Question

A sample containing \(\mathrm{NaCl}, \mathrm{Na}_{2} \mathrm{SO}_{4},\) and \(\mathrm{NaNO}_{3}\) gives the following elemental analysis: 32.08 percent \(\mathrm{Na}\), 36.01 percent \(\mathrm{O}, 19.51\) percent \(\mathrm{Cl} .\) Calculate the mass percent of each compound in the sample.

Short answer

Calculate each compound's mass by solving the system of equations, then find its percentage of the total.

Step-by-step solution

Define Variables and Equations

Let the masses of \(\mathrm{NaCl}, \mathrm{Na}_{2} \mathrm{SO}_{4},\) and \(\mathrm{NaNO}_{3}\) in the sample be \(x, y,\) and \(z\) respectively. We know the total mass percentages of \(\mathrm{Na},\) \(\mathrm{O},\) and \(\mathrm{Cl}\) are given. We can use these percentages to set up equations based on the masses contributed by each compound to each element's total mass percentage.

Write the Equation for Sodium

The total percentage of sodium is 32.08%. Each compound contributes to this based on its sodium content: \(\mathrm{NaCl}\) contributes \(23/58.44\), \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) contributes \(46/142.04\), and \(\mathrm{NaNO}_{3}\) contributes \(23/85\). So, the equation for sodium is: \[ \frac{23}{58.44}x + \frac{46}{142.04}y + \frac{23}{85}z = 0.3208 \]

Write the Equation for Oxygen

The total percentage of oxygen is 36.01%. Only \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{NaNO}_{3}\) contribute to this. \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) contributes \(64/142.04\) and \(\mathrm{NaNO}_{3}\) contributes \(48/85\). So, the equation for oxygen is: \[ \frac{64}{142.04}y + \frac{48}{85}z = 0.3601 \]

Write the Equation for Chlorine

The total percentage of chlorine is 19.51%. Only \(\mathrm{NaCl}\) contributes to this, with a percentage of \(35.45/58.44\) from the chlorine atom. So, the equation for chlorine is: \[ \frac{35.45}{58.44}x = 0.1951 \]

Solve the System of Equations

From the equation for chlorine, solve for \(x\) in terms of percentages: \(x = 0.1951 \times \frac{58.44}{35.45}\). Substitute the value of \(x\) from Step 4 into the sodium and oxygen equations from Step 2 and Step 3 to solve for \(y\) and \(z\). The system of equations might require a linear algebra approach or use of substitution methods.

Calculate Mass Percent Composition

Substitute the values of \(x, y,\) and \(z\) back to find the mass percentages for each compound. The mass percentage of each compound is found by dividing the mass from each equation by the total sample mass and then converting to percentage. These are the required mass percentages for each compound in the sample.

Key concepts

Elemental Analysis

Elemental analysis is like a detective work done to find out what elements are present in a compound and in what quantities. It's essential for figuring out the composition of substances in a sample. In our exercise, the sample contains three compounds: \(\mathrm{NaCl}\), \(\mathrm{Na}_{2} \mathrm{SO}_{4}\), and \(\mathrm{NaNO}_{3}\). We know the elemental makeup in percentages for sodium (Na), oxygen (O), and chlorine (Cl). This analysis aids in determining how much of each compound makes up the whole sample.
Having these elemental percentages allows us to set up equations that relate the known percentages to the unknown masses of the compounds. For example, sodium is found in all three compounds, while only sodium sulfate and sodium nitrate contribute to the oxygen percentage. Chlorine is exclusive to \(\mathrm{NaCl}\).
This foundational step is critical for solving problems where you need to link the mass of each compound in a sample to the relative mass of its elements.

Mass Percent Composition

Mass percent composition is a way of describing how much of each substance is present in a compound or mixture. It's expressed as a percentage of the whole mass. For our sample, we need to calculate how much each compound contributes to the total mass by the percentages given for each element.
Think of it like cutting a cake into several pieces. Each piece represents a compound, and the size of each piece depends on how much of certain elements it contains. For instance, in our solution, sodium's contribution comes through fractions derived from its role in each of the compounds. We express these contributions as equations.

• The mass percent is calculated by dividing the mass of an individual element in a compound by the total mass of the compound, then multiplying by 100 for a percentage.
• Understanding mass percent composition allows us to translate elemental analysis data into practical information about compound composition.

This helps chemists and students understand the makeup of mixtures and their properties.

System of Equations

A system of equations is a set of equations with multiple variables that you solve together. In chemical analysis, it helps solve for unknown quantities, like the masses of compounds in a sample.
For example, in this exercise, we created a system of equations from the elemental percentages:

• The sodium equation accounts for how sodium is included in each compound.
• The oxygen equation includes oxygen's contribution from \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) and \(\mathrm{NaNO}_{3}\).
• The chlorine equation only involves \(\mathrm{NaCl}\).

This system provides a mathematical pathway to solve for the unknown masses \(x\), \(y\), and \(z\). We can use substitution or elimination methods to find these values effectively.
This approach is common in chemistry where it's crucial to quantify a mix of substances precisely using their elemental fingerprints.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. It also applies to mixtures like in this exercise, where it helps us determine the amount of each substance present.
By using stoichiometry, we relate the mass of individual compounds with the masses of elements present in them.

• We calculate how elements contribute to the mass of each compound, translating it into useful information such as the percentage composition.
• In our solution, stoichiometry allowed us to relate the elemental makeup to the mass percentages of \(\mathrm{NaCl}\), \(\mathrm{Na}_{2} \mathrm{SO}_{4}\), and \(\mathrm{NaNO}_{3}\).

Stoichiometry is essential in chemistry for understanding reactions and compositions, ensuring precise measurements and compositions in chemical processes.