Question

Gallium (Ga) consists of two naturally occurring isotopes with masses of \(68.926\) and \(70.925\) amu. (a) How many protons and neutrons are in the nucleus of each isotope? Write the complete atomic symbol for each, showing the atomic number and mass number. (b) The average atomic mass of Ga is \(69.72\) amu. Calculate the abundance of each isotope.

Short answer

Gallium (Ga) has an atomic number of 31, so both of its naturally occurring isotopes have 31 protons. The two isotopes have mass numbers 69 and 71, so the first isotope (^69Ga_31) has 38 neutrons and the second isotope (^71Ga_31) has 40 neutrons. Using the average atomic mass (69.72 amu) and isotope masses (68.926 and 70.925 amu), we can calculate that the abundance of ^69Ga_31 is 60.25% and the abundance of ^71Ga_31 is 39.75%.

Step-by-step solution

Determine the Atomic Number of Gallium

To find the atomic number (Z) of an element, we can look it up in the periodic table. For Gallium (Ga), the atomic number is 31.

Calculate the Number of Protons and Neutrons in Each Isotope

Since Gallium has an atomic number (Z) of 31, both isotopes have 31 protons. To find the number of neutrons for each isotope, we can round the mass numbers (A) to the nearest whole numbers and subtract the number of protons: For the first isotope (mass = 68.926 amu): Number of neutrons = Mass number (A) - Number of protons = 69 - 31 = 38 neutrons. For the second isotope (mass = 70.925 amu): Number of neutrons = Mass number (A) - Number of protons = 71 - 31 = 40 neutrons.

Write the Complete Atomic Symbols

For each isotope, we will write the atomic symbol with the atomic number (Z) as a subscript to the left and the mass number (A) as a superscript to the left. For the first isotope (68.926 amu): Isotope symbol = ^69Ga_31 For the second isotope (70.925 amu): Isotope symbol = ^71Ga_31

Calculate the Abundance of Each Isotope

We can use the average atomic mass of Gallium (69.72 amu) and the masses of the two isotopes (68.926 amu and 70.925 amu) to find the relative abundance of each isotope. Let x be the abundance of the first isotope, and (1-x) be the abundance of the second isotope. Then, the equation for the average atomic mass is: - Average Atomic Mass = (mass of first isotope × abundance of first isotope) + (mass of second isotope × abundance of second isotope) - 69.72 = (68.926 × x) + (70.925 × (1-x)) We can now solve this equation for x: - 69.72 = 68.926x + 70.925 - 70.925x - 69.72 - 70.925 = -1.999x - x = 0.6025 The abundance of the first isotope (^69Ga_31) is 60.25%, and the abundance of the second isotope (^71Ga_31) is 100 - 60.25 = 39.75%.