Question

Three isotopes of argon occur in nature- \(\frac{36}{18} \mathrm{Ar}, \frac{38}{18} \mathrm{Ar}\) , and \(_{18}^{40} \mathrm{Ar}\) . Calculate the average atomic mass of argon to two decimal places, given the following relative atomic masses and abundances of each of the isotopes: argon \(-36(35.97 \mathrm{u} ; 0.337 \%),\) argon \(-38(37.96 \mathrm{u} ; 0.063 \%)\) and argon \(-40(39.96 \mathrm{u} ; 99.600 \%)\)

Short answer

The average atomic mass of argon is 39.94 u.

Step-by-step solution

- Identify the given information

List the relative atomic masses and their abundances for each isotope.- Argon-36: Relative atomic mass = 35.97 u, Abundance = 0.337%- Argon-38: Relative atomic mass = 37.96 u, Abundance = 0.063%- Argon-40: Relative atomic mass = 39.96 u, Abundance = 99.600%

- Convert the abundances from percentages to decimals

Divide the percentage abundances by 100 to convert them to decimal form.- Argon-36: 0.337% = 0.00337- Argon-38: 0.063% = 0.00063- Argon-40: 99.600% = 0.99600

- Calculate the contribution of each isotope to the average atomic mass

Multiply the relative atomic mass of each isotope by its abundance (in decimal form).- Contribution from Argon-36: 35.97 u × 0.00337 = 0.1211189 u- Contribution from Argon-38: 37.96 u × 0.00063 = 0.0231148 u- Contribution from Argon-40: 39.96 u × 0.99600 = 39.80016 u

- Sum the contributions

Add the contributions from each isotope to find the total average atomic mass of argon.Total Average Atomic Mass = 0.1211189 u + 0.0231148 u + 39.80016 u = 39.9443937 u

- Round to two decimal places

Round the total average atomic mass to two decimal places.Rounded Average Atomic Mass = 39.94 u

Key concepts

isotopes

Isotopes are different forms of the same element. They have the same number of protons but different numbers of neutrons. This results in different atomic masses. For argon, we have three isotopes: \(\frac{36}{18} Ar\), \(\frac{38}{18} Ar\), and \(\frac{40}{18} Ar\). They are chemically identical but differ in mass due to their neutron count. Understanding isotopes is essential for calculating average atomic mass.

relative atomic mass

Relative atomic mass refers to the mass of an isotope compared to one-twelfth of the mass of carbon-12. For example, argon-36 has a relative atomic mass of 35.97 u. This number is essential when averaging the atomic masses of isotopes. Each isotope's mass contributes depending on its relative abundance.

abundance

Abundance tells us how much of each isotope exists compared to the total amount of the element. It's usually given as a percentage. For instance, argon-40 has an abundance of 99.600%. Knowing the abundance is necessary for determining the average atomic mass. We convert these percentages to decimals for calculations by dividing by 100.

decimal conversion

When dealing with percentages in scientific calculations, converting to decimals makes the math easier. To do this, divide the percentage by 100. For argon-36, an abundance of 0.337% becomes 0.00337. This step is crucial for simplifying your calculations later on.

step-by-step solution

Let's walk through the solution:
1. Identify the given information about each isotope's relative atomic mass and abundance.
2. Convert the percentage abundances to decimal form.
3. Calculate each isotope's contribution by multiplying its relative atomic mass with its abundance. For instance, 35.97 u for argon-36 is multiplied by 0.00337 to get 0.1211189 u.
4. Add up all the contributions to find the total average atomic mass. Summing the values gives us 39.9443937 u.
5. Round the total to two decimal places to get 39.94 u.
By following these steps, you can easily calculate the average atomic mass of an element with multiple isotopes.