Problem 49 - General Chemistry: Principles and Modern Applications

Question

A mass spectrum of germanium displayed peaks at mass numbers \(70,72,73,74,\) and \(76,\) with relative heights of \(20.5,27.4,7.8,36.5,\) and \(7.8,\) respectively. (a) In the manner of Figure \(2-14,\) sketch this mass spectrum. (b) Estimate the weighted-average atomic mass of germanium, and state why this result is only approximately correct.

Step-by-Step Solution

Answer

The weighted-average atomic mass of germanium is approximately 73.6 atomic mass units.

Step 1: Sketching the Mass Spectrum

Draw the axis of the graph. The x-axis represents the mass numbers and the y-axis represents the relative abundances. Plot the given points \((70,20.5), (72,27.4), (73,7.8), (74,36.5)\), and \((76,7.8)\). Connect the points with vertical lines from the x-axis.

Step 2: Calculating the Weighted Average

Calculate the relative percentage for each mass number by dividing the relative height by the sum of all relative heights. Multiply each relative percentage by its corresponding mass number. This gives: \((20.5/100)*70+(27.4/100)*72+(7.8/100)*73+(36.5/100)*74+(7.8/100)*76.\)

Step 3: Sum to find the Weighted Average

Add up the products to obtain the estimated weighted-average atomic mass of germanium.

Key Concepts

Weighted-average atomic mass

The weighted-average atomic mass of an element is a calculated average that factors in the masses of the isotopes of the element and their relative abundances. It is not simply the average of the isotopic masses, but rather a sum that considers how much each isotope contributes to the total. This is crucial because elements like germanium have several naturally occurring isotopes, each with different masses and abundances.
The formula for calculating the weighted-average atomic mass is: \[ \text{Weighted-average atomic mass} = \sum (\text{mass of the isotope} \times \text{relative abundance of the isotope}) \]

• First, determine the percentage abundance of each isotope.
• Then, multiply the mass number of each isotope by its relative abundance in decimal form.
• Finally, sum the products to get the total weighted-average atomic mass.

This method ensures that the atomic mass accurately reflects the contribution of each isotope. It’s a key concept in understanding the atomic structure and properties of elements.

Germanium isotopes

Germanium is an interesting element because it naturally occurs in several different isotopic forms. Its mass spectrum typically shows peaks corresponding to different mass numbers: 70, 72, 73, 74, and 76. Each of these numbers represents a unique isotope of germanium, which are forms of the element that differ in their number of neutrons.
These isotopes are:

• Germanium-70 with mass number 70
• Germanium-72 with mass number 72
• Germanium-73 with mass number 73
• Germanium-74 with mass number 74
• Germanium-76 with mass number 76

Each isotope has its own set of natural abundance levels, which is why the mass spectrum exhibits different peak heights. Understanding the isotopic composition helps in applications ranging from materials science to semiconductor technology.

Relative abundance

Relative abundance is a measure of how common or rare an isotope is within a sample of an element. It is usually expressed as a percentage, indicating how much of a substance consists of a specific isotope compared to the total amount of the element.
In the context of the mass spectrum for germanium, the relative abundance of each isotope is reflected in the height of the peaks:

• Germanium-70 has a relative abundance of 20.5%.
• Germanium-72 is more abundant at 27.4%.
• Germanium-73 and Germanium-76 both share the same low relative abundance of 7.8%.
• Germanium-74 has the highest relative abundance at 36.5%.

These percentages are critical for calculating the weighted-average atomic mass and for understanding the physical and chemical behavior of germanium. They highlight the balance of isotopes that contribute to the element's overall properties.