Question

Argon has three naturally occurring isotopes: \(0.3336 \%\) \({ }^{36} \mathrm{Ar}, 0.063 \%{ }^{38} \mathrm{Ar}\), and \(99.60 \%{ }^{40} \mathrm{Ar}\). Estimate the average atomic mass of argon. If the masses of the isotopes are \(35.968 \mathrm{u}, 37.963 \mathrm{u},\) and \(39.962 \mathrm{u},\) respectively, calculate the average atomic mass of natural argon.

Short answer

The average atomic mass of argon is approximately 39.945 u.

Step-by-step solution

Understand the problem

To find the average atomic mass of argon, we need to use the isotopic composition and the masses of each isotope. The average atomic mass is a weighted average, where each isotope's mass is multiplied by its abundance and then all these products are summed.

Convert percentages to decimals

Convert each of the isotopic abundances from a percentage to a decimal for calculation. For \(^{36}\mathrm{Ar}\), use \(0.3336\% = 0.003336\); for \(^{38}\mathrm{Ar}\), \(0.063\% = 0.00063\); and for \(^{40}\mathrm{Ar}\), \(99.60\% = 0.996\).

Calculate contribution of each isotope

Calculate the contribution to the average atomic mass for each isotope. Multiply the mass of each isotope by its abundance (in decimal form). The contributions are:- \(^{36}\mathrm{Ar}: 35.968 \mathrm{u} \times 0.003336 = 0.1199 \mathrm{u}\)- \(^{38}\mathrm{Ar}: 37.963 \mathrm{u} \times 0.00063 = 0.0239 \mathrm{u}\)- \(^{40}\mathrm{Ar}: 39.962 \mathrm{u} \times 0.996 = 39.8012 \mathrm{u}\)

Sum up the contributions

Add up all the contributions from each isotope to find the weighted average atomic mass of argon. Sum: \(0.1199 \mathrm{u} + 0.0239 \mathrm{u} + 39.8012 \mathrm{u} = 39.945 \mathrm{u}\).

Key concepts

Atomic Mass Calculation

To grasp how we calculate the atomic mass of an element like argon, we need to understand what an atomic mass represents. The atomic mass of an element is a measurement that reflects the mass of all its isotopes, considering their respective abundances in nature. Isotopes are different forms of an element's atoms that have the same number of protons but a different number of neutrons. This variation in neutron number leads to different atomic masses for each isotope.

Calculating atomic mass essentially involves finding a weighted average of the isotopes' masses, which gives us a value that reflects the mass of an atom of the element as it naturally occurs. In our example, integrating each isotope's mass with its natural abundance allows us to estimate the average atomic mass of argon accurately.

Natural Abundance

Natural abundance refers to the relative proportion of each isotope present in nature. In simpler terms, it tells us how often each isotope occurs naturally, usually expressed as a percentage. Because the atomic mass of an element incorporates all its isotopes, knowing their natural abundance is crucial to ensuring our calculations accurately reflect real-world atoms.

In the case of argon, we see three isotopes:

• the abundance of ³⁶ Ar is 0.3336%,
• the abundance of ³⁸ Ar is 0.063%,
• and finally, the abundance of ⁴⁰ Ar is 99.60%.

These percentages indicate that ⁴⁰ Ar is the most prevalent isotope of argon. To use these abundances effectively in calculations, we first convert them from percentages to decimals. This conversion allows consistent use in the mathematical operations needed to calculate the weighted average mass.

Weighted Average

The concept of a weighted average is a mathematical technique used to find an average that accounts for different levels of contribution or frequency from each component. When dealing with isotopes, not all isotopes occur in equal proportions. Thus, each one's mass must be weighed by its natural abundance to calculate the atomic mass accurately.

For argon, we calculate a weighted average by multiplying each isotope's mass by its decimal abundance. Then, we sum these values to find the total average. This process can be outlined with the formula:

• ³⁶ Ar: 35.968 u × 0.003336 = 0.1199 u,
• ³⁸ Ar: 37.963 u × 0.00063 = 0.0239 u,
• ⁴⁰ Ar: 39.962 u × 0.996 = 39.8012 u.

Adding these contributions gives us the final weighted average atomic mass: 39.945 u. This represents the average atomic mass of argon, reflecting its isotopic makeup in the natural world.