Question

The balanced equation for the reaction of chromium metal and chlorine gas is $$ 2 \mathrm{Cr}(s)+3 \mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{CrCl}_{3}(x) $$ What is the limiting reactant when each of the following sets of quantities of reactants is mixed? (a) \(4 \mathrm{Cr}\) atoms and \(6 \mathrm{Cl}_{2}\) molecules (b) \(6 \mathrm{Cr}\) atoms and \(10 \mathrm{Cl}_{2}\) molecules (c) \(12 \mathrm{Cr}\) atoms and \(20 \mathrm{Cl}_{2}\) molecules

Short answer

(a) No limiting reactant (b) Cr is the limiting reactant (c) Cr is the limiting reactant

Step-by-step solution

Identify Ratios from Equation

From the balanced equation, we can see that the required ratio of Cr to Cl2 for the reaction to occur is 2:3. This ratio will be used to compare with the provided quantities of Cr and Cl2 in each set.

Determine Limiting Reactant for (a)

In set (a), there are 4 atoms of Cr and 6 molecules of Cl2, which matches exactly with the 2:3 ratio from the balanced equation. So, neither of the reactants will be left over, meaning there isn't a limiting reactant for set (a).

Determine Limiting Reactant for (b)

In set (b), there are 6 atoms of Cr and 10 molecules of Cl2. Based on the 2:3 ratio from the balanced equation, for 6 Cr atoms, we would need 9 Cl2 molecules. But since there are 10 Cl2 molecules, Cr will be the limiting reactant because we have more Cl2 than needed for the complete reaction with 6 Cr atoms.

Determine Limiting Reactant for (c)

In set (c), there are 12 atoms of Cr and 20 molecules of Cl2. Based on the 2:3 ratio from the balanced equation, for 12 Cr atoms, we would need 18 Cl2 molecules. But since there are 20 Cl2 molecules, Cr will again be the limiting reactant since there are more Cl2 molecules than required for a complete reaction with 12 Cr atoms.

Key concepts

Stoichiometry

Stoichiometry is a core concept in chemistry that deals with the calculation of reactants and products in chemical reactions. It helps us understand how much of each substance is involved in a reaction.
It's like using a recipe to determine how many ingredients you need to bake a cake. When you look at a chemical equation, stoichiometry tells you the exact proportions required to ensure that all the reactants are perfectly used up.
For example, in the given reaction, stoichiometry tells us the amount of chromium (Cr) and chlorine gas (Cl extsubscript{2}) needed to produce chromium chloride (CrCl extsubscript{3}). This is determined by balancing the chemical equation, ensuring that atoms are conserved from reactants to products.

• This balanced equation is crucial for calculating quantities in a reaction.
• It involves the use of ratios derived from the balanced equation.
• Stoichiometry allows us to find the limiting reactant by comparing reactant quantities to these ratios.

With stoichiometry, we can predict outcomes, plan for production of chemicals, or even control reactions in a laboratory setting.

Chemical Equations

Chemical equations are symbolic representations of chemical reactions. They show the substances involved as well as the changes they undergo during the reaction.
In our exercise, the equation \(2 \text{Cr}(s) + 3 \text{Cl}_{2}(g) \rightarrow 2 \text{CrCl}_{3}(x)\) serves as our roadmap.
This equation is balanced - meaning the number of atoms for each element is the same on both sides of the equation. This is what ensures the law of conservation of mass is kept intact.
Balanced chemical equations help us understand how reactants convert into products. Without balancing, you could not accurately predict how much product you can make or how much of each reactant is needed.
The coefficients in the equation (like the '2' before Cr and Cl extsubscript{3}) tell us how many units of each reactant combine and the amount of product formed. Having a balanced equation gives the necessary ratios needed for calculating potential yields and consumption of reactants.

• They're essential for finding the limiting reactant.
• Guide the stoichiometry calculations.
• Show the states of each substance (solid \(s\), gas \(g\), etc.).

Every chemical equation expresses a specific reaction, providing a concise way to represent these transformations.

Reactant Ratios

Reactant ratios are derived from the coefficients in a balanced chemical equation, showcasing how reactants combine at a molecular level.
In the reaction \(2 \text{Cr}(s) + 3 \text{Cl}_{2}(g) \rightarrow 2 \text{CrCl}_{3}(x)\), the ratio is 2:3 between chromium and chlorine gas.
This ratio helps us compare with the actual amounts we have in a reaction and identify the limiting reactant. The limiting reactant is the one that runs out first, stopping the reaction.
In our exercise, the ratios are applied to find which reactant is limiting under varying conditions:

• For set (a) with 4 Cr atoms and 6 Cl extsubscript{2} molecules, the 2:3 ratio is perfectly met so neither reactant limits the reaction.
• For set (b) with 6 Cr atoms and 10 Cl extsubscript{2} molecules, Cr is the limiting reactant because we have more Cl extsubscript{2} than the ratio requires.
• Similarly, in set (c) with 12 Cr atoms and 20 Cl extsubscript{2} molecules, Cr runs out first.

Understanding these ratios allows chemists to control reactions more efficiently, ensure resources are used wisely and predict the maximum possible yield.