Problem 173 - Chemistry The Science in Context

Question

In some aquatic ecosystems, nitrate (NO \(_{3}^{-}\) ) is converted to nitrite \(\left(\mathrm{NO}_{2}^{-}\right),\) which then decomposes to nitrogen and water. As an example of this second reaction, consider the decomposition of ammonium nitrite: $$ \mathrm{NH}_{4} \mathrm{NO}_{2}(a q) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell) $$ What would be the change in pressure in a sealed \(10.0 \mathrm{L}\) vessel due to the formation of \(\mathrm{N}_{2}\) gas when the ammonium nitrite in \(1.00 \mathrm{L}\) of \(1.0 \mathrm{MNH}_{4} \mathrm{NO}_{2}\) decomposes at \(25^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The change in pressure in the sealed 10.0 L vessel is 2.467254 bar.

Step 1: Find the moles of ammonium nitrite and nitrogen gas

First, let's calculate the number of moles of ammonium nitrite (NH4NO2). The solution has a concentration of 1.0 M and a volume of 1.00 L, so we can multiply these values to obtain: moles of NH4NO2 = concentration × volume = 1.0 mol/L × 1.00 L = 1.00 mol From the balanced chemical equation, we can see that 1 mole of NH4NO2 produces 1 mole of N2 gas. Thus, the number of moles of N2 gas produced is equal to the moles of NH4NO2: moles of N2 = moles of NH4NO2 = 1.00 mol

Step 2: Calculate the initial pressure

Since the reaction hasn't started yet, there is initially no nitrogen gas in the vessel. We can assume the initial pressure inside the vessel only comes from the trapped air. We are not given the partial pressure of nitrogen gas in the air, but we are asked to find the change in pressure due to the formation of N2 from the decomposition, so we could also assume the initial pressure is 0. We will subtract the initial pressure from the final pressure to find the change in pressure later.

Step 3: Calculate the final pressure

Use the ideal gas law formula, with the constant R = 8.314 J/(mol·K), the given temperature (25°C = 298.15 K), and the volume of the vessel (10.0 L = 0.010 m³) to determine the final pressure: PV = nRT P = nRT/V P = (1.00 mol)(8.314 J/(mol·K))(298.15 K)/(0.010 m³) P = 246.7254 J/m³ 1 J/m³ is equal to 10^(-5) bar. Therefore, the final pressure is: P = 246.7254 J/m³ × 10^(-5) bar/J/m³ = 2.467254 bar

Step 4: Calculate the change in pressure

Now we will subtract the initial pressure from the final pressure to obtain the change in pressure: ΔP = P_final - P_initial Since we assumed the initial pressure is 0: ΔP = 2.467254 bar - 0 bar = 2.467254 bar The change in pressure in the sealed 10.0 L vessel due to the formation of N2 gas when the ammonium nitrite decomposes at 25°C is 2.467254 bar.

Key Concepts

Ideal Gas Law

The ideal gas law is a cornerstone of chemistry and physics, providing a clear relationship between the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas. It’s given by the equation:

\[ PV = nRT \]
Where R is the ideal gas constant, with a value of 8.314 J/(mol·K) in SI units. To understand the influence of gaseous products in a chemical reaction, you can apply the ideal gas law to calculate the final pressure of gas in a container. When solving problems involving the ideal gas law:

• Always pay attention to unit consistency. Convert temperature to Kelvin and volumes to cubic meters if necessary.
• Make use of the fact that this equation allows you to find any of the four variables provided the other three are known.

In our specific example with ammonium nitrite decomposition, the law allows us to predict the pressure exerted by the nitrogen gas after the reaction in a sealed vessel, assuming ideal behavior.

Stoichiometry

Stoichiometry involves the calculation of reactants and products in chemical reactions. In stoichiometry, a balanced chemical equation serves as the roadmap for these calculations, letting us know precisely how many moles of each substance are involved.

In the given example of ammonium nitrite decomposing to nitrogen and water, each mole of ammonium nitrite yields a mole of nitrogen gas, which indicates a one-to-one molar ratio. Understanding these ratios is crucial because they allow you to predict the amount of product formed from a given quantity of reactant. To improve your stoichiometric calculations, consider these tips:

• Ensure the chemical equation is balanced before performing any stoichiometric calculations.
• Convert all your quantities to moles, as this will simplify using the mole ratios from the balanced equation.

This forms the foundation for determining how much of a chemical is needed or produced in a given reaction.

Molar Concentration

Molar concentration, often represented as molarity and denoted by M, is a measure of the concentration of a solute in a solution. It’s defined as the number of moles of solute divided by the volume of solution in liters. Its unit is mol/L.

In our exercise, we’re given that ammonium nitrite has a molarity of 1.0 M in 1.00 L of solution. This directly tells us that there is 1.00 mole of ammonium nitrite present in that volume since:
\[ \text{molar concentration} = \frac{\text{moles of solute}}{\text{volume of solution (L)}} \]
To successfully handle problems involving molar concentrations:

• Ensure you correctly relate volume to molarity to find the number of moles needed for your calculations.
• Recognize that molar concentration doesn’t change if the chemical reaction occurs in a closed system.

Remember, molarity helps transition quantities from the macroscopic world we see to the molecular world of atoms and molecules where the reaction actually happens.

Balanced Chemical Equation

A balanced chemical equation is vital for studying any chemical reaction because it satisfies the Law of Conservation of Mass, which states that matter cannot be created or destroyed in an isolated system. The equation reflects a quantitative relationship between reactants and products, with the same number of atoms of each element present on both sides of the equation.

The equation for the decomposition of ammonium nitrite to nitrogen and water is:
\[ \text{NH}_4\text{NO}_2 (aq) \rightarrow N_2(g) + 2H_2O(l) \]
This equation is already balanced with one nitrogen (N) molecule and two water molecules produced per one ammonium nitrite molecule decomposed. To improve your understanding of chemical equations, remember:

• Check that each element’s atom count is the same on both sides of the equation.
• Use coefficients to balance the equation instead of altering chemical formulas.

A balanced equation is crucial as it forms the basis for stoichiometry, and therefore, for solving problems involving chemical reactions.