Problem 55 - Chemistry: The Central Science

Question

Magnesium can be used as a "getter" in evacuated enclosures to react with the last traces of oxygen. (The magnesium is usually heated by passing an electric current through a wire or ribbon of the metal.) If an enclosure of \(0.452 \mathrm{~L}\) has a partial pressure of \(\mathrm{O}_{2}\) of \(3.5 \times 10^{-6}\) torr at \(27^{\circ} \mathrm{C}\), what mass of magnesium will react according to the following equation? $$ 2 \mathrm{Mg}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{MgO}(s) $$

Step-by-Step Solution

Answer

\( n = \frac{(3.5 \times 10^{-6} \text{ torr})\left(\frac{1 \text{ atm}}{760 \text{ torr}}\right)(0.452 \text{ L})}{(0.0821 \text{ L*atm/mol*K})(27 + 273.15)^{\circ} \text{K}}\) \( n = 6.48 \times 10^{-9} \text{ mol O}_2\)

Step 1: Convert the given conditions for oxygen gas to moles

First, let's use the Ideal Gas Law, \(PV = nRT\), to find the moles of O2 in the evacuated enclosure. Here, 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. We have: - P = 3.5 x 10^{-6} torr (convert torr to atm: 1 atm = 760 torr) - V = 0.452 L - R = 0.0821 L*atm/mol*K (ideal gas constant) - T = 27°C (convert to Kelvin: add 273.15) Now, solve for n (number of moles of O2): \( n = \frac{PV}{RT}\)

Key Concepts

Stoichiometry

Stoichiometry is the study of the quantitative relationships or ratios between reactants and products in a chemical reaction. In chemical equations, stoichiometry allows us to understand how much of each reactant is needed to produce a desired amount of product. For instance, when considering stoichiometry in our reaction of magnesium with oxygen, the balanced chemical equation tells us that 2 moles of magnesium (\( \mathrm{ Mg } \)) react with 1 mole of oxygen (\( \mathrm{ O }_2 \)) to form 2 moles of magnesium oxide (\( \mathrm{ MgO } \)).

This balanced equation is crucial because it informs us that the ratio of magnesium to oxygen is 2:1, meaning two atoms of magnesium are needed to react completely with each molecule of oxygen gas. Stoichiometry simplifies the calculation of how much magnesium is required to react with a certain quantity of oxygen. It helps in predicting the quantities of reactants consumed and products formed in a chemical reaction.

• The stoichiometric coefficients (the numbers before substances in the chemical equation) indicate the proportion of substances.
• Knowing the amounts of reactants, stoichiometry helps in calculating the theoretical yield of a reaction.
• It also aids in determining the limiting reactant, which limits the amount of product formed.

Understanding stoichiometry is essential for conducting effective and efficient chemical reactions, especially in experimental and industrial settings.

Chemical Reactions

Chemical reactions are processes in which substances, the reactants, undergo transformations to form new substances, the products. These reactions are represented by balanced chemical equations, showing both reactants and products with their respective stoichiometric coefficients. In the magnesium-oxygen reaction given in the problem, the chemical equation is \(2 \ \mathrm{ Mg }(s) + \mathrm{ O }_2(g) \rightarrow 2 \ \mathrm{ MgO }(s)\).

A reaction like this involves breaking bonds in the reactants and forming new bonds in the products. This process involves energy changes and is often driven by conditions such as heat or pressure. In our example, magnesium reacts vigorously with oxygen when heated, showing that heat is necessary for this reaction to proceed.

• Reactions can be exothermic (releasing heat) or endothermic (absorbing heat).
• The state of each substance (solid, liquid, gas) is crucial in understanding reaction dynamics.
• Reaction conditions like temperature and pressure can significantly influence the rate and extent of reactions.

Understanding chemical reactions is fundamental to predicting how different substances will behave and interact under various conditions.

Partial Pressure

Partial pressure is the pressure exerted by a single component of a mixture of gases. Each gas in a mixture behaves independently and contributes to the total pressure proportionally to its amount. In our problem, the partial pressure of oxygen is provided, and it is essential to determine how much magnesium will react with this amount of oxygen.

The concept of partial pressure derives from Dalton's Law of Partial Pressures, which states that the total pressure of a gaseous mixture is equal to the sum of the partial pressures of individual gases. This principle is particularly helpful when dealing with gas reactions, where each gas's behavior is analyzed separately.

• Partial pressure is affected by the mole fraction of the gas in the mixture.
• It is used to calculate the amount of gas using the Ideal Gas Law, \(PV = nRT\).
• Knowing the partial pressure helps in determining how gases will diffuse or react.

In the exercise, by knowing the partial pressure of \(\mathrm{O}_2\), you can calculate the moles of \(\mathrm{O}_2\) using the Ideal Gas Law, which is crucial for further stoichiometric calculations.