Problem 53 - Chemistry: An Atoms-Focused Approach

Question

Copper in nature is a mixture of \(69.17 \%\) copper- 63 \((62.9296 \text { amul })\) and \(30.83 \%\) copper \(-65(64.9278 \text { amu }) .\) Use this information to calculate the average atomic mass of copper.

Step-by-Step Solution

Answer

Answer: The average atomic mass of copper is approximately 63.5334 amu.

Step 1: Write down the given information

Copper has two isotopes, copper-63 and copper-65. Their masses and relative abundances are given as follows: - Copper-63: mass = \(62.9296 \,\text{amu}\) and abundance = \(69.17 \%\) - Copper-65: mass = \(64.9278 \,\text{amu}\) and abundance = \(30.83 \%\)

Step 2: Convert percentages into decimals

To use the relative abundances in calculations, we need to convert the percentage values into decimal values. We do this by dividing each percentage by 100. - Copper-63: \(69.17 \% = \frac{69.17}{100} = 0.6917\) - Copper-65: \(30.83 \% = \frac{30.83}{100} = 0.3083\)

Step 3: Calculate the weighted average

Now that we have the decimal values of the relative abundances, we can calculate the weighted average atomic mass of copper. We do this by multiplying each isotope's mass by its relative abundance and then summing up the results. Average Atomic Mass of Copper \(= (0.6917 * 62.9296) + (0.3083 * 64.9278)\)

Step 4: Perform the calculation

Now we simply need to evaluate the expression: \(= (0.6917 * 62.9296) + (0.3083 * 64.9278)\) \(= 43.5068 + 20.0266\) \(= 63.5334\)

Step 5: Write down the result

The average atomic mass of copper is approximately 63.5334 amu.

Key Concepts

Isotopes

Atoms of the same element can have different numbers of neutrons. These variants are known as isotopes. Isotopes have the same number of protons, which defines the element, but they differ in their neutron count. This difference leads to varying atomic masses among isotopes of a single element. In the case of copper, there are two naturally occurring isotopes, copper-63 and copper-65. While both have 29 protons, copper-63 has 34 neutrons and copper-65 has 36 neutrons.

These isotopes can have slightly different physical properties due to their mass difference. However, chemically, they behave the same since chemical properties are determined mostly by electron configuration rather than atomic mass. Copper exists as a mixture of these isotopes in nature, contributing to its overall atomic mass.

Weighted Average

The concept of a weighted average is crucial when dealing with mixtures of isotopes like copper. A weighted average takes into account not just the value of each item, in this case, the atomic mass of each isotope, but also how prevalent or abundant each item is, represented by their respective abundances in nature.

To calculate the weighted average atomic mass, you multiply the atomic mass of each isotope by its fractional abundance. Fractional abundance is simply the percentage abundance converted to a decimal. For copper's naturally occurring isotopes:

• Copper-63: 62.9296 amu, 69.17% which is 0.6917 as a decimal, and
• Copper-65: 64.9278 amu, 30.83% which is 0.3083 as a decimal.

The formula used is \[\text{Average Atomic Mass} = (\text{mass of isotope 1} \times \text{fractional abundance of isotope 1}) + (\text{mass of isotope 2} \times \text{fractional abundance of isotope 2})\] resulting in the final average atomic mass due to these isotopic contributions.

Atomic Mass Unit (amu)

The atomic mass unit, abbreviated as amu, is a standard unit of mass used to express atomic and molecular weights. It's defined as one twelfth of the mass of a carbon-12 atom, which is approximately equal to 1.66 x 10\(^{-24}\) grams. This unit is useful because it provides a convenient scale for dealing with the tiny masses of atoms and molecules.

In the context of isotopic calculations, like those of copper, the amu allows scientists to effectively communicate and compare atomic masses. For example, when we say copper-63 has a mass of 62.9296 amu and copper-65 has a mass of 64.9278 amu, these values are in terms of this standardized unit. These precise measurements are crucial for calculating the weighted average to find the overall atomic mass of the element copper.