Iron has a reported average atomic mass of 55.85 amu. If iron has two isotopes, one of which has 29 neutrons and an isotopic abundance of 57.50%, what is the mass of the other isotope?
The Correct Answer and Explanation is :
To find the mass of the other isotope of iron, we can use the weighted average formula for atomic mass. The atomic mass of an element is a weighted average of the masses of its isotopes, where the abundance of each isotope is considered.
Step 1: Understand the given data
We are given:
- The average atomic mass of iron (Fe) = 55.85 amu.
- One isotope has 29 neutrons, with an isotopic abundance of 57.50%.
- We need to determine the mass of the other isotope.
Step 2: Determine the number of protons and neutrons in the first isotope
The atomic number of iron (Fe) is 26, meaning it has 26 protons. The first isotope has 29 neutrons, so its mass number (total number of protons and neutrons) is:
[
\text{Mass number} = 26 + 29 = 55 \, \text{amu}.
]
Thus, the first isotope has a mass of approximately 55 amu.
Step 3: Use the weighted average formula for atomic mass
The atomic mass is a weighted average of the two isotopes:
[
\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 (expressed as decimals),
- ( m_1 ) and ( m_2 ) are the masses of the isotopes.
We know:
- ( f_1 = 57.50\% = 0.575 ),
- ( m_1 = 55 \, \text{amu} ),
- The average atomic mass = 55.85 amu.
Since the total abundance is 100%, the abundance of the second isotope, ( f_2 ), is:
[
f_2 = 100\% – 57.50\% = 42.50\% = 0.425.
]
Now, we can use the weighted average formula to solve for the mass of the second isotope, ( m_2 ):
[
55.85 = (0.575 \times 55) + (0.425 \times m_2),
]
[
55.85 = 31.625 + (0.425 \times m_2),
]
[
55.85 – 31.625 = 0.425 \times m_2,
]
[
24.225 = 0.425 \times m_2,
]
[
m_2 = \frac{24.225}{0.425} = 57.
]
Step 4: Conclusion
The mass of the second isotope is approximately 57 amu.
Explanation
Isotopes are atoms of the same element with the same number of protons but a different number of neutrons, which leads to a different mass. In this case, the average atomic mass of iron (55.85 amu) is a weighted average of the masses of two isotopes, one with a mass of 55 amu and the other with a mass of 57 amu. By using the known isotopic abundance and atomic mass, we can calculate the mass of the second isotope.