Determining the Abundances of Rubidium's Isotopes

Rubidium, an element with an atomic number of 37, naturally exists as a mixture of two isotopes: 85Rb and 87Rb. The atomic masses of these isotopes are 84.9117 u and 86.9085 u, respectively. The average atomic mass of rubidium is given as 85.4678 u. This article explains the process to determine the abundance of each isotope using the given information.

Understanding the Problem

Given:

Mass of 85Rb: 84.9117 u Mass of 87Rb: 86.9085 u Average atomic mass of rubidium: 85.4678 u

We need to find the relative abundance of each isotope. The concept of atomic mass and percentage abundance is central to this calculation. The formula used to find the average atomic mass is based on the product of the mass of each isotope and its abundance.

Formulating the Equation

The average atomic mass can be represented as:

average mass (abundance of Rb-85 * mass of Rb-85) (abundance of Rb-87 * mass of Rb-87)

Let x fraction of Rb-85. Then, the fraction of Rb-87 is (1 - x).

The equation for the average atomic mass is:

85.4678 x * 84.9117 (1 - x) * 86.9085

Solving the Equation

Substituting the known values:

85.4678 x * 84.9117 (1 - x) * 86.9085

Expanding the equation:

85.4678 84.9117x 86.9085 - 86.9085x

Combining like terms:

85.4678 86.9085 - 2.9968x

Rearranging to solve for x:

2.9968x 86.9085 - 85.4678

2.9968x 1.4407

x ≈ 0.480

This means the abundance of Rb-85 is approximately 48.0%.

To find the abundance of Rb-87:

1 - x 1 - 0.480 ≈ 0.520

So the abundance of Rb-87 is approximately 52.0%.

Summary of Abundances

Rb-85: ~48.0%

Rb-87: ~52.0%

Alternative Methods to Determine Isotope Abundances

Another approach can be used to arrive at the same result. We can set up the equation in terms of y, where y is the relative abundance of one isotope. Let:

y relative abundance of Rb-85 100 - y relative abundance of Rb-87

The average atomic mass equation becomes:

84.9117y 86.9085(100 - y) 85.4678 * 100

Solving for y:

84.9117y 8690.85 - 86.9085y 8546.78

1.9968y 144.07

y ≈ 72.15

Thus, the relative abundance of Rb-85 is approximately 72.15%, and that of Rb-87 is approximately 27.85%.

Conclusion

By understanding and applying the principles of atomic mass and relative abundance, we can accurately determine the isotopic composition of rubidium. Both methods yield consistent results, providing a comprehensive approach to solving similar problems.