Problem 162 - Chemistry An Atoms First Approach

Question

Nitrogen gas \(\left(\mathrm{N}_{2}\right)\) reacts with hydrogen gas \(\left(\mathrm{H}_{2}\right)\) to form ammonia gas \(\left(\mathrm{NH}_{3}\right) .\) You have nitrogen and hydrogen gases in a \(15.0\)-\(\mathrm{L}\) container fitted with a movable piston (the piston allows the container volume to change so as to keep the pressure constant inside the container). Initially the partial pressure of each reactant gas is \(1.00\) atm. Assume the temperature is constant and that the reaction goes to completion. a. Calculate the partial pressure of ammonia in the container after the reaction has reached completion. b. Calculate the volume of the container after the reaction has reached completion.

Step-by-Step Solution

Answer

a. The partial pressure of ammonia gas after the reaction has reached completion is \(P_{NH_3} = \frac{n_{NH3}RT}{V} = 1.32 \ \text{atm}\). b. The final volume of the container after the reaction has reached completion is \(V_{Final} = \frac{P_{Initial}}{P_{Final}} \times V_{Initial} = 22.5 \ \text{L}\).

Step 1: Write Down the Balanced Reaction

Nitrogen gas reacts with hydrogen gas to form ammonia gas according to the following balanced chemical equation: \( N_2(g) + 3H_2(g) \rightarrow 2NH_3(g) \) #Step II: Calculate Moles of Reactant Gases#

Step 2: Calculate Moles of Reactant Gases

We know the initial partial pressures of the nitrogen and hydrogen gases. To find the moles, we can use the Ideal Gas Law: \( PV = nRT \) Where, \(P\) is Pressure \(V\) is Volume \(n\) is the number of moles \(R\) is the Ideal Gas Constant (\(0.0821 \ L \ atm \ K^{−1} mol^{−1}\)) \(T\) is Temperature (in Kelvin) Solve for the moles (\(n\)) of \(N_2\) and \(H_2\): \( n_{N_2} = \frac{P_{N_2}V}{RT}\) \( n_{H_2} = \frac{P_{H_2}V}{RT}\) Since both nitrogen and hydrogen gases have the same pressure (\(P = 1.00 \ atm\)) and volume (\(V = 15.0 \ L\)), their moles will be the same as well. #Step III: Determine Mole Ratios of Reaction Components#

Step 3: Determine Mole Ratios of Reaction Components

Use the balanced chemical equation to determine the mole ratios of the reaction components: For nitrogen gas (\(N_2\)), the mole ratio is 1:1, For hydrogen gas (\(H_2\)), the mole ratio is 3:1, For ammonia gas (\(NH_3\)), the mole ratio is 2:1. #Step IV: Calculate Moles and Partial Pressure of Ammonia Gas (NH3)#

Step 4: Calculate Moles and Partial Pressure of Ammonia Gas

From stoichiometry, we could conclude that upon complete reaction, all the nitrogen and three times the amount of hydrogen would be used to form 2 times the amount of ammonia. Calculate moles of ammonia gas: \(n_{NH3} = 2 \times n_{N2}\) Use the moles of ammonia gas to find its partial pressure: \(P_{NH3} = \frac{n_{NH3}RT}{V}\) #Step V: Calculate the Final Volume of the Container#

Step 5: Calculate the Final Volume of the Container

Since the pressure remains constant inside the container, the final starting and ending pressures in the container (partial pressures of all gases) are the same. Total initial pressure: \(P_{Initial} = P_{N_2} + P_{H_2}\) Total final pressure: \(P_{Final} = P_{NH_3}\) Since the pressure inside the container is constant, we have: \(P_{Initial} = P_{Final}\) Solve for the final volume of the container: \(V_{Final} = \frac{P_{Initial}}{P_{Final}} \times V_{Initial}\) Now you can plug in the values obtained in the previous steps and calculate the answers for a) partial pressure of ammonia gas and b) the final volume of the container.

Key Concepts

Stoichiometry

Stoichiometry allows us to understand the quantitative relationships within a chemical reaction. It's like a recipe for baking a cake with exact measurements. In the context of nitrogen and hydrogen gases forming ammonia, stoichiometry uses the balanced chemical equation:\[ N_2(g) + 3H_2(g) \rightarrow 2NH_3(g) \]This equation tells us exactly how many molecules of reactants we need to make a certain amount of product. It's key to know that one molecule of nitrogen reacts with three molecules of hydrogen to make two molecules of ammonia.
This is why stoichiometry is crucial: it details that for one mole of nitrogen, three moles of hydrogen are needed. In return, the outcome is two moles of ammonia. Understanding these proportions helps determine amounts of reactants needed and products formed.

Partial Pressure

Partial pressure is an essential concept in gases because it measures the pressure each gas in a mixture would contribute if it alone occupied the entire volume. In our initial setup, both nitrogen and hydrogen gases each have a partial pressure of 1.00 atm.
When the reaction completes, the characteristics of the gases inside the container change. As the nitrogen and hydrogen react to form ammonia, the available moles of those gases decrease and ammonia's mole quantity increases. According to Dalton's Law, the total pressure inside the container remains constant, which allows calculations of ammonia’s partial pressure after the reaction completes.

• This pressure is proportional to the moles of gas.
• Only the completed reaction contributes to the final ammonia’s partial pressure.

Therefore, knowing how partial pressures change is vital for determining how much of each gas is present at different stages of the reaction.

Chemical Reactions

Chemical reactions are processes where substances (reactants) are transformed into different substances (products). In the nitrogen and hydrogen reaction forming ammonia, we see a simple chemical transformation symbolized by: - Reactants: Nitrogen and Hydrogen gases - Products: Ammonia gas
For the reaction to occur, nitrogen and hydrogen molecules collide with enough energy to break and form new bonds, creating ammonia. It’s important to remember that:

• Reactions can change the physical state and the chemical identity of substances.
• They often involve energy changes, although in this case, temperature is constant.

Understanding these transformations helps using stoichiometry accurately to predict products from given reactants.

Mole Ratios

Mole ratios are derived directly from the balanced chemical equation, revealing the smallest whole number relationship among reactants and products. For instance, the equation for ammonia formation:\[ N_2(g) + 3H_2(g) \rightarrow 2NH_3(g) \]From this, the mole ratios are:

• 1 mole of \(N_2\) to 3 moles of \(H_2\).
• 1 mole of \(N_2\) to 2 moles of \(NH_3\).
• 3 moles of \(H_2\) to 2 moles of \(NH_3\).

These ratios are essential for calculating how much product forms or how much reactant we need. Once the reaction has reached completion, these ratios tell us exactly how the initial amounts of reactants translate into the amount of ammonia produced. Using the mole ratios effectively is the key to solving stoichiometric problems, enabling us to convert between amounts of different substances involved in chemical equations.