Question

At elevated temperatures, sodium chlorate decomposes to produce sodium chloride and oxygen gas. A \(0.8765\)-\(\mathrm{g}\) sample of impure sodium chlorate was heated until the production of oxygen gas ceased. The oxygen gas collected over water occupied \(57.2 \mathrm{mL}\) at a temperature of \(22^{\circ} \mathrm{C}\) and a pressure of \(734\) torr. Calculate the mass percent of \(\mathrm{NaClO}_{3}\) in the original sample. (At \(22^{\circ} \mathrm{C}\) the vapor pressure of water is \(19.8\) torr.)

Short answer

The mass percent of sodium chlorate in the original sample is calculated as follows: 1. Calculate the moles of O₂ using the ideal gas law: \(n(O_2) = \frac{(734\,\text{torr} - 19.8\,\text{torr})\cdot 57.2\,\text{mL}}{62.364\,\text{L⋅torr/mol⋅K} \cdot (22^\circ C + 273.15)}\) 2. Find moles of sodium chlorate that decomposed: \(n(\text{NaClO}_3) = n(O_2) \cdot \frac{2}{3}\) 3. Calculate the mass of sodium chlorate in the original sample: \(\text{mass}_{\text{NaClO}_3} = n(\text{NaClO}_3)\cdot106.44\,\text{g/mol}\) 4. Calculate the mass percent of sodium chlorate in the original sample: \(\text{mass percent}_{\text{NaClO}_3} = \frac{\text{mass}_{\text{NaClO}_3}}{0.8765\,\text{g}} \cdot 100\) Thus, the mass percent of \(\text{NaClO}_3\) in the original sample is \(\text{mass percent}_{\text{NaClO}_3}\).

Step-by-step solution

Calculate moles of O₂ gas using the ideal gas law

First, we need to find the moles of oxygen gas (O₂) generated by the decomposition. The ideal gas law is given by: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. The partial pressure of the oxygen gas can be found by: P(O₂) = P(total) - P(water vapor) We're given that the total pressure is 734 torr and the vapor pressure of the water at the given temperature is 19.8 torr. Temperature can be converted to Kelvin by adding 273.15 to the Celsius value. We'll use R = 62.364 L⋅torr/mol⋅K, which is suitable for pressure in torr. With that, we can solve for the moles of O₂:

Find the moles of sodium chlorate that decomposed

The balanced decomposition reaction of sodium chlorate is: 2NaClO₃ -> 2NaCl + 3O₂ From the stoichiometry of the reaction, we observe that 2 moles of sodium chlorate produces 3 moles of oxygen gas. Using this, we can find out the moles of sodium chlorate (NaClO3) decomposed: moles of NaClO3 = (moles of O₂) × (2 moles of NaClO3 / 3 moles of O₂)

Calculate the mass of sodium chlorate in the original sample

Now that we have the moles of sodium chlorate that decomposed, we can determine its mass by multiplying the number of moles with the molar mass of NaClO₃, which is 106.44 g/mol: mass of NaClO3 = moles of NaClO3 × 106.44 g/mol

Calculate the mass percent of sodium chlorate in the original sample

The mass percent of sodium chlorate in the original sample can be calculated using the formula: mass percent of NaClO3 = (mass of NaClO3 / mass of original sample) × 100 Plug in the values calculated in the previous steps to get the final mass percent of sodium chlorate in the original sample.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that relates the four properties of gas—pressure (P), volume (V), number of moles (n), and temperature (T)—through the equation PV = nRT.

In our exercise, the ideal gas law was used to calculate the moles of oxygen gas produced in the decomposition reaction. The pressure was adjusted for water vapor, a common practice when collecting gases over water, to only account for the gas of interest, in this case, oxygen. The volume given is measured in liters, and temperature is converted to Kelvin, as absolute temperature is a requirement for the calculations. The gas constant (R) changes based on the units used for pressure; here, we use 62.364 L⋅torr/mol⋅K suitable for torr.

It's interesting to note that the ideal gas law assumes gas molecules do not attract or repel each other and do not take up any space. While this isn't entirely true for real gases, the ideal gas law provides a close approximation under many conditions, particularly those involving gases at relatively high temperature and low pressure, like in this exercise.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe. One mole of any substance contains Avogadro's number of entities (6.022 × 10²³) and corresponds to its formula weight expressed in grams.

In our calculation, we determine the moles of oxygen which allows us to derive the moles of sodium chlorate through stoichiometry. This concept is crucial because it allows us to convert between the mass of a substance and the number of particles or entities (atoms, molecules, ions) it contains. Understanding the mole concept is vital when dealing with chemical reactions because it provides a method for quantifying substances involved.

Chemical Reaction Balancing

Balancing a chemical reaction is the process of ensuring that the number of atoms for each element is the same on both the reactant and product sides of the equation. It's a fundamental skill in stoichiometry as it ensures the law of conservation of mass is met.

In our exercise's decomposition reaction, the balanced equation was: 2NaClO₃ -> 2NaCl + 3O₂. This balance tells us the stoichiometric ratios required to convert between moles of reactants and products. Without a balanced equation, we would not be able to accurately perform stoichiometric calculations, such as those needed to determine the mass percent purity of the sodium chlorate sample.

Mass Percent Calculation

The mass percent of a component in a mixture is a way of expressing its concentration, determined by the formula: mass percent = (mass of component / total mass of mixture) × 100.

This calculation was the final step in our exercise to find the purity of the sodium chlorate sample. By calculating the mass of sodium chlorate that decomposed, we can compare it to the total mass of the original sample to get a percentage. This value gives us insight into the composition of the sample, which is particularly important in real-world applications where purity can affect the performance of a product or the accuracy of a scientific experiment.

Decomposition Reactions

A decomposition reaction is a type of chemical reaction where one compound breaks down into two or more simpler substances. In our exercise, sodium chlorate decomposes when heated to produce sodium chloride and oxygen gas.

Decomposition reactions are involved in many natural and industrial processes, such as the breakdown of organic matter or the production of metals from their ores. They can be triggered by heat, light, or chemical catalysts, among other factors. In this scenario, knowing the specifics of the decomposition behavior of sodium chlorate is essential for predicting the outcome and understanding the products formed, which is foundational to accurately performing related stoichiometry calculations.