Question

A flask of volume \(5.00 \mathrm{~L}\) is evacuated and \(43.78 \mathrm{~g}\) of solid dinitrogen tetroxide, \(\mathrm{N}_{2} \mathrm{O}_{4}\), is introduced at \(-196^{\circ} \mathrm{C}\). The sample is then warmed to \(25^{\circ} \mathrm{C}\), during which time the \(\mathrm{N}_{2} \mathrm{O}_{4}\) vaporizes and some of it dissociates to form brown \(\mathrm{NO}_{2}\) gas. The pressure slowly increases until it stabilizes at \(2.96\) atm. (a) Write a balanced equation for the reaction. (b) If the gas in the flask at \(25^{\circ} \mathrm{C}\) were all \(\mathrm{N}_{2} \mathrm{O}_{4}\), what would the pressure be? (c) If all the gas in the flask converted into \(\mathrm{NO}_{2}\), what would the pressure be? (d) What are the mole fractions of \(\mathrm{N}_{2} \mathrm{O}_{4}\) and \(\mathrm{NO}_{2}\) once the pressure stabilizes at \(2.96 \mathrm{~atm}\) ?

Short answer

Balanced reaction: 2 N2O4(g) -> 4 NO2(g). Pressure if only N2O4 were present at 25°C: calculated using Ideal Gas Law with moles of N2O4. Pressure if all turned into NO2: calculated as twice the moles of N2O4. Mole fractions can be derived from the stabilized pressure and the Ideal Gas Law, using x for the fraction of N2O4 that did not dissociate and 2(1-x) for NO2.

Step-by-step solution

Write a balanced equation for the reaction

The reaction that describes the dissociation of dinitrogen tetroxide (N_2O_4) into nitrogen dioxide (NO_2) is as follows: 2 N_2O_4(g) -> 4 NO_2(g).The equation explains that one mole of N_2O_4 dissociates into two moles of NO_2.

Calculate moles of N2O4

Use the molecular weight of N_2O_4 (Molar mass = 92.02 g/mol) to determine the moles of N_2O_4 introduced into the flask: moles N_2O_4 = 43.78g / 92.02 g/mol.

Calculate pressure if only N2O4 were present

Using the Ideal Gas Law (PV=nRT), where P is pressure, V is volume, n is number of moles, R is the gas constant (0.0821 L·atm/mol·K), and T is temperature in Kelvin. First, convert 25°C to Kelvin: T = 25 + 273.15. Then, solve for pressure, P: P = (moles of N2O4) * R * T / V.

Calculate pressure if all N2O4 converted to NO2

Since the equation shows that 1 mole of N_2O_4 gives 2 moles of NO_2, there would be twice as many moles of gas. Using Ideal Gas Law again: P = (2 * moles of N2O4) * R * T / V.

Calculate the mole fractions of N2O4 and NO2

Let x be the fraction of N_2O_4 that remains as N_2O_4 and (1-x) be the fraction that converts to NO_2. Total moles = x * (original moles of N2O4) + 2 * (1-x) * (original moles of N2O4). Use the stabilized pressure (2.96 atm) and the Ideal Gas Law to solve for x. The mole fraction of N_2O_4 is x and of NO_2 is 2*(1-x).

Key concepts

Dinitrogen Tetroxide Dissociation

Understanding the chemical behavior of dinitrogen tetroxide (\r\(N_2O_4\)) is critical when studying chemical equilibrium and reaction dynamics. \r
Dinitrogen tetroxide is a colorless gas at room temperature and can readily dissociate into nitrogen dioxide (\r\(NO_2\)), a brown gas, when conditions such as temperature change. This transformation is an example of a reversible chemical equilibrium.
In particular, the dissociation reaction can be represented by the chemical equation\r\[2 N_2O_4(g) \rightarrow 4 NO_2(g)\]. \r
The significance of this balanced equation lies in the stoichiometry - it exhibits that two molecules of \r\(N_2O_4\) decompose to form four molecules of \r\(NO_2\), essentially doubling the amount of gas particles. This has direct implications when considering factors such as pressure changes and gas concentrations in a closed system, as in the textbook's flask example. \r
Moreover, the extent of this reaction is influenced by conditions like temperature and pressure, and it can reach a state of dynamic equilibrium where the rates of the forward and reverse reactions are equal, ensuring constant concentrations of both \r\(N_2O_4\) and \r\(NO_2\) over time.

Ideal Gas Law

The ideal gas law is a fundamental equation in understanding how gases behave under different conditions. \r
It is articulated as:\r\[PV = nRT\], \r
where \r\(P\) represents the pressure, \r\(V\) denotes volume, \r\(n\) is the number of moles of the gas, \r\(R\) is the universal gas constant, and \r\(T\) is the absolute temperature in Kelvin.
The versatility of this law allows us to interconnect these variables and deduce one when the others are known. In the context of a chemical reaction like dinitrogen tetroxide dissociation, the ideal gas law provides a way to calculate the theoretical pressures that would be exerted by the initial and potential products of the reaction if they occupied the flask alone, assuming ideal gas behavior.
Moreover, by observing the changes in pressure, it is possible to infer the progress and extent of the dissociation of \r\(N_2O_4\) inside the flask. These calculated pressures are vital for determining other parameters, such as mole fractions, and for assessing how closely real gases behave to the ideal model. The Real behavior of gases can deviate from ideality due to intermolecular forces and the volumes occupied by gas molecules, especially under high pressure or low temperatures.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture of substances. \r
It can be defined as the ratio of the number of moles of a particular substance to the total number of moles of all substances present in the mixture. Represented by the symbol \r\(X\), the mole fraction of a substance \r\(A\) is mathematically described as:\r\[X_A = \frac{n_A}{n_{total}}\], \r
where \r\(n_A\) is the number of moles of the substance and \r\(n_{total}\) is the total number of moles of all substances in the mixture.
Mole fractions are dimensionless and always add up to 1 for all components in the mixture. In the case of dinitrogen tetroxide dissociating into nitrogen dioxide, calculating the mole fractions of \r\(N_2O_4\) and \r\(NO_2\) once the pressure stabilizes involves determining the extent of the dissociation and the total moles of gas in the system. The mole fraction is particularly useful because it remains unaffected by changes in temperature or pressure, providing a convenient measure of concentration for comparisons and calculations in chemical equations and reaction dynamics.