Question

Naturally occurring copper is a mixture of \(69.17 \%\) Cu-63 with a mass of 62.93 amu and \(30.83 \%\) Cu-65 with a mass of 64.93 amu. What is the atomic mass of copper?

Short answer

The atomic mass of copper is approximately 63.54 amu.

Step-by-step solution

Understand the Problem

We need to calculate the average atomic mass of copper, considering its isotopic composition: Cu-63 and Cu-65. Atomic mass is calculated using the isotopic masses and their relative abundances.

Write the Formula

The formula to calculate the atomic mass (\( \text{Atomic Mass} = (\text{Fraction of Cu-63} \times \text{Mass of Cu-63}) + (\text{Fraction of Cu-65} \times \text{Mass of Cu-65}) \) .

Convert Percentages to Fractions

Convert the percentages of each isotope into fractions by dividing them by 100: - Cu-63: \( \frac{69.17}{100} = 0.6917 \) - Cu-65: \( \frac{30.83}{100} = 0.3083 \).

Substitute Values into the Formula

Substitute the fractions and the isotopic masses into the formula: \[ \text{Atomic Mass} = (0.6917 \times 62.93) + (0.3083 \times 64.93) \].

Perform Calculations

Calculate the contributions of each isotope to the atomic mass: - Cu-63 contribution: \( 0.6917 \times 62.93 = 43.5341 \) - Cu-65 contribution: \( 0.3083 \times 64.93 = 20.0044 \).

Add Contributions

Add the contributions of each isotope to find the atomic mass of copper: \[ \text{Atomic Mass} = 43.5341 + 20.0044 = 63.5385 \, \text{amu} \].

Key concepts

Isotopic Composition

Copper in nature exists as a mix of different isotopes. Isotopes are atoms with the same number of protons but a different number of neutrons. This means they have different masses. For copper, the two main isotopes are Cu-63 and Cu-65. Isotopic composition refers to the proportion of each isotope present in a sample. Understanding this composition is crucial because it allows us to calculate properties like the average atomic mass by considering both the mass and abundance of each isotope present.

Percent to Fraction Conversion

When calculating the average atomic mass, it's essential to use fractions rather than percentages. This is because the formula for atomic mass requires the fractional abundance of each isotope. To convert a percentage to a fraction, simply divide the percentage by 100.
For instance:

• Cu-63 has an abundance of 69.17%, which converts to the fraction 0.6917 by dividing by 100.
• Cu-65 has an abundance of 30.83%, which converts to the fraction 0.3083.

Converting percentages to fractions ensures the total abundance sums up to 1, which is a key step in determining the correct average atomic mass.

Average Atomic Mass

The average atomic mass of an element is a weighted average of the masses of its isotopes, based on their natural abundance. To find it, we multiply the mass of each isotope by its fractional abundance and then sum these values.
The formula is:

• \( \text{Atomic Mass} = (\text{Fraction of Cu-63} \times \text{Mass of Cu-63}) + (\text{Fraction of Cu-65} \times \text{Mass of Cu-65}) \)

Performing these calculations gives us the average atomic mass, providing an understanding of how much each isotope contributes to the atomic mass of the element. This average helps chemists and scientists in their calculations related to copper and its compounds.

Copper Isotopes

Copper, often denoted as Cu in the periodic table, has two stable isotopes: Cu-63 and Cu-65. Each isotope has unique properties due to differences in their neutron count. Cu-63, with a mass of 62.93 amu, is more abundant in nature compared to Cu-65, which has a mass of 64.93 amu. This difference in mass and abundance affects the overall calculated atomic mass of copper.
The existence of these isotopes is taken into account when determining the atomic mass listed in the periodic table for copper, providing a reliable and average value that scientists can use in further studies and applications.