Problem 95 - Chemistry: The Central Science in SI Units, Global Edition

Question

The element chromium (Cr) consists of four naturally occurring isotopes with atomic masses 49.9460,51.9405,52.9407 , and \(53.9389 \mathrm{u}\). The relative abundances of these four isotopes are \(4.3,83.8,9.5,\) and \(2.4 \%\), respectively. From these data, calculate the atomic weight of chromium.

Step-by-Step Solution

Answer

The atomic weight of chromium is approximately \(51.999 \mathrm{u}\).

Step 1: Convert relative abundances to decimals

To use the relative abundances in the atomic weight calculation, we need to convert them from percentages to decimals. We do this by dividing each percentage by 100: - 4.3% = 0.043 - 83.8% = 0.838 - 9.5% = 0.095 - 2.4% = 0.024

Step 2: Multiply each isotope's atomic mass by its relative abundance

Now, we can multiply the atomic mass of each isotope by its relative abundance (in decimal form) to find the weighted contribution of each isotope to the atomic weight of chromium: - 49.9460 u * 0.043 = 2.147978 u - 51.9405 u * 0.838 = 43.526990 u - 52.9407 u * 0.095 = 5.0292665 u - 53.9389 u * 0.024 = 1.2943336 u

Step 3: Add the weighted contributions together

Finally, we add the weighted contributions of each isotope to find the atomic weight of chromium: Atomic weight = 2.147978 u + 43.526990 u + 5.0292665 u + 1.2943336 u = 51.9985681 u

Step 4: Round the result

Although the atomic weight we calculated is very precise, it is not necessary to report it with that many decimal places. Let's round it to 3 decimal places: Atomic weight = 51.999 u The atomic weight of chromium is approximately \(51.999 \mathrm{u}\).

Key Concepts

Isotopes

Isotopes are different forms of the same element. They have identical numbers of protons but different numbers of neutrons in the nucleus. This difference in neutron number gives each isotope a unique atomic mass.
For instance, chromium has four naturally occurring isotopes with distinct atomic masses:

• 49.9460 u
• 51.9405 u
• 52.9407 u
• 53.9389 u

Each of these isotopes contributes to the overall characteristics of chromium, especially its atomic weight. Understanding isotopes is crucial when discussing atomic weight because different isotopes of the same element vary in abundance and mass.
Isotopes are essential in fields like chemistry and physics, where the understanding of elemental compositions and reactions is vital. They also play key roles in applications like radioactive dating and medical diagnostics.

Relative Abundance

Relative abundance refers to how often an isotope occurs in nature compared to other isotopes of the same element. It is usually expressed as a percentage but can also be converted to a decimal for calculations.
For chromium, the naturally occurring isotopes have relative abundances of 4.3%, 83.8%, 9.5%, and 2.4%. In calculations, these percentages can be expressed as decimals as follows:

• 4.3% becomes 0.043
• 83.8% becomes 0.838
• 9.5% becomes 0.095
• 2.4% becomes 0.024

Using these decimal values, we can determine how much each isotope contributes to the element’s overall atomic weight. A higher relative abundance means an isotope significantly impacts the element's atomic weight.
Calculating atomic weight from isotopes' relative abundance helps scientists understand the element's average mass and predict its behavior in chemical reactions.

Element Chromium

Chromium, symbolized as Cr, is a transition metal widely used in various industries because of its corrosion resistance and shiny appearance. It is primarily known for its role in creating stainless steel and as a dazzling component in chrome plating.
The element consists of multiple isotopes, contributing to its atomic weight. As previously calculated, the atomic weight of chromium is approximately 51.999 u, derived from the weighted average of its isotopes:

• 49.9460 u
• 51.9405 u
• 52.9407 u
• 53.9389 u

Understanding chromium's isotopic composition is essential in applications like metallurgy, where precise knowledge of elemental composition influences the material properties.
Moreover, in environmental science, knowing about chromium's isotopes can assist in tracing sources of pollution and studying their impacts on ecological systems.