Question

Nitrogen can react with steam to form ammonia and nitrogen oxide gases. A 20.0-L sample of nitrogen at \(173^{\circ} \mathrm{C}\) and \(772 \mathrm{~mm} \mathrm{Hg}\) is made to react with an excess of steam. The products are collected at room temperature \(\left(25^{\circ} \mathrm{C}\right)\) into an evacuated flask with a volume of \(15.0 \mathrm{~L}\). (a) Write a balanced equation for the reaction. (b) What is the total pressure of the products in the collecting flask after the reaction is complete? (c) What is the partial pressure of each of the products in the flask?

Short answer

Question: Write the balanced equation for the reaction between nitrogen and steam and calculate the total and partial pressures of the products in the collecting flask. Answer: The balanced equation for the reaction between nitrogen and steam is: $$N_2(g) + 2H_2O(g) \rightarrow 2NH_3(g) + NO(g)$$. The total pressure of the products in the collecting flask is approximately 2.1345 atm. The partial pressure of ammonia in the flask is approximately 1.4234 atm, and the partial pressure of nitrogen oxide is approximately 0.7111 atm.

Step-by-step solution

(a) Balanced Equation

First, we need to write a balanced equation for the reaction between nitrogen and steam (water vapor) to form ammonia and nitrogen oxide gases. The unbalanced equation is: $$N_2 + H_2O \to NH_3 + NO$$ Balancing the equation, we have: $$N_2 + 2H_2O \to 2NH_3 + NO$$ The balanced equation is: $$ N_2(g) + 2H_2O(g) \rightarrow 2NH_3(g) + NO(g) $$

(b) Calculate the Total Pressure of Products

First, we will convert the initial nitrogen pressure given in mm Hg to atmospheres (1 atm = 760 mm Hg). Initial Pressure of Nitrogen(\(P_N\)): $$P_N = \frac{772 \mathrm{~mmHg}}{760 \mathrm{~mmHg/atm}} = 1.0158 \mathrm{~atm}$$ Next, let's convert the temperature in degrees Celsius to Kelvin: Initial Temperature(\(T_1\)): $$T_1 = 173^{\circ} C + 273.15 = 446.15 \mathrm{K}$$ Final Temperature(\(T_2\)): $$T_2 = 25^{\circ} C + 273.15 = 298.15 \mathrm{K} $$ Now we need to find the moles of nitrogen(\(n_N\)) using Ideal Gas Law equation. Assume the volume (\(V_1\)) of nitrogen is 20.0 L. $$PV = nRT$$ $$n_N = \frac{P_NV_1}{RT_1}$$ We can now plug in the values and R (Ideal Gas Constant) = 0.0821 L atm / K mol. $$n_N = \frac{1.0158 \mathrm{~atm} \times 20.0\mathrm{~L}}{0.0821\mathrm{~L~atm/mol~K} \times 446.15\mathrm{~K}} = 0.5469 \mathrm{~mol}$$ Using stoichiometry, we can find the moles of ammonia (\(n_{NH_3}\)) and nitrogen oxide (\(n_{NO}\)) formed: $$n_{NH_3} = 2 \times n_N = 2 \times 0.5469\mathrm{~mol} = 1.0938\mathrm{~mol}$$ $$n_{NO} = n_N = 0.5469\mathrm{~mol}$$ Now, we will find the total moles of products and then calculate the total pressure. Total moles of products (\(n_{total}\)) = \(1.0938\mathrm{~mol} + 0.5469\mathrm{~mol}= 1.6407\mathrm{~mol}\) The volume of the collecting flask (\(V_2\)) is given as 15.0 L. We will use this volume and the total moles of products(\(n_{total}\)) to find the total pressure of the products in the flask(\(P_{total}\)) using the Ideal Gas Law. $$ P_{total} = \frac{n_{total}RT_2}{V_2}$$ $$ P_{total} = \frac{1.6407\mathrm{~mol} \times 0.0821 \mathrm{~L~atm/mol~K} \times 298.15 \mathrm{~K}}{15.0\mathrm{~L}} ≈ 2.1345\mathrm{~atm} $$ Total pressure of products in the collecting flask after the reaction is ≈ 2.1345 atm.

