Problem 116 - Chemistry Matter and Change

Question

Magnesium Isotopes Compute the mass number, \(X,\) of the third isotope of magnesium given that the respective abundances of the naturally occurring isotopes are: \(79.0 \%, 10 \%,\) and 11\(\%\) for \(_{12}^{24} \mathrm{Mg}\) \(\underset{12}{25} \mathrm{Mg}, \underset{12}{X} \mathrm{Mg}\) The relative atomic mass of magnesium is 24.305 amu.

Step-by-Step Solution

Answer

The mass number of the third isotope of magnesium is approximately 35, represented as \(_{12}^{35}\mathrm{Mg}\).

Step 1: Write down the atomic mass equation for magnesium

The atomic mass of an element is the weighted average of the masses of its isotopes. The equation for the average atomic mass of magnesium is: Average_atomic_mass_Mg = (mass_Mg24 * abundance_Mg24) + (mass_Mg25 * abundance_Mg25) + (mass_MgX * abundance_MgX) Here we are given the relative atomic mass of magnesium = 24.305, and the respective abundances of the isotopes: - mass_Mg24 = 24 - abundance_Mg24 = 0.79 - mass_Mg25 = 25 - abundance_Mg25 = 0.10 - abundance_MgX = 0.11 Now we can use this information to find the mass number of the third isotope (mass_MgX).

Step 2: Write down the average atomic mass equation using the given values

Using the given values, we write down the atomic mass equation as follows: 24.305 = (24 * 0.79) + (25 * 0.10) + (mass_MgX * 0.11)

Step 3: Simplify the equation and solve for mass_MgX

To find the mass number of the third isotope (mass_MgX), first simplify the equation by multiplying the mass numbers with their corresponding abundances: 24.305 = (18.96) + (2.5) + (0.11 * mass_MgX) Now, subtract the sum of the first two terms from the total average atomic mass: (24.305 - 18.96 - 2.5) = 0.11 * mass_MgX Perform the subtraction: 3.845 = 0.11 * mass_MgX Now, divide by 0.11 to find mass_MgX: mass_MgX = 3.845 / 0.11 mass_MgX ≈ 34.9545 Since mass numbers are integers, we round the value of mass_MgX to the nearest whole number: mass_MgX ≈ 35

Step 4: State the mass number of the third isotope

The mass number of the third isotope of magnesium is approximately 35. So, the third isotope of magnesium can be represented as \(_{12}^{35}\mathrm{Mg}\).

Key Concepts

Atomic Mass

The concept of atomic mass is central to understanding elements and their isotopes. Atomic mass refers to the mass of an atom, which is primarily determined by the sum of the protons and neutrons in its nucleus. Since electrons have negligible mass, they do not significantly contribute to the atomic mass.
The atomic mass is often expressed in atomic mass units (amu), where one amu is defined as one-twelfth the mass of a carbon-12 atom. This standard helps in comparing the atomic masses of different elements. When discussing atoms with multiple isotopes, we rely on the weighted average of these isotopic masses, often referred to as the atomic weight.
In calculations, the weighted average enables us to consider the relative abundances of different isotopes. This is crucial because naturally occurring elements, like magnesium, often exist as a mixture of isotopes, each contributing to the final atomic mass. Hence, the atomic mass of an element like magnesium is not a straight sum, but a careful average that reflects the isotope distribution.

Isotope Abundance

Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons, resulting in different mass numbers. In nature, elements usually occur as a mixture of isotopes, and the isotope abundance tells us how common each isotope is in nature.
For example, in the case of magnesium, it has isotopes with mass numbers like 24, 25, and a third unknown mass number. The percentage indicating the isotope abundance, such as 79% for magnesium-24, represents its relative occurrence among all magnesium atoms.
This abundance is crucial when calculating properties like the relative atomic mass because it determines how much weight each isotope should contribute to the average. By combining these abundances with the respective masses, scientists can accurately compute elements' relative atomic mass, bridging the gap between microscopic details and macroscopic measurable quantities.

Weighted Average

Understanding the concept of a weighted average is essential when dealing with atomic masses and isotopes. A weighted average takes into account both the value of each item and its relative importance, in this case, its abundance.
To calculate the weighted average atomic mass of an element, you multiply the mass of each isotope by its corresponding abundance (expressed as a fraction), and then sum these values.
In the example of magnesium isotopes, the weighted average is used to find the resultant average atomic mass that matches the known relative atomic mass. This means multiplying each isotope’s mass by its fractional abundance and summing the results to derive a comprehensive average. The process effectively balances each isotope by how common it is, allowing us to determine values like the accurately predicted atomic mass of complex makeup elements.

Relative Atomic Mass

Relative atomic mass is an average of the masses of all isotopes of an element, as they occur in nature, weighted by their abundance. It's often referred to as atomic weight and is commonly expressed in atomic mass units (amu).
In the context of magnesium isotopes, the relative atomic mass takes into account the constants: the individual isotopic masses and their respective abundances. This value isn't always a whole number because it is a weighted average, considering all naturally occurring isotopes of that element.
This relative atomic mass is significant because it simplifies the comparative mass calculations for chemists and physicists. It allows standard tables to be developed, which provide the necessary averages used in practical calculations across many scientific domains. By understanding relative atomic mass, students can appreciate how diverse isotopes contribute to the properties of elements as we know them.