Calculate the atomic mass of an element if 60.4% of the atoms have a mass of 68.9257 amu and the rest have a mass of 70.9249 amu:
The Correct Answer and Explanation is:
To calculate the atomic mass of an element given its isotopic composition, we use the weighted average formula:Atomic Mass=(f1×m1)+(f2×m2)\text{Atomic Mass} = (f_1 \times m_1) + (f_2 \times m_2)Atomic Mass=(f1×m1)+(f2×m2)
Where:
- f1f_1f1 and f2f_2f2 are the fractional abundances of the isotopes
- m1m_1m1 and m2m_2m2 are the masses of the isotopes
Given:
- 60.4% of the atoms (or 0.604 as a fraction) have a mass of 68.9257 amu
- The rest (100% – 60.4% = 39.6%, or 0.396 as a fraction) have a mass of 70.9249 amu
Substituting into the formula:Atomic Mass=(0.604×68.9257)+(0.396×70.9249)\text{Atomic Mass} = (0.604 \times 68.9257) + (0.396 \times 70.9249)Atomic Mass=(0.604×68.9257)+(0.396×70.9249)Atomic Mass=41.6471+28.0943=69.7414 amu\text{Atomic Mass} = 41.6471 + 28.0943 = \boxed{69.7414 \text{ amu}}Atomic Mass=41.6471+28.0943=69.7414 amu
Explanation
The atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes. Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons, resulting in different masses.
In this problem, the element has two isotopes. One isotope has a mass of 68.9257 amu and makes up 60.4 percent of the natural abundance. The other isotope has a mass of 70.9249 amu and makes up the remaining 39.6 percent. To find the atomic mass, we convert these percentages into decimal fractions because the formula for atomic mass requires fractional abundances. This gives us 0.604 and 0.396 respectively.
We then multiply each isotopic mass by its respective fractional abundance. This gives us the contribution of each isotope to the average atomic mass. For the lighter isotope, the contribution is approximately 41.6471 amu. For the heavier isotope, the contribution is approximately 28.0943 amu.
Adding these two contributions gives a final atomic mass of about 69.7414 amu. This value represents the average mass of a single atom of the element, considering how often each isotope occurs in nature.
Atomic masses listed on the periodic table are calculated using this method. They reflect a weighted average, not a simple mean, and this is why atomic masses often include decimal places even though the masses of individual isotopes are whole numbers or close to them.
