Question

A lunar surface probe analyzed a sample of soil and found that \(58 \%\) of the atoms it contained were oxygen \((0\), atomic weight \(=16 \mathrm{~g} / \mathrm{mole}), 18 \%\) were silicon \((\mathrm{Si}\), atomic weight \(=\) \(28 \mathrm{~g} /\) mole \(), 9 \%\) were aluminum (Al, atomic weight \(=27\) \(\mathrm{g} /\) mole), and \(15 \%\) consisted of other elements with an average atomic weight of \(30 \mathrm{~g} /\) mole. Determine the percent oxygen by weight in this sample.

Short answer

The percent oxygen by weight in the lunar soil sample is approximately \(43.6\%\).

Step-by-step solution

Calculate the weight of each element in the sample

First, we'll calculate the weight of each element present in the soil sample. We're not given the overall mass of the sample, so let's assume the sample consists of 100 atoms. This will help simplify the calculation as the given percentages directly correspond to the number of atoms of each element. Number of Oxygen atoms = \(58\%\) of 100 = 58 atoms Number of Silicon atoms = \(18\%\) of 100 = 18 atoms Number of Aluminum atoms = \(9\%\) of 100 = 9 atoms Number of other elements atoms = \(15\%\) of 100 = 15 atoms Now, we'll calculate the weight by multiplying the number of atoms of each element with their respective atomic weights: Weight of Oxygen = 58 atoms * 16 g/mole = 928 g/mole Weight of Silicon = 18 atoms * 28 g/mole = 504 g/mole Weight of Aluminum = 9 atoms * 27 g/mole = 243 g/mole Weight of other elements = 15 atoms * 30 g/mole = 450 g/mole

Calculate total weight of the sample

Now that we have the weight of each type of element in the sample, we can find the total weight of the sample by summing up the weights of each element: Total weight = Weight of Oxygen + Weight of Silicon + Weight of Aluminum + Weight of other elements Total weight = \(928 g/mole + 504 g/mole + 243 g/mole + 450 g/mole = 2125 g/mole\)

Calculate the weight percentage of Oxygen

Now that we have the total weight of the sample and the weight of oxygen in it, we can find the weight percentage of oxygen by dividing the weight of oxygen by the total weight and then multiplying the result by 100: Percent Oxygen by weight = \(\frac{Weight of Oxygen}{Total weight} \times 100\) Percent Oxygen by weight = \(\frac{928 g/mole}{2125 g/mole} \times 100\) Percent Oxygen by weight = \(43.6\%\) So, the percent oxygen by weight in the lunar soil sample is approximately \(43.6\%\).

Key concepts

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships of the reactants and products in a chemical reaction. It involves using balanced chemical equations to calculate the masses, volumes, and number of moles of compounds involved within a reaction. When analyzing lunar soil composition, as in our given exercise, stoichiometry principles are used to determine the proportion of each element in the sample relative to the number of atoms present.

For instance, in the analysis of the lunar soil, stoichiometry is applied by assuming a simplified model where the soil sample consists of 100 atoms. This makes calculations straightforward because each percentage can be directly translated into the number of atoms, which lays the foundation for determining the actual mass of each element within the soil sample. Such an approach demystifies the process and assists in visualizing the elemental distribution in a tangible way. To implement stoichiometry in practical scenarios, steps generally involve identifying the relevant elements, converting percentages into actual quantities (often assuming a fixed total quantity, such as 100 atoms), and then using the known atomic weights to find the masses involved.

Molecular Weight Calculation

Molecular weight calculation, also known as molar mass determination, is a fundamental concept in chemistry that involves adding up the atomic weights of the atoms in a molecule to obtain the molecule's total weight. The unit commonly used is grams per mole (g/mol), indicating how much one mole (Avogadro’s number of particles) of a substance weighs.

In the context of our lunar soil problem, we perform this calculation for pure elements, which simplifies to using the atomic weight of each constituent element. The atomic weights are multiplied by the number of atoms of each element (determined from the percentage of atoms present) to calculate the total weight of each element in the soil sample. For example, with oxygen having an atomic weight of 16 g/mol and its atoms comprising 58% of the soil sample, the calculation for oxygen's weight would be 58 atoms times 16 g/mol.

Percent Composition

Percent composition is a measure that expresses what fraction of a sample's total mass is made up of a particular element or compound. It is a particularly useful metric when analyzing the make-up of mixtures or compounds, such as lunar soil. Calculating percent composition allows researchers and students alike to determine the abundance of elements within a sample, which can have implications for understanding the sample's origin, potential usefulness, and its physical and chemical properties.

In our exercise, the percent composition of oxygen by weight is found by dividing the total weight of oxygen by the overall mass of the lunar soil sample and then multiplying by 100 to get a percentage. The result is the percent by weight of oxygen in the soil sample, which reflects how much oxygen contributes to the overall mass of the lunar soil. This is an essential calculation in many fields, including geology, material science, and planetary science.