Problem 35 - Chemistry The Central Science

Question

Only two isotopes of copper occur naturally: 63 Cu (atomic mass \(=62.9296\) amu; abundance 69.17\(\%\) and 65 (atomic mass \(=64.9278\) amu; abundance 30.83\(\%\) ). Calculate the atomic weight (average atomic mass) of copper.

Step-by-Step Solution

Answer

The average atomic mass (atomic weight) of copper is approximately 63.56 amu.

Step 1: Convert percentages to decimals

First, we need to convert the given abundances of the copper isotopes from percentages to decimals. To do this, just divide the percentages by 100. 63Cu Abundance (in decimal) = 69.17% / 100 = 0.6917 65Cu Abundance (in decimal) = 30.83% / 100 = 0.3083

Step 2: Calculate the atomic weight of copper

Now that we have the abundances as decimals, we can plug the values into the formula mentioned above: \(Atomic\:weight = (f_{63Cu} * m_{63Cu}) + (f_{65Cu} * m_{65Cu})\) Plugging the given values: Atomic weight = (0.6917 * 62.9296 amu) + (0.3083 * 64.9278 amu) Calculate the result: Atomic weight = 43.5281952 amu + 20.0323334 amu Atomic weight = 63.5605286 amu

Step 3: Round the result

Now, we'll round the atomic weight to an appropriate number of decimal places. In this case, rounding to two decimal places seems appropriate: Atomic weight (rounded) = 63.56 amu So, the average atomic mass (atomic weight) of copper is approximately 63.56 amu.

Key Concepts

Isotopic Abundance

Understanding isotopic abundance is crucial to grasping the concept of average atomic mass. Isotopic abundance refers to the percentage of a specific isotope present in a natural sample of an element. Each isotope of an element has a slightly different mass, but the abundance of each isotope contributes to the overall average atomic mass that we encounter on the periodic table. For instance, if an element has two isotopes that occur naturally, the one with higher abundance will have a more significant impact on the average atomic mass calculation.

When these abundances are given as percentages, they need to be converted into decimals before they can be used in calculations. This is simply done by dividing the percentage value by 100. The resulting decimal represents the fraction of the element that is composed of that isotope and is essential for accurate atomic weight determination.

Atomic Weight Determination

Determining atomic weight is a fundamental practice in chemistry that involves calculating the weighted average mass of all the naturally occurring isotopes of an element. This weight is not just a simple average, but rather a weighted one that takes into account the isotopic abundances, which signifies how commonly each isotope occurs in nature.

The formula for atomic weight can be illustrated as follows: \[Atomic\ weight = (f_1 * m_1) + (f_2 * m_2) + ... + (f_n * m_n)\], where \(f\) represents the fraction (isotopic abundance in decimal form) of each isotope and \(m\) denotes the mass of each isotope. When plugging in the numbers, it’s imperative to maintain precision with the decimal places, and rounding off should only be performed at the final step to ensure accurate results.

Copper Isotopes

Copper is an excellent example to illustrate the concept of atomic weight determination because it has two isotopes, namely Copper-63 and Copper-65, with their specific masses and distinct isotopic abundances. Copper-63 has a mass of 62.9296 amu and is more abundant, making up about 69.17% of natural copper, while Copper-65 has a mass of 64.9278 amu and comprises about 30.83%.

To calculate the average atomic mass of copper, it is necessary to account for the contribution of both isotopes based on their masses and respective natural abundances. The exercise involves multiplying the abundance in decimal form by the isotopic mass and then summing these products to yield the atomic weight. Accuracy in this process is paramount because the smallest errors in calculation can lead to incorrect results. This precise atomic weight, found on the periodic table, is fundamental to various applications in science, from understanding the properties of copper to conducting accurate chemical reactions.

Applying the Concept to Copper

The detailed calculation shown in the exercise provides a clear approach to this determination. Initially, we start with the percentage abundances which are converted into fractions, and these fractions are then utilized alongside the isotopic masses to compute the atomic weight, which in the case of copper, is approximately 63.56 amu after proper rounding. This demonstrates the practical aspect of isotopic abundances and atomic weight calculation in chemistry.