Question

As discussed in the chapter, the atomic mass of argon is greater than that of potassium. This observation created a problem in the early development of the periodic table because it meant that argon should be placed after potassium. (a) How was this difficulty resolved? (b) From the following data, calculate the average atomic masses of argon and potassium: Ar-36 (35.9675 amu, 0.337 percent), \(\mathrm{Ar}-38(37.9627 \mathrm{amu}, 0.063\) percent \()\) Ar- \(40(39.9624\) amu, 99.60 percent), \(\mathrm{K}-39(38.9637\) amu, 93.258 percent \(), \mathrm{K}-40(39.9640 \mathrm{amu}, 0.0117\) percent \()\) K-41 \((40.9618\) amu, 6.730 percent).

Short answer

The issue was resolved by using atomic number order in the periodic table, and the calculated average atomic masses are approximately 39.9485 amu for argon and 39.1071 amu for potassium.

Step-by-step solution

Understand the Issue

The atomic mass of argon is about 39.948 amu, while that of potassium is approximately 39.0983 amu. According to atomic mass, argon should come before potassium, but this contradicts the order based on chemical properties in the periodic table.

Resolving the Difficulty

The difficulty was resolved by recognizing that the periodic table is arranged by atomic number rather than atomic mass. Argon has a lower atomic number of 18 compared to potassium, which has an atomic number of 19. Thus, argon is placed before potassium in the table.

Calculate Weighted Average of Argon's Atomic Mass

Use the formula for average atomic mass, which is a weighted average based on isotopic abundance.\[\text{Average atomic mass of Ar} = (35.9675 \times 0.00337) + (37.9627 \times 0.00063) + (39.9624 \times 0.9960)\]

Compute Weighted Average of Argon's Atomic Mass

Calculate each term separately:- For \(\text{Ar-36}\): \(35.9675 \times 0.00337 = 0.1212 \text{ amu}\)- For \(\text{Ar-38}\): \(37.9627 \times 0.00063 = 0.0239 \text{ amu}\)- For \(\text{Ar-40}\): \(39.9624 \times 0.9960 = 39.8034 \text{ amu}\)Add these results to find the weighted average: \[0.1212 + 0.0239 + 39.8034 = 39.9485 \text{ amu}\]

Calculate Weighted Average of Potassium's Atomic Mass

Apply the same weighted average formula for potassium:\[\text{Average atomic mass of K} = (38.9637 \times 0.93258) + (39.9640 \times 0.000117) + (40.9618 \times 0.06730)\]

Compute Weighted Average of Potassium's Atomic Mass

Calculate each contribution:- For \(\text{K-39}\): \(38.9637 \times 0.93258 = 36.3456 \text{ amu}\)- For \(\text{K-40}\): \(39.9640 \times 0.000117 = 0.0047 \text{ amu}\)- For \(\text{K-41}\): \(40.9618 \times 0.06730 = 2.7568 \text{ amu}\)Add these to find the weighted average:\[36.3456 + 0.0047 + 2.7568 = 39.1071 \text{ amu}\]

Compare Results

The calculated average atomic masses are approximately 39.9485 amu for argon and 39.1071 amu for potassium. These results show that argon does indeed have a greater atomic mass than potassium, consistent with the standard values.

Key concepts

Isotopic Abundance

Isotopic abundance is a crucial concept in understanding atomic mass. Elements can exist in different forms known as isotopes, which have the same number of protons but differ in the number of neutrons. These isotopes contribute differently to the overall atomic mass of an element because they have different masses.

To calculate the average atomic mass of an element, we use isotopic abundance. This value reflects the proportion of each isotope found in nature. Mathematically, the average atomic mass is a weighted average. It takes into account the mass of each isotope and its respective abundance. For example, in the case of argon, we have isotopes Ar-36, Ar-38, and Ar-40, each contributing a different percentage to the total atomic mass.

• Ar-36: Mass is 35.9675 amu, with an abundance of 0.337%
• Ar-38: Mass is 37.9627 amu, with an abundance of 0.063%
• Ar-40: Mass is 39.9624 amu, with an abundance of 99.60%

Understanding how isotopic abundance affects atomic mass helps us see why calculated values may differ from simple averages.

Periodic Table Arrangement

The arrangement of the periodic table is based on atomic number, not atomic mass. When the periodic table was first developed, some elements appeared out of order if arranged by atomic mass alone. This caused confusion since elements like argon and potassium would switch places if atomic mass was the organizing principle.

The problem was resolved by understanding that chemical properties are more closely linked to atomic numbers. The atomic number of an element represents the number of protons in the nucleus, which determines its chemical behavior. For instance, although argon has a higher atomic mass than potassium, it is placed before potassium in the periodic table because argon’s atomic number is 18 and potassium’s is 19.

• Atomic Number of Argon: 18
• Atomic Number of Potassium: 19

This principle ensures that the periodic table accurately reflects element relationships and properties.

Argon Atomic Mass

Argon’s atomic mass is a weighted average based on its isotopic composition. The majority of argon's atomic mass comes from the most abundant isotope, Ar-40. The calculation involves multiplying the mass of each isotope by its abundance and summing these values.

Here’s how it works for argon:

• For Ar-36: Multiply its mass (35.9675 amu) by its abundance (0.337%), giving approximately 0.1212 amu.
• For Ar-38: Multiply its mass (37.9627 amu) by its abundance (0.063%), resulting in about 0.0239 amu.
• For Ar-40: Multiply its mass (39.9624 amu) by its abundance (99.60%), which is the largest contribution, providing about 39.8034 amu.

Add these contributions to get the total atomic mass of argon, approximately 39.9485 amu. This calculated atomic mass matches closely with the standard atomic mass of argon.

Potassium Atomic Mass

Much like argon, potassium’s atomic mass is derived from its isotopic abundances. Potassium primarily consists of three isotopes: K-39, K-40, and K-41. Each contributes to the overall atomic mass of potassium based on its prevalence and specific mass.

The calculation process for potassium involves:

• K-39: With a mass of 38.9637 amu and abundance of 93.258%, it provides a major part of the atomic mass, about 36.3456 amu.
• K-40: Though rare, contributing only 0.0117% abundance, it adds a small amount of 0.0047 amu.
• K-41: Having a mass of 40.9618 amu and 6.730% abundance, it adds significantly, about 2.7568 amu.

When these contributions are combined, the weighted average atomic mass of potassium is approximately 39.1071 amu. This highlights how isotopic compositions influence the average atomic mass of elements.