Question

Naturally occurring zirconium exists as five stable isotopes: \({ }^{90} \mathrm{Zr}\) with a mass of \(89.905 \mathrm{u}(51.45 \%) ;{ }^{91} \mathrm{Zr}\) with a mass of \(90.906 \mathrm{u}\) \((11.22 \%) ;{ }^{92} \mathrm{Zr}\) with a mass of \(91.905 \mathrm{u}(17.15 \%) ;{ }^{94} \mathrm{Zr}\) with a mass of \(93.906 \mathrm{u}(17.38 \%)\); and \({ }^{96} \mathrm{Zr}\) with a mass of \(95.908 \mathrm{u}\) \((2.80 \%)\). Calculate the average mass of zirconium.

Short answer

The average mass of zirconium is approximately 91.224 u.

Step-by-step solution

Determine the Weighted Mass of Each Isotope

To find the average atomic mass of zirconium, calculate the weighted mass of each isotope by multiplying the mass of each isotope by its relative natural abundance. Remember to convert the percentages to fractions by dividing by 100.

Calculate the Total Average Mass

Sum the weighted masses of all the isotopes to obtain the total average atomic mass of zirconium.

Express the Result

The average atomic mass of zirconium is the total average mass calculated in Step 2.

Key concepts

Isotopes of Elements

Isotopes are versions of an element's atoms that differ in the number of neutrons in their nuclei, leading to different mass numbers. However, isotopes of an element still have the same number of protons, which defines the element itself. For example, the element carbon has several isotopes, including carbon-12 and carbon-14, where the numbers indicate the mass number of each isotope. Isotopes can be stable or unstable (radioactive).

Understanding isotopes is crucial in fields such as chemistry, physics, and geology, and they have practical applications in dating archaeological finds, medical diagnostic techniques, and nuclear power generation.

Relative Atomic Mass

The relative atomic mass, also known as the atomic weight, is an average mass of the atoms of an element, measured in atomic mass units (u), and it takes into account the mass and abundance of each isotope present in a naturally occurring sample. Unlike the mass number, which is an integer and represents the sum of protons and neutrons, the relative atomic mass often has a decimal value and reflects the isotopic composition of the element as it is found on Earth. The relative atomic mass is essential in calculating molar masses and performing stoichiometric calculations in chemistry.

Weighted Average

A weighted average accounts for the varying significance of individual values, assigning more weight to some than others. It's particularly relevant when calculating the average atomic mass of an element, as each isotope's mass is multiplied by its relative abundance in the natural world. This method ensures the result represents the average mass of an atom that one would likely encounter. To calculate this weighted average, each weight (or natural abundance) needs to be expressed as a decimal before multiplication.

The concept of a weighted average is widely used beyond chemistry, including in statistics, finance, and economics, because it provides a more accurate reflection of the 'real-world' average when uniform distribution cannot be assumed.

Natural Abundance

Natural abundance refers to the proportion of a particular isotope relative to the total number of isotopes for an element found in nature. Isotopes occur in varying amounts, often depicted as percentages. This natural distribution must be understood to determine the average atomic mass accurately. For instance, if an element has one very common isotope and several rare ones, the common isotope will heavily influence the element's average atomic mass. This concept is invaluable when scientists determine the relative atomic mass of elements and it highlights that not all atoms of an element are identical in mass.