Calculate the atomic mass of titanium. The five titanium isotopes have atomic masses and relative abundances of 45.953 amu (8.00%), 46.952 amu (7.30%), 47.948 amu (73.80%), 48.948 amu (5.50%), and 49.945 amu (5.40%).
The Correct Answer and Explanation is:
To calculate the atomic mass of titanium, we use the formula for the weighted average of the atomic masses of its isotopes, based on their relative abundances. The formula is:Atomic mass=∑(isotope mass×abundance fraction)\text{Atomic mass} = \sum \left( \text{isotope mass} \times \text{abundance fraction} \right)Atomic mass=∑(isotope mass×abundance fraction)
The abundances need to be expressed as fractions, so we divide the percentage by 100. Let’s apply this to the five isotopes of titanium provided:
- Titanium-45:
- Atomic mass = 45.953 amu
- Abundance = 8.00% = 0.0800
- Contribution to atomic mass = 45.953×0.0800=3.6762445.953 \times 0.0800 = 3.6762445.953×0.0800=3.67624
- Titanium-46:
- Atomic mass = 46.952 amu
- Abundance = 7.30% = 0.0730
- Contribution to atomic mass = 46.952×0.0730=3.4284946.952 \times 0.0730 = 3.4284946.952×0.0730=3.42849
- Titanium-47:
- Atomic mass = 47.948 amu
- Abundance = 73.80% = 0.7380
- Contribution to atomic mass = 47.948×0.7380=35.3979847.948 \times 0.7380 = 35.3979847.948×0.7380=35.39798
- Titanium-48:
- Atomic mass = 48.948 amu
- Abundance = 5.50% = 0.0550
- Contribution to atomic mass = 48.948×0.0550=2.6881448.948 \times 0.0550 = 2.6881448.948×0.0550=2.68814
- Titanium-49:
- Atomic mass = 49.945 amu
- Abundance = 5.40% = 0.0540
- Contribution to atomic mass = 49.945×0.0540=2.6961049.945 \times 0.0540 = 2.6961049.945×0.0540=2.69610
Now, summing up these contributions:Atomic mass=3.67624+3.42849+35.39798+2.68814+2.69610=47.88695 amu\text{Atomic mass} = 3.67624 + 3.42849 + 35.39798 + 2.68814 + 2.69610 = 47.88695 \, \text{amu}Atomic mass=3.67624+3.42849+35.39798+2.68814+2.69610=47.88695amu
Therefore, the atomic mass of titanium is approximately 47.89 amu.
Explanation:
The atomic mass of an element is a weighted average of the masses of its naturally occurring isotopes, where the weight of each isotope is determined by its relative abundance. In this case, titanium has five isotopes with different masses and relative abundances. By multiplying the mass of each isotope by its fractional abundance, we get the contribution of each isotope to the total atomic mass. Summing these contributions gives the atomic mass of titanium.
