Question

Although naturally occurring potassium consists mostly of the isotope of mass number \(39(93.25 \%)\), isotopes of mass number \(41(6.73 \%)\) and \(40(0.01 \%)\) also are present. Write the nuclear symbol for each of the isotopes of potassium. How many neutrons are present in each isotope? Is the average atomic mass of potassium ( \(39.10 \mathrm{~g}\) ) consistent with the relative abundances of the isotopes?

Short answer

The nuclear symbols for the isotopes of potassium are: Isotope 1: \(_{19}^{39}\textrm{K}\) with 20 neutrons Isotope 2: \(_{19}^{41}\textrm{K}\) with 22 neutrons Isotope 3: \(_{19}^{40}\textrm{K}\) with 21 neutrons The calculated weighted average atomic mass of potassium is 39.098 g, which is consistent with the given average atomic mass of 39.10 g.

Step-by-step solution

Write the nuclear symbol for each isotope of potassium

First, we need to write the nuclear symbol for each isotope of potassium. The atomic number of potassium (K) is 19, which represents the number of protons. The isotopes have mass numbers 39, 41, and 40. So, the nuclear symbols for these isotopes are: Isotope 1: \(_{19}^{39}\textrm{K}\) Isotope 2: \(_{19}^{41}\textrm{K}\) Isotope 3: \(_{19}^{40}\textrm{K}\)

Calculate the number of neutrons in each isotope

To calculate the number of neutrons in each isotope, we subtract the number of protons (atomic number) from the mass number: Isotope 1 (\(_{19}^{39}\textrm{K}\)): Number of neutrons = 39-19 = 20 Isotope 2 (\(_{19}^{41}\textrm{K}\)): Number of neutrons = 41-19 = 22 Isotope 3 (\(_{19}^{40}\textrm{K}\)): Number of neutrons = 40-19 = 21

Check if the average atomic mass of potassium matches the given information

To check if the average atomic mass of potassium is consistent with the isotopes' relative abundance, we have to compare the given average atomic mass (39.10 g) with the calculated weighted average atomic mass, using the given relative abundances: Average atomic mass = (mass_isotope_1 x relative_abundance_1) + (mass_isotope_2 x relative_abundance_2) + (mass_isotope_3 x relative_abundance_3) Calculating the weighted average atomic mass, we get: Average atomic mass = (39 x 0.9325) + (41 x 0.0673) + (40 x 0.0001) ≈ 39.098 Comparing the given average atomic mass (39.10 g) with our calculated weighted average atomic mass (39.098 g), we can conclude that the average atomic mass of potassium is consistent with the relative abundances of the isotopes.

Key concepts

Nuclear Symbols

Nuclear symbols help identify isotopes of an element and provide crucial information about atomic composition. A nuclear symbol is written in a specific format:

• The element's symbol is in the center, representing the chemical element.
• The atomic number, which is the number of protons, is written as a subscript on the lower left.
• The mass number, representing the sum of protons and neutrons, is on the upper left as a superscript.

Based on the given exercise, potassium, denoted by the symbol 'K', has an atomic number of 19. The isotopes of potassium with different mass numbers are presented using nuclear symbols:

• \( _{19}^{39}\textrm{K} \) for the isotope with mass number 39
• \( _{19}^{41}\textrm{K} \) for mass number 41
• \( _{19}^{40}\textrm{K} \) for mass number 40

These symbols encapsulate vital information about each isotope of potassium, making it easier to distinguish among them.

Neutron Calculation

Understanding how to calculate the number of neutrons in an isotope is essential in chemistry and physics. Neutrons are neutral particles found in an atom's nucleus along with protons. The mass number of an isotope equals the total number of protons and neutrons.

To find the number of neutrons in a given isotope, you subtract the atomic number (the number of protons) from the mass number. This formula can be expressed as: \[\text{Number of neutrons} = \text{Mass number} - \text{Atomic number}\]Applying this to the potassium isotopes:

• For \( _{19}^{39}\textrm{K} \), the neutron calculation is 39 - 19 = 20
• For \( _{19}^{41}\textrm{K} \), we get 41 - 19 = 22
• For \( _{19}^{40}\textrm{K} \), it is 40 - 19 = 21

These calculations help us understand not just the basic composition of each isotope but also the subtle differences that give each one its unique properties.

Average Atomic Mass

The average atomic mass of an element is an important concept that provides insights into the element's composition in nature. It represents the weighted average of all the isotopes' mass numbers, adjusted according to their natural abundance.

To calculate the average atomic mass, you need to multiply the mass of each isotope by its relative abundance (expressed as a decimal), and then sum up these values for all isotopes. The formula looks like this: \[\text{Average atomic mass} = (\text{mass isotope 1} \times \text{abundance 1}) + (\text{mass isotope 2} \times \text{abundance 2}) + \cdots\]For potassium, the calculation, using the given isotopic masses and abundances, is:

• \(39\times 0.9325 = 36.4075\)
• \(41\times 0.0673 = 2.7593\)
• \(40\times 0.0001 = 0.0040\)

Adding these results gives an average atomic mass of approximately 39.098, which aligns closely with the provided value of 39.10. This match indicates consistency, reflecting the accurate distribution of isotopes in naturally occurring potassium.