(c) Calculate Partial Pressures

We can now find the partial pressure of each product in the flask by using the mole fractions and the total pressure. Mole fraction is calculated as the moles of an individual component divided by the total moles. Mole fraction of ammonia (\(x_{NH_3}\)): $$x_{NH_3} = \frac{n_{NH_3}}{n_{total}} = \frac{1.0938\mathrm{~mol}}{1.6407\mathrm{~mol}} ≈ 0.6668$$ Mole fraction of nitrogen oxide (\(x_{NO}\)): $$x_{NO} = \frac{n_{NO}}{n_{total}} = \frac{0.5469\mathrm{~mol}}{1.6407\mathrm{~mol}} ≈ 0.3332$$ Partial pressure of ammonia (\(P_{NH_3}\)): $$P_{NH_3} = x_{NH_3} \times P_{total} = 0.6668 \times 2.1345\mathrm{~atm} ≈ 1.4234\mathrm{~atm}$$ Partial pressure of nitrogen oxide (\(P_{NO}\)): $$P_{NO} = x_{NO} \times P_{total} = 0.3332 \times 2.1345\mathrm{~atm} ≈ 0.7111\mathrm{~atm}$$ The partial pressure of ammonia in the flask is approximately 1.4234 atm, and the partial pressure of nitrogen oxide is approximately 0.7111 atm.

Key concepts

Understanding Stoichiometry

Stoichiometry is the mathematical relationship between the amounts of reactants and products in a chemical reaction. It is based on the conservation of mass and the concept that matter is neither created nor destroyed during a chemical reaction.

When solving stoichiometry problems, the first step is to write a balanced chemical equation. This provides the ratio of moles of each reactant and product involved. In the provided exercise, the balanced equation for the reaction between nitrogen and steam forming ammonia and nitrogen oxide was established, which is critical for further calculations. Using the coefficients from the balanced equation, we determine the mole ratio of reactants to products. This relationship allows us to predict the amount of products formed from a given quantity of reactants or vice versa.

For example, in the given problem, the balanced equation tells us that 1 mole of nitrogen reacts with 2 moles of water to produce 2 moles of ammonia and 1 mole of nitrogen oxide. With the mole ratio, we move on to use the initial conditions to calculate how much product can be formed from the given reactants. This step is essential for understanding the quantitative aspect of chemistry and predicting the outcome of the reaction.

Applying the Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and the amount of an ideal gas. It is commonly expressed as: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of the gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Understanding this law is important for predicting how gases will behave under different conditions. Converting units to match the gas constant used is necessary—pressure to atmospheres and temperature to Kelvin—as illustrated in the exercise. In the exercise solution, the Ideal Gas Law was utilized to determine the moles of nitrogen initially present. By identifying the moles and applying stoichiometry principles, we can calculate the amounts of the products. It's critical to note that the Ideal Gas Law assumes gases behave ideally, which is a good approximation under many conditions but may not hold for all gases at high pressures or low temperatures.

Calculating Partial Pressures

Partial pressure is the pressure that an individual gas component in a mixture of gases would exert if it occupied the entire volume of the mixture at the same temperature. Dalton's Law of partial pressures states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas in the mixture.

To calculate the partial pressure of each gas, as shown in the exercise, we start by determining the mole fraction of each gas. The mole fraction is the ratio of the number of moles of a particular gas to the total number of moles in the mixture. Multiplying the mole fraction of each gas by the total pressure yields the partial pressure of that gas.

In practical terms, knowing partial pressures is essential for industries that use gas mixtures, such as in respiratory therapy, scuba diving, and the synthesis of chemicals. Understanding how to manipulate and calculate partial pressures is key to controlling and predicting the behavior of gas mixtures in various scientific applications.