Question

Suppose you have \(5.00 \mathrm{~g}\) of powdered magnesium metal, \(1.00 \mathrm{~L}\) of \(2.00 \mathrm{M}\) potassium nitrate solution, and \(1.00 \mathrm{~L}\) of \(2.00 \mathrm{M}\) silver nitrate solution. (a) Which one of the solutions will react with the magnesium powder? (b) What is the net ionic equation that describes this reaction? (c) What volume of solution is needed to completely react with the magnesium? (d) What is the molarity of the \(\mathrm{Mg}^{2+}\) ions in the resulting solution?

Short answer

(a) Magnesium will react with the silver nitrate (AgNO₃) solution. (b) The net ionic equation is: \(Mg(s) + 2 Ag^{+}(aq) \rightarrow Mg^{2+}(aq) + 2 Ag(s)\) (c) The volume of AgNO₃ solution needed to completely react with the magnesium is 0.2056 L (or 205.6 mL). (d) The molarity of Mg²⁺ ions in the resulting solution is 1.00 M.

Step-by-step solution

(Determine the Reacting Solution)

First, let's determine which one of the given solutions will react with the magnesium powder. Since magnesium is an active metal, it will only react with solutions containing a metal cation that is less active than magnesium. Comparing the activities of potassium (K) and silver (Ag) in the activity series, we find that silver is less active than magnesium. Therefore, magnesium will react with the silver nitrate (AgNO₃) solution.

(Balanced Chemical Equation and Net Ionic Equation)

Now let's write the balanced chemical equation and the net ionic equation for the reaction between magnesium and silver nitrate: Balanced chemical equation: \[Mg(s) + 2 AgNO_{3}(aq) \rightarrow Mg(NO_{3})_{2}(aq) + 2 Ag(s)\] Net Ionic Equation: \[Mg(s) + 2 Ag^{+}(aq) \rightarrow Mg^{2+}(aq) + 2 Ag(s)\]

(Calculate the Volume of Solution Needed)

To find the volume of silver nitrate solution needed to completely react with the magnesium, we'll use stoichiometry. First, we need to find the moles of magnesium: \[moles \thinspace of \thinspace Mg = \frac{5.00 \thinspace g \thinspace Mg}{1 \thinspace mol \thinspace Mg / 24.305 \thinspace g \thinspace Mg} \approx 0.2056 \thinspace mol \thinspace Mg\] According to the balanced chemical equation, 1 mole of Mg reacts with 2 moles of AgNO₃. So, moles of AgNO₃ needed for the reaction: \[moles \thinspace of \thinspace AgNO_{3} = 0.2056 \thinspace mol \thinspace Mg \times \frac{2 \thinspace mol \thinspace AgNO_{3}}{1 \thinspace mol \thinspace Mg} = 0.4112 \thinspace mol \thinspace AgNO_{3}\] Now using the concentration of AgNO₃, we can find the volume of the solution required: \[volume \thinspace of \thinspace AgNO_{3}\thinspace solution = \frac{0.4112 \thinspace mol \thinspace AgNO_{3}}{2.00 \thinspace M \thinspace AgNO_{3}} = 0.2056 \thinspace L (or \thinspace 205.6 \thinspace mL)\]

(Calculate the Molarity of Mg²⁺ Ions)

Finally, let's calculate the molarity of Mg²⁺ ions in the resulting solution. The moles of Mg(NO₃)₂ formed are equal to the moles of magnesium reacted, which is 0.2056 mol. The total volume of the resulting solution is the volume of AgNO₃ solution that was reacted (205.6 mL). \[Molarity \thinspace of \thinspace Mg^{2+} = \frac{0.2056 \thinspace mol \thinspace Mg(NO_{3})_{2}}{0.2056 \thinspace L} = 1.00 \thinspace M\] So, the molarity of Mg²⁺ ions in the resulting solution is 1.00 M.

Key concepts

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. It allows us to predict how much of each reactant is needed to produce a certain amount of product, or vice versa.

When working with stoichiometry, the first step is always to write a balanced chemical equation, which represents the conservation of mass in a reaction. From the balanced equation, we can determine the mole ratio of the reactants and products. Using this ratio, along with the molar masses of reactants, we can convert between mass, moles, and even volume of gases (at standard temperature and pressure).

In the exercise solution, stoichiometry is applied to calculate the volume of silver nitrate solution needed to completely react with 5.00 g of magnesium. The balanced equation tells us that one mole of magnesium reacts with two moles of silver nitrate, meaning the reaction ratio is 1:2. From the weight of magnesium given, we can calculate the moles of magnesium and then use the mole ratio to find the required moles of silver nitrate. Dividing this by the molarity gives us the volume of solution needed.

Net Ionic Equation

A net ionic equation represents the chemical species that are involved in a reaction at the ionic level and excludes spectator ions that don't participate in the chemical change. To write a net ionic equation, one must first understand the chemical species present in a reaction, which ones react, and which are simply present in the solution, not participating in the reaction itself.

To arrive at the net ionic equation, begin with a balanced molecular equation and then break down all the soluble ionic compounds into their constituent ions. Identify the ions that change their oxidation state or composition during the reaction - these are the ones included in the net ionic equation.

In our exercise example, magnesium metal (Mg) reacts with silver nitrate (AgNO₃) in which silver ions (Ag⁺) and magnesium form the core reacting species. The net ionic equation simplifies the reaction to show only these changes, excluding the nitrates (NO₃⁻) since they remain unchanged and are considered spectator ions.

Activity Series of Metals

The activity series of metals is a list that ranks metals by their reactivity. Metals at the top of the list, such as alkali metals, are more reactive and can displace hydrogen from water and acids. As you move down the list, the metals become less reactive. Metals lower on the list cannot displace hydrogen from water but can displace cations from solutions of compounds from metals below them.

The activity series is key in predicting the outcomes of single replacement reactions, where one metal displaces another from a compound. In our exercise, we compare magnesium (Mg) to potassium (K) and silver (Ag). Magnesium ranks higher than silver on the activity series, which means it can displace silver ions from their compounds, like silver nitrate, but it cannot displace potassium ions because potassium is above it in the series.

This property is used to decide which of the two given solutions, potassium nitrate or silver nitrate, will react with magnesium powder. As per the activity series, magnesium will react with silver and not with potassium, thereby determining the correct reactant and guiding the student to the accurate solution for the problem.