Question

Two isotopes of copper are naturally occurring, with \(\frac{63}{29} \mathrm{Cu}\) at \(69.09 \%(62.93 \mathrm{amu})\) and \({ }_{29}^{65} \mathrm{Cu}\) at \(30.91 \%(64.93 \mathrm{amu})\). What is the atomic mass of copper?

Short answer

The atomic mass of copper is 63.54 amu.

Step-by-step solution

- Identify Given Data

Identify the isotopes and their respective abundances and atomic masses. Isotope 1: \ \ \ \ \( \frac{63}{29} \mathrm{Cu} \) \ - Abundance: \( 69.09 \% \) \ - Atomic Mass: \( 62.93 \ amu \) \ \ \ \ Isotope 2: \ \ \( {}_{29}^{65} \mathrm{Cu} \) \ - Abundance: \( 30.91 \% \) \ - Atomic Mass: \( 64.93 \ amu \)

- Convert Percentages to Decimals

Convert the percentage abundances of each isotope to decimals by dividing by 100.\ \ Isotope 1: \( 69.09 \% \) becomes \( 0.6909 \) \ \ Isotope 2: \( 30.91 \% \) becomes \( 0.3091 \)

- Calculate the Weighted Average

Calculate the atomic mass of copper by multiplying the relative abundance of each isotope by its atomic mass and then summing these values. \ \ Weighted Average = \( 0.6909 \times 62.93 \ \text{amu} \ \) + \( 0.3091 \times 64.93 \ \text{amu} \)

- Perform Multiplication

Perform the multiplication for each isotope. \ \ Isotope 1: \( 0.6909 \times 62.93 \ \approx 43.47 \ \text{amu} \) \ \ Isotope 2: \( 0.3091 \times 64.93 \ \approx 20.07 \ \text{amu} \)

- Sum the Values

Add the calculated values from each isotope to find the atomic mass of copper. \ Weighted Average = \( 43.47 \ \text{amu} \) + \( 20.07 \ \text{amu} \) \ = \( 63.54 \ \text{amu} \)

Key concepts

isotopes of copper

Copper is a chemical element that has more than one form, known as isotopes. Isotopes of an element have the same number of protons but different numbers of neutrons. For copper, there are two naturally occurring isotopes: \( \frac{63}{29} \mathrm{Cu} \) and \( {}_{29}^{65} \mathrm{Cu} \). These isotopes are identified by their mass numbers, which are written as superscripts (63 and 65 in this case).

The first isotope is \( \frac{63}{29} \mathrm{Cu} \), which means it has 29 protons and 34 neutrons, giving it a mass number of 63. The second isotope is \( {}_{29}^{65} \mathrm{Cu} \), with 29 protons and 36 neutrons, making its mass number 65.

Isotopes are important because they affect the atomic mass calculation, which we'll explore in more detail below.

weighted average

The atomic mass of an element that has more than one isotope is calculated as a weighted average. A weighted average takes into account both the mass and the relative abundance of each isotope.

This means that we do not simply average the masses of the isotopes; instead, we multiply the mass of each isotope by its relative abundance (expressed as a decimal) and then sum these values.

For example, to find the atomic mass of copper, we follow these steps:

• Multiply the atomic mass of each isotope by its relative abundance
• Add the resulting values

This weighted average method gives us a more accurate representation of the atomic mass of an element as it exists in nature.

relative abundance

Relative abundance is the percentage of a particular isotope present in a natural sample of an element. It is usually expressed as a percentage but needs to be converted to a decimal for calculations.

For copper, the relative abundances of its isotopes are:

• \( \frac{63}{29} \mathrm{Cu} \) - 69.09%
• \( {}_{29}^{65} \mathrm{Cu} \) - 30.91%

To use these percentages in calculations, we convert them to decimals by dividing by 100:

• \( \frac{63}{29} \mathrm{Cu} \) - 69.09% becomes 0.6909
• \( {}_{29}^{65} \mathrm{Cu} \) - 30.91% becomes 0.3091

This step is essential for calculating the weighted average and, ultimately, the atomic mass of copper.

atomic mass unit (amu)

The atomic mass unit (amu) is the standard unit used to express the masses of atoms and molecules. One amu is defined as one-twelfth of the mass of a carbon-12 atom.

This unit is very useful in atomic mass calculations because it allows us to express the mass of atoms in a way that is easy to compare and work with.

In the context of our exercise, the atomic masses of \( \frac{63}{29} \mathrm{Cu} \) and \( {}_{29}^{65} \mathrm{Cu} \) are given in amu units (62.93 amu and 64.93 amu, respectively).

By performing weighted average calculations with these atomic masses, we can find the overall atomic mass of copper, which is expressed in the same unit: amu. This standardized unit makes it straightforward to communicate and calculate atomic mass values across different elements and isotopes.