Problem 44 - General Chemistry: Principles and Modern Applications

Question

\begin{tabular}{l} There are four naturally occurring isotopes of \\ \hline \end{tabular} chromium. Their masses and percent natural abundances are \(49.9461 \mathrm{u}, 4.35 \% ; 51.9405 \mathrm{u}, 83.79 \% ; 52.9407 \mathrm{u}\) \(9.50 \% ;\) and \(53.9389 \mathrm{u}, 2.36 \% .\) Calculate the weightedaverage atomic mass of chromium.

Step-by-Step Solution

Answer

The weighted average atomic mass of chromium is approximately \(51.996 \mathrm{u}\)

Step 1: Understand the concept of weighted average

A weighted average is typically used when the average of a list of numbers must account for different levels of significance or importance of the numbers. In this case, the different masses of the isotopes are not all equally common, indicated by their different percent abundances.

Step 2: Convert the percentages into decimals

Before proceeding, one should convert the percent natural abundances into decimal form. This is done by simply dividing each percentage by 100. Thus, these will become: 0.0435, 0.8379, 0.0950, and 0.0236, respectively.

Step 3: Multiply each mass by its fraction and sum the products

To find the weighted average atomic mass of chromium, multiply each isotope's atomic mass by its abundance, then sum these products. The calculations are as follows: \((49.9461 \mathrm{u} * 0.0435) + (51.9405 \mathrm{u} * 0.8379) + (52.9407 \mathrm{u} * 0.0950) + (53.9389 \mathrm{u} * 0.0236)\)

Step 4: Calculate

Performing these calculations gives a sum of \(51.996 \mathrm{u}\)

Key Concepts

Isotopes

Atoms of the same element can have different numbers of neutrons, and these variations are known as isotopes. Isotopes possess the same number of protons but can differ in mass due to the difference in the number of neutrons. For example, chromium has four naturally occurring isotopes, as mentioned in the problem statement, each with a unique mass.

Understanding isotopes is critical as they can vary widely in abundance. This is why the resulting atomic mass of an element is not a simple average but rather a weighted one, taking into account each isotope's occurrence. This means that the element's atomic mass that appears on the periodic table comes from these natural variations in isotope mass and abundance.

Weighted Average

When dealing with isotopes, it's important to understand how to calculate a weighted average. In real-world scenarios, not all numbers contribute equally to an average. Instead, some numbers are more significant than others, based on their frequency or importance.

In the case of isotopes, each isotope has its mass, but these masses are not simply averaged. Instead, they are weighted by their natural abundance. This means you give more weight or importance to isotopes that are more common.

• Convert the percentage of each isotope's abundance into decimal form by dividing by 100.
• Multiply the mass of each isotope by its abundance in decimal form.
• Add the products together to get the weighted average atomic mass.

This way, you ensure that more common isotopes have a greater impact on the final atomic mass of the element.

Percent Natural Abundances

Percent natural abundance tells us how frequently an isotope appears relative to all isotopes of a given element. In the context of atomic mass calculations, it is crucial because it expresses the proportion of each isotope in a naturally occurring mixture.

To use percent natural abundances in calculations, convert these percentages into decimals by dividing by 100. This transformation lets you use them as multipliers when calculating weighted averages.

• For instance, an isotope with a 50% abundance is represented as 0.50 when calculating.
• This conversion allows for straightforward multiplication with the corresponding isotope masses.

These values allow you to accurately reflect each isotope's contribution to the average mass, providing a realistic view of the element's atomic composition as found in nature.