.

Understanding lithium's atomic structure and isotope composition is crucial for applications leveraging its unique properties.