Problem 31 - Chemistry: The Central Science

Question

Only two isotopes of copper occur naturally, \({ }^{63} \mathrm{Cu}\) (atomic mass \(=62.9296\) amu; abundance \(69.17 \%\) ) and \({ }^{65} \mathrm{Cu}\) (atomic mass \(=64.9278\) amu; abundance \(\left.30.83 \%\right)\). Calculate the atomic weight (average atomic mass) of copper.

Step-by-Step Solution

Answer

The atomic weight (average atomic mass) of copper is approximately 63.5650 amu. This is calculated by converting the given abundances to decimals, multiplying the atomic masses of both isotopes by their respective decimal abundances, and then summing the weighted atomic masses of both isotopes.

Step 1: Identify known values

The known values in this exercise are the atomic masses and abundances of the two copper isotopes: 1. \({ }^{63} \mathrm{Cu}\): atomic mass = 62.9296 amu, abundance = 69.17% 2. \({ }^{65} \mathrm{Cu}\): atomic mass = 64.9278 amu, abundance = 30.83%

Step 2: Convert percentages to decimals

To use the abundances in calculations, we need to convert the percentages to decimals. To do this, divide each percentage by 100. 1. 69.17% = 0.6917 2. 30.83% = 0.3083

Step 3: Calculate the weighted atomic mass for each isotope

Multiply the atomic mass of each isotope by its respective decimal abundance. 1. Weighted atomic mass of \({ }^{63} \mathrm{Cu}\) = 62.9296 amu × 0.6917 = 43.5500 amu 2. Weighted atomic mass of \({ }^{65} \mathrm{Cu}\) = 64.9278 amu × 0.3083 = 20.0150 amu

Step 4: Calculate the atomic weight (average atomic mass) of copper

Add the weighted atomic masses of both isotopes together to find the atomic weight of copper: Atomic weight = Weighted atomic mass of \({ }^{63} \mathrm{Cu}\) + Weighted atomic mass of \({ }^{65} \mathrm{Cu}\) Atomic weight = 43.5500 amu + 20.0150 amu = 63.5650 amu The atomic weight (average atomic mass) of copper is approximately 63.5650 amu.

Key Concepts

Atomic Mass

Atomic mass refers to the mass of an isotope of an element, usually expressed in atomic mass units (amu). Each isotope of an element has a different number of neutrons, affecting its mass. For example, copper has isotopes like \({ }^{63} \mathrm{Cu} \) and \({ }^{65} \mathrm{Cu} \), each with specific atomic masses of 62.9296 amu and 64.9278 amu, respectively. Understanding atomic mass is critical because it serves as a foundation for calculating the average atomic mass, which is more reflective of the element's general weight as found in nature.

• Atomic mass is specific to each isotope.
• Expressed in atomic mass units (amu).
• Affects the calculation of average atomic mass.

Grasping the concept of atomic mass is essential for understanding the composition and behavior of chemical elements.

Natural Abundance

Natural abundance refers to the relative proportion of an isotope as it occurs naturally on Earth. In other words, it is the percentage of a particular isotope among all isotopes of that element found in nature. For copper, the isotopes \(^{63} \mathrm{Cu}\) and \(^{65} \mathrm{Cu}\) have natural abundances of 69.17% and 30.83%, respectively. These percentages indicate how prevalent each isotope is in a natural sample of copper.

• Essential for calculating average atomic mass.
• Expressed as a percentage, which must be converted to a decimal for calculations.
• Indicates the isotopic composition of an element's natural sample.

Understanding natural abundance helps chemists and students predict and calculate the properties of elements as they exist in the natural world.

Average Atomic Mass

Average atomic mass is the weighted average of all the isotopes of an element, taking into account their natural abundances and atomic masses. It is what you typically see on the periodic table as the atomic weight of the element.To calculate it, you must:

• Convert the natural abundances from percentages to decimals.
• Multiply each isotope's atomic mass by its decimal abundance for the weighted mass.
• Add these weighted masses together to get the average atomic mass.

For copper, the calculation involves using \(^{63} \mathrm{Cu}\) and \(^{65} \mathrm{Cu}\), leading to an average atomic mass of approximately 63.5650 amu. This average gives a comprehensive picture of the element's overall mass in natural samples and is vital for accurate scientific calculations and predictions.

Copper Isotopes

Copper isotopes, such as \(^{63} \mathrm{Cu}\) and \(^{65} \mathrm{Cu}\), are variants of the copper atom that have different numbers of neutrons. Although they have the same number of protons, which defines the element, the difference in neutrons gives them different atomic masses. These isotopes are key to understanding copper's place in nature and are critical for calculating copper's average atomic mass. Copper has two main isotopes:

• \(^{63} \mathrm{Cu}\): 62.9296 amu with 69.17% abundance.
• \(^{65} \mathrm{Cu}\): 64.9278 amu with 30.83% abundance.

These isotopes contribute differently to the average atomic mass because of their varied abundances, impacting copper's overall chemical properties and reactions. Understanding copper isotopes is of great importance for fields such as material science and chemistry, where precise measurements are critical.