Question

Nitrogen gas reacts with hydrogen gas to form ammonia gas \(\left(\mathrm{NH}_{3}\right) .\) Consider the following illustration representing the original reaction mixture in a \(15.0 \mathrm{L}\) container (the numbers of each molecule shown are relative numbers):Assume this reaction mixture goes to completion. The piston apparatus allows the container volume to change in order to keep the pressure constant at 1.00 atm. Assume ideal behavior and constant temperature. a. What is the partial pressure of ammonia in the container when the reaction is complete? b. What is the mole fraction of ammonia in the container when the reaction is complete? c. What is the volume of the container when the reaction is complete?

Short answer

The partial pressure of ammonia when the reaction is complete is 1.00 atm, as it is the only gas present in the container. The mole fraction of ammonia is 1.00 (100%), as both reactants are completely consumed in the reaction. Finally, the volume of the container when the reaction is complete can be calculated using the Ideal Gas Law equation with constant pressure and temperature, resulting in \( V_\text{final} = \frac{6\ molecules \times R \times T}{1.00 \mathrm{atm}} \).

Step-by-step solution

Reaction equation and stoihiometry

First, let's write the balanced equation for this reaction: \[ N_2 (g) + 3H_2 (g) \rightarrow 2NH_3 (g) \] From this equation, we can determine the stoichiometric coefficients for each reactant and product: 1 for N2, 3 for H2, and 2 for NH3.

Initial condition and limiting reagent

Based on the given illustration, we have the following initial number of molecules: \[ N_2 = 3\ molecules \] \[ H_2 = 9\ molecules \] To determine the limiting reagent, we can compare the current ratio of molecules to the stoichiometric coefficients. The initial ratio of nitrogen to hydrogen is \(\frac{3}{9} = \frac{1}{3}\), which matches exactly the stoichiometric coefficients of \(1:3\), meaning both reactants will be completely consumed in the reaction.

Moles of ammonia produced

According to the stoichiometry of the reaction, we have: \[ 1\ N_2 \xrightarrow{} 2\ NH_3 \] \[ 3\ N_2 \xrightarrow{} 6\ NH_3 \] So, 6 molecules of ammonia will be produced when the reaction is complete.

Mole fractions and partial pressure

Since both reactants are completely consumed, the reaction mixture will only consist of ammonia, thus making it the only gas present in the container. Hence, the mole fraction of ammonia will be 1.00 (100%). The partial pressure of ammonia will be equal to the total pressure inside the container, which is given as 1 atm.

Volume calculations

Now, we need to find the volume of the container when the reaction is complete. Since the piston apparatus keeps the pressure constant throughout the reaction, we will use the Ideal Gas Law equation, \(PV = nRT\), for both initial and final conditions. Initial condition: \[ P_\text{initial}V_\text{initial} = (n_\text{N2} + n_\text{H2})RT \] Final condition: \[ P_\text{final}V_\text{final} = n_\text{NH3}RT \] Where the final pressure is equal to the initial pressure: \[P_\text{initial} = P_\text{final} = 1.00 \mathrm{atm}\] We can rearrange the final condition equation to find the final volume: \[ V_\text{final} = \frac{n_\text{NH3}RT}{P_\text{final}} \] Given the initial volume, temperature, and number of moles of each gas, we can find the final volume of the container, when the reaction is complete: \[ V_\text{final} = \frac{6\ molecules \times R \times T}{1.00 \mathrm{atm}} \] Note that this fraction will have the correct units, since the volume will be proportional to the number of molecules, and the pressure and temperature are kept constant. So, the volume of the container when the reaction is complete can be found using this equation.

Key concepts

Partial Pressure of Gases

Understanding the partial pressure of gases is crucial when studying chemical reactions. Partial pressure refers to the pressure that a single gas within a mixture of gases would exert if it occupied the entire volume on its own. It's an important concept in chemistry because it helps us predict how gases will behave when they are part of a mixture.

For instance, in the given exercise, nitrogen and hydrogen gases react to form ammonia, and we need to determine the partial pressure of ammonia when the reaction is complete. Since ammonia is the only gas present at the end of the reaction, its partial pressure is equal to the total pressure within the container. This situation simplifies our calculations because the partial pressure of a single gas in a container (when it's the only gas present) is the same as the total pressure, which in this case is maintained at 1.00 atm.

Mole Fraction

The mole fraction is another essential concept in stoichiometry, representing the proportion of a particular substance within a mixture. It is calculated by dividing the number of moles of the substance by the total number of moles in the mixture. Represented by the symbol \(\chi\) (chi), the mole fraction is a way to express concentrations without relying on volume or mass.

In the given exercise solution, all reactants have reacted completely to form the product, ammonia. Therefore, the mole fraction of ammonia is 1.00 because there are no other gases left in the container. Represented mathematically, the mole fraction of ammonia \(\chi_{\mathrm{NH}_3}\) would be \(\frac{n_{\mathrm{NH}_3}}{n_{\mathrm{NH}_3}} = 1\). This represents a pure gas sample and indicates that the mole fraction can provide insight into the composition of the gas mixture resulting from the reaction.

Ideal Gas Law

The Ideal Gas Law, \(PV = nRT\), is a fundamental equation in understanding the behavior of gases under certain conditions. It relates the pressure (P), volume (V), number of moles (n), the gas constant (R), and temperature (T) of an ideal gas. Ideal gases are hypothetical gases that perfectly fit all the assumptions of the kinetic-molecular theory.

In the exercise, the Ideal Gas Law is used to calculate the final volume of the container after the reaction completes. Because the pressure is held constant by a piston apparatus and the temperature remains unchanged, we can infer that any change in the volume of the container is directly proportional to the number of moles of gas present. After the reaction, only ammonia gas is present, and its volume can be calculated knowing the number of moles, the ideal gas constant, and the fixed temperature and pressure conditions. The equation shows us that, with the right data, we can predict how much space a certain amount of gas will occupy under constant conditions, which is crucial in various scientific and industrial processes.