Question

A reaction of 1 liter of chlorine gas \(\left(\mathrm{Cl}_{2}\right)\) with 3 liters of fluorine gas \(\left(\mathrm{F}_{2}\right)\) yields 2 liters of a gaseous product. All gas volumes are at the same temperature and pressure. What is the formula of the gaseous product?

Short answer

The formula of the gaseous product is \(ClF_3\).

Step-by-step solution

Moles and Volume Relationship

At the same temperature and pressure, the volume of the gases is directly proportional to the number of moles. Therefore, we can find the mole ratios of the reactants and product directly from the given volumes. Using the given volumes: Number of moles of Cl2 = 1 liter Number of moles of F2 = 3 liters

Write a Balanced Equation

Let's denote the gaseous product as ClxFy. From the reactants and product ratios, we can write the following equation: \(Cl_2 + F_2 \rightarrow Cl_x F_y\) The total volume of the reactants is 1 + 3 = 4 liters, while the total volume of the product is 2 liters. It means that two moles of reactants give one mole of product. Therefore, we can balance the equation as: \(1/2 Cl_2 + 3/2 F_2 \rightarrow Cl_x F_y\)

Determine the Formula of the Gaseous Product

From the balanced equation, we can see that: 1/2 mole of Cl2 provides x moles of Cl (to form the product). 3/2 moles of F2 provides y moles of F (to form the product). To find x and y, we'll use the proportion of Cl and F in the product: \(x/1/2 = y/3/2\) Solving this equation, we get: \(x = 1\) \(y = 3\) So, the formula of the gaseous product is \(ClF_3\).

Key concepts

Stoichiometry

Stoichiometry is basically the math behind chemistry reactions. It allows us to understand the relationships between reactants and products in a chemical reaction.
It's all about measuring the quantities of reactants (the stuff you start with) and products (the stuff you end up with) and finding how much of each you'll need or get.
In this exercise, you're working with gas volumes, which are tied to moles (a way to measure chemical amounts) through stoichiometry.

The key takeaway with stoichiometry is to use the balanced equation of the reaction, as it shows the mole ratios of reactants and products.

• If you know the amount of either reactants or products, you can figure out how much of the others you'll need or produce.
• Since gas volumes at constant temperature and pressure are directly proportional to moles, the simplicity of this problem arises: volumes can be treated as moles.

So, when 1 liter of \( Cl_{2} \) reacts with 3 liters of \( F_{2} \), resulting in 2 liters of gaseous product, you're essentially given a stoichiometry problem using volumes instead of direct moles. This exercise showcases the concept of proportional relationships supported by the stoichiometry foundation.

Gas Laws

The fundamental idea here is Avogadro’s Law, one of the essential gas laws, which connects the volume of gases to the amount of gas (in moles) when the temperature and pressure remain constant.
Avogadro's Law states that equal volumes of gases, at the same temperature and pressure, contain an equal number of moles. This is the link between gases and stoichiometry in our problem.

When applying Avogadro’s Law in stoichiometry:

• You're assuming that all gases behave ideally. That is, the volume of gas is directly related to the number of moles of gas.
• That's why you can equate the gas volumes to moles when balancing chemical reactions.

So, 1 liter of \( Cl_{2} \) and 3 liters of \( F_{2} \) mean you have a 1:3 mole ratio. Whereas for the product, 2 liters result from the reacted gases signifying another pair of mole relationships that helped us deduce the product in stepwise calculation. Therefore, gas laws act as a bridge between the theory (stoichiometry) and practical application in real-world scenarios.

Balanced Equations

The concept of balanced equations is central to stoichiometry and chemistry as a whole.
A balanced equation ensures that the same number of each type of atom appears on both sides of the equation, reflecting the conservation of mass.
In a balanced reaction, you'll see the proportions of reactants leading to products.

In this exercise:

• We write an initial reaction based on what we know: \( Cl_{2} + F_{2} \rightarrow Cl_{x} F_{y} \).
• Next, balance the equation according to the given volumes, which relate to moles due to Avogadro's relationship.

With this 1:3 volume ratio, it's here calculated that: \( \frac{1}{2} Cl_{2} + \frac{3}{2} F_{2} \rightarrow ClF_{3} \), deducing the molecular formula for the product.
Balanced equations don't just allow us to describe chemical changes; they let us predict, plan and measure the outcomes of reactions, making them key pillars of chemistry and practical lab work alike.