Question

Gallium, which has an atomic mass of 69.723 amu, has two naturally occurring isotopes, Ga-69 and Ga-71. Which isotope occurs in greater abundance? Explain.

Short answer

The isotope Ga-69 occurs in greater abundance since its fraction (0.6385) is larger than that of Ga-71 (0.3615). This is determined by considering the atomic mass of Gallium (69.723 amu) and using the weighted average of the masses of its isotopes, Ga-69 and Ga-71.

Step-by-step solution

Identify the information given

We know the following information: 1. Atomic mass of Gallium (Ga) = 69.723 amu (atomic mass units) 2. Two naturally occurring isotopes: Ga-69 and Ga-71

Determine the abundance of each isotope

Let x be the fraction of Ga-69 and (1-x) be the fraction of Ga-71 in the natural sample of Gallium. Given that the atomic mass is the weighted average of the masses of the isotopes, we can write an equation as follows: 69.723 = 69x + 71(1-x)

Solve for the fraction 'x'

To solve the equation, follow these steps: 1. Distribute 71: \(69.723 = 69x + 71 - 71x\) 2. Simplify and combine like terms: \(69.723 - 71 = -2x\) 3. Divide by -2: \(x = \frac{69.723 - 71}{-2} = 0.6385\)

Compare the abundance of each isotope

Since we solved for the fraction x, we can compare the abundance of each isotope: 1. Fraction of Ga-69 (x) = 0.6385 2. Fraction of Ga-71 (1-x) = 1 - 0.6385 = 0.3615 Since the fraction of Ga-69 (0.6385) is greater than the fraction of Ga-71 (0.3615), Ga-69 occurs in greater abundance.

Conclusion

Based on the calculation, Ga-69 is the isotope that occurs in greater abundance.

Key concepts

Understanding Atomic Mass

Atomic mass is an incredibly important concept in chemistry. It is not just a regular mass, but the average mass of all the isotopes of an element, taking into account their abundance. Each element is made of atoms, and these atoms can have different forms known as isotopes. For example, Gallium has isotopes like Ga-69 and Ga-71. These isotopes have varying masses due to differences in their number of neutrons.
To get the atomic mass of an element like Gallium, we don't just add up the masses of its isotopes and divide by the number of isotopes. Instead, we calculate a **weighted average**, combining these isotopic masses with their respective abundances.
For Gallium, this gives us an atomic mass of 69.723 amu (atomic mass units). Understanding this concept helps in predicting the element's behavior and chemical interactions.

How Weighted Average Works

Calculating the weighted average is essential when dealing with atomic masses of elements. Unlike a simple average, a weighted average takes into account the "weight" or significance of each number in the set.
In our example of Gallium, the atomic mass of 69.723 amu is determined by the masses of its isotopes, Ga-69 and Ga-71, as well as their respective abundances in nature.
To calculate this, you would multiply the mass of each isotope by its relative abundance. Then, you sum these values to get the atomic mass. Hence:

• Ga-69 contributes to the average mass based on its proportion in nature.
• Ga-71 does the same, but proportionally less, if its abundance is lower.

The weighted average ensures that more abundant isotopes have a greater impact on the atomic mass. This is why Ga-69's greater abundance results in a higher influence on the atomic mass.

The Meaning of Abundance

Abundance in chemistry refers to how common or pervasive an isotope is in nature. It is usually expressed as a fraction or a percentage of the total atoms of an element.
When calculating atomic mass, knowing the abundance of each isotope is crucial. In our Gallium example, Gallium-69 is more abundant than Gallium-71, with fractional abundances of 0.6385 and 0.3615 respectively.
This means that in any sample of Gallium, about 63.85% of the atoms will be Ga-69, while 36.15% will be Ga-71. The concept of abundance helps explain why certain isotopes contribute more or less to the atomic mass and affects how we use this element in scientific and industrial applications.

Gallium's Unique Properties

Gallium is a fascinating element with some unique properties influenced by its isotopic composition. With an average atomic mass of 69.723 amu, predominated by its more abundant isotope Ga-69, Gallium exhibits specific physical and chemical behaviors that are essential in various industries.
It melts just above room temperature, making it useful in high-temperature thermometers and LEDs. The predominance of Ga-69 means that this isotope largely defines Gallium’s interaction in these applications.
Though less abundant, Ga-71 also plays a role particularly in nuclear medicine and research. Understanding Gallium's isotopes allows scientists and engineers to exploit its unique properties effectively.