Given data

The key idea is Average Molar Mass Calculation. The molar mass of naturally occurring argon is the weighted average of the molar masses of its isotopes, with each isotope's contribution proportional to its fractional abundance.

1. Convert the given percentage abundances to fractional abundances:

   For $^{36}Ar$: $0.337\% = 0.00337$

   For $^{38}Ar$: $0.063\% = 0.00063$

   For $^{40}Ar$: $99.600\% = 0.99600$

2. Calculate the weighted sum of the isotopic molar masses:

   Molar Mass $= (35.96755\ g\ mol^{-1} \times 0.00337) + (37.96272\ g\ mol^{-1} \times 0.00063) + (39.9624\ g\ mol^{-1} \times 0.99600)$

3. Perform the multiplication and summation: …

When we refer to the molar mass of an element, such as argon, we are typically talking about the average molar mass of its naturally occurring isotopes. Elements in nature are often a mixture of different isotopes, which are atoms of the same element with different numbers of neutrons and thus different masses. The periodic table lists these average molar masses.

To calculate this average molar mass, we use a weighted average. Each isotope's molar mass is "weighted" by its relative abundance in nature. This means that isotopes present in higher percentages contribute more to the overall average molar mass.

Here's how to calculate the molar mass of naturally occurring argon:

1. Understand the given data:

   We are provided with the isotopic molar mass and the natural abundance for each of argon's three main isotopes:

   Isotope	Isotopic molar mass ($M_i$)	Abundance ($P_i$)
   $^{36}Ar$	$35.96755\ g\ mol^{-1}$	$0.337\%$
   $^{38}Ar$	$37.96272\ g\ mol^{-1}$	$0.063\%$
   $^{40}Ar$	$39.9624\ g\ mol^{-1}$	$99.600\%$

2. Convert percentage abundances to fractional abundances:

   For the weighted average calculation, we need to express the abundances as decimal fractions (where $100\%$ corresponds to $1$). To do this, divide each percentage by $100$.

   • For $^{36}Ar$: $f_{36} = \frac{0.337}{100} = 0.00337$
   • For $^{38}Ar$: $f_{38} = \frac{0.063}{100} = 0.00063$
   • For $^{40}Ar$: $f_{40} = \frac{99.600}{100} = 0.99600$
   Tip

   Always check that your fractional abundances sum up to $1$ (or very close to $1$ due to rounding). In this case, $0.00337 + 0.00063 + 0.99600 = 1.00000$, which confirms our conversions are correct.

3. Apply the weighted average formula:

   The average molar mass ($M_{avg}$) is calculated by summing the products of each isotope's molar mass ($M_i$) and its corresponding fractional abundance ($f_i$).

   $M_{avg} = \sum (M_i \times f_i)$

   For argon, this expands to:

   $M_{avg} = (M_{36} \times f_{36}) + (M_{38} \times f_{38}) + (M_{40} \times f_{40})$

4. Calculate the contribution of each isotope:

   Multiply the isotopic molar mass by its fractional abundance for each isotope.

   • For $^{36}Ar$: $35.96755\ g\ mol^{-1} \times 0.00337 = 0.121170085\ g\ mol^{-1}$
   • For $^{38}Ar$: …

Method: Weighted Average (Isotopic Abundance) Method

This is the standard method for calculating the average atomic mass (or molar mass) of an element from its naturally occurring isotopes.

Steps

1. Convert each abundance percentage to a decimal fraction

   Divide each percentage by 100.

   • $^{36}\text{Ar}$: $0.337\% \rightarrow 0.00337$
   • $^{38}\text{Ar}$: $0.063\% \rightarrow 0.00063$
   • $^{40}\text{Ar}$: $99.600\% \rightarrow 0.99600$

2. Multiply each isotopic mass by its decimal abundance

   This gives the "contribution" of each isotope to the average.

   • $^{36}\text{Ar}$: $35.96755 \times 0.00337 = 0.12121$
   • $^{38}\text{Ar}$: $37.96272 \times 0.00063 = 0.02392$
   • $^{40}\text{Ar}$: $39.9624 \times 0.99600 = 39.8026$

3. Sum all contributions

   $\text{Molar mass} = 0.12121 + 0.02392 + 39.8026$ …

Here are the common mistakes students make when calculating the molar mass of naturally occurring argon isotopes from the given data, along with how to avoid each.

1. Forgetting to convert percentage abundance to decimal form

Mistake:

Using the abundance as a whole number (e.g., 0.337 instead of 0.00337) in the weighted average formula.

Why it’s wrong:

The formula for average atomic mass is:

Fractional abundance = $\frac{\text{percentage}}{100}$.

How to avoid:

Always divide each percentage by 100 before multiplying. Write the fractional abundances clearly:

• $^{36}\text{Ar}$: $0.00337$
• $^{38}\text{Ar}$: $0.00063$
• $^{40}\text{Ar}$: $0.99600$

2. Using the wrong number of significant figures

Mistake:

Rounding intermediate steps too early or reporting the final answer with too many/few decimal places.

Why it’s wrong:

The isotopic masses are given to 5–6 decimal places. The final molar mass should reflect the precision of the most abundant isotope (here, $^{40}\text{Ar}$ has 4 decimal places).

How to avoid:

Carry at least 5 decimal places in all intermediate calculations. Round the final answer to 4 decimal places (matching the least precise isotopic mass).

3. Misplacing the decimal point when multiplying

Mistake:

A calculation like $35.96755 \times 0.00337$ is done incorrectly (e.g., forgetting the leading zeros in 0.00337).

Why it’s wrong:

This leads to an answer that is off by a factor of 10 or 100.

How to avoid:

Write the multiplication step explicitly and check the number of decimal places. Use a calculator carefully, or break it down:

4. Adding the contributions without checking the sum of abundances

Mistake:

Assuming the percentages add to 100% without verifying.

Why it’s wrong:

If the sum is not exactly 100%, the weighted average will be slightly off.

How to avoid:

Quickly check: $0.337 + 0.063 + 99.600 = 100.000\%$ — it’s correct here. Always do this check before calculating.

5. Confusing “isotopic molar mass” with “mass number”

Mistake:

Using 36, 38, and 40 (the mass numbers) instead of the given precise isotopic masses. …