Gallium has two naturally occurring isotopes, 69Ga and 71Ga. The average atomic mass of galllium is 69.70 amu. Which of the following statements concerning the relative abundance of each isotope is correct?
a) The abundance of 71Ga is greater than 69Ga.
b) The abundance of 71Ga is less than 69Ga.
c) The abundance of 71Ga is exactly equal to the abundance of 69Ga.
d) Based on the atomic mass, only 69Ga occurs naturally.
e) Not enough data is provided to determine the correct answer.
The correct answer and explanation is :
The correct answer is:
b) The abundance of ⁷¹Ga is less than ⁶⁹Ga.
Explanation:
The atomic mass of an element is the weighted average of the atomic masses of its naturally occurring isotopes, considering their relative abundances. This is given by:
[
\text{Average Atomic Mass} = (f_1 \times m_1) + (f_2 \times m_2)
]
where:
- ( f_1 ) and ( f_2 ) are the fractional abundances of the isotopes,
- ( m_1 ) and ( m_2 ) are the atomic masses of the isotopes.
For gallium:
- Isotope ⁶⁹Ga has a mass of 68.93 amu,
- Isotope ⁷¹Ga has a mass of 70.92 amu,
- The average atomic mass of gallium is 69.70 amu.
Since 69.70 amu is closer to 68.93 amu than to 70.92 amu, this indicates that 69Ga is more abundant than 71Ga. If both isotopes were equally abundant, the average atomic mass would be closer to the midpoint between the two values (around 69.92 amu). However, since it is lower than that, it suggests that 69Ga has a higher abundance than 71Ga.
Mathematically, we can estimate the relative abundances:
Let ( x ) be the fraction of 69Ga, so the fraction of 71Ga is ( 1 – x ).
[
(68.93x) + (70.92(1-x)) = 69.70
]
Solving for ( x ), we find that the abundance of ⁶⁹Ga is about 60.1% and the abundance of ⁷¹Ga is about 39.9%.
Thus, the abundance of 71Ga is less than that of 69Ga, making option (b) the correct choice.