To find the natural abundance of each isotope, we can set up a system of equations using the known atomic masses and the average atomic mass of silver. Let the natural abundance of $^{107}\text{Ag}$ be $x$ (expressed as a fraction), and the abundance of $^{109}\text{Ag}$ will be $1 - x$ since the total abundance must equal 100%.

Given:

• Mass of $^{107}\text{Ag} = 106.90$ u
• Mass of $^{109}\text{Ag} = 108.90$ u
• Average atomic mass of silver = 107.87 u

Step 1: Set up the equation for the average atomic mass

The average atomic mass is a weighted average of the two isotopes based on their natural abundances. This can be written as:

$\text{Average atomic mass} = (x \times 106.90) + [(1 - x) \times 108.90]$

Substitute the given average atomic mass:

$107.87 = (x \times 106.90) + [(1 - x) \times 108.90]$

Step 2: Simplify the equation

$107.87 = 106.90x + 108.90(1 - x)$ $107.87 = 106.90x + 108.90 - 108.90x$ $107.87 = 108.90 - 2x$

Step 3: Solve for $x$

$107.87 - 108.90 = -2x$ $-1.03 = -2x$ $x = \frac{1.03}{2} = 0.515$

So, the natural abundance of $^{107}\text{Ag}$ is approximately 0.515, or 51.5%.

Step 4: Calculate the natural abundance of $^{109}\text{Ag}$

Since the total abundance must add up to 100%:

$1 - 0.515 = 0.485$

So, the natural abundance of $^{109}\text{Ag}$ is approximately 0.485, or 48.5%.

Final Answer:

• $^{107}\text{Ag}$: 51.5%
• $^{109}\text{Ag}$: 48.5%

Would you like more details on any step?

Here are 5 related questions you might find interesting:

1. How do we derive the equation for a weighted average of isotopes?
2. What are the physical implications of different isotopes of an element?
3. How are the atomic masses of elements determined experimentally?
4. Can isotopic composition affect the chemical properties of an element?
5. How does mass spectrometry help in determining isotopic abundances?

Tip: When solving weighted average problems, always ensure that the sum of the fractional abundances equals 1 (or 100% if expressed as percentages).