Problem 31 - General Chemistry: Principles and Modern Applications

Question

The following data on isotopic masses are from a chemical handbook. What is the ratio of each of these masses to that of\(^{12}_{6} \mathrm{c}\) ?\( \)^{17}_{35} \mathrm{CL}\( , 34.96885 \mathrm{u} ;\) (b) \(_{12}^{26} \mathrm{Mg}\) 25.98259 u ;\(^{86}_{222} \mathrm{Rn}\) , 222.0175 u.

Step-by-Step Solution

Answer

The ratios of the isotopic masses to the mass of \(^{12}_{6}\mathrm{C}\) are approximately 2.914 for \(^{35}_{17}\mathrm{Cl}\), approximately 2.165 for \(^{26}_{12}\mathrm{Mg}\), and approximately 18.501 for \(^{222}_{86}\mathrm{Rn}\).

Step 1: Calculate the Ratio for Chlorine

Specifically, for the first isotope \(^{35}_{17}\mathrm{Cl}\), calculate its ratio to the mass of \(^{12}_{6}\mathrm{C}\) by dividing its given mass by 12. This gives: \[\frac{34.96885\mathrm{u}}{12\mathrm{u}}\]

Step 2: Calculate the Ratio for Magnesium

Similarly, for the second isotope \(^{26}_{12}\mathrm{Mg}\), the ratio to the mass of \(^{12}_{6}\mathrm{C}\) can can be found by dividing its given mass by 12. This gives: \[\frac{25.98259\mathrm{u}}{12\mathrm{u}}\]

Step 3: Calculate the Ratio for Radon

Lastly, for the third isotope \(^{222}_{86}\mathrm{Rn}\), the ratio to the mass of \(^{12}_{6}\mathrm{C}\) can be found by dividing its given mass by 12. This gives: \[\frac{222.0175\mathrm{u}}{12\mathrm{u}}\]

Key Concepts

Isotopic Masses

Isotopic masses refer to the mass of a specific isotope of an element. Isotopes are different forms of a given element, distinguished by having different numbers of neutrons. This results in different atomic masses, although they maintain the same number of protons. Isotopic masses are usually expressed in atomic mass units (amu), allowing for precise calculations when dealing with chemical reactions or isotopic ratios.
For example, in the given exercise, we're provided with the isotopic masses of chlorine ( ^{35}_{17} Cl), magnesium ( ^{26}_{12} Mg), and radon ( ^{222}_{86} Rn).
To find their ratios to the carbon-12 standard, knowing their specific isotopic masses helps us understand the proportionate mass of each isotope compared to a common reference. This provides a scale for measuring the various isotopes relative to each other.

Carbon-12 Standard

The carbon-12 standard is a key concept in chemistry as it provides a baseline for comparing isotopic masses. Carbon-12, or ^{12}_{6} C, is an isotope of carbon with a mass of exactly 12 atomic mass units by definition. This standardization allows chemists to easily compare other isotopes' masses.
By using carbon-12 as a comparison, calculations become more manageable and accurate since carbon is abundant and its isotope is highly stable, making it reliable. Each atomic mass provided in the problem is divided by 12 u, which represents the uniform mass of carbon-12. This method simplifies the process of understanding isotopic mass ratios and is fundamental to studying atomic masses.

Atomic Mass Unit

The atomic mass unit (amu) is the unit used to express the mass of an atomic particle. It is defined relative to the carbon-12 standard. One atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom. This makes the amu a very small unit, suitable for measuring subatomic particles.
The amu is crucial because it provides a consistent and practical way of expressing the masses of atoms and isotopes. Without such a standard, understanding and communication between scientists would be less efficient, requiring constant explanations or conversions of measurements.

• An amu allows for uniform measurement of isotopic mass.
• It is aligned with the carbon-12 standard for consistency.
• Atomic mass units facilitate easier computation in chemical equations.

Through utilizing amu, isotopic masses can be easily compared and utilized in scientific research and practical applications.