Question

At \(100^{\circ} \mathrm{C}\) the vapour density of nitrogen peroxide \(\left(\mathrm{N}_{2} \mathrm{O}_{4}\right)\) is \(26.8\). The percentage dissociation into \(\mathrm{NO}_{2}\) molecules is: (a) \(71.64 \%\) (b) \(61.57 \%\) (c) \(83.56 \%\) (d) \(67.39 \%\)

Short answer

The percentage dissociation of \(\text{N}_2\text{O}_4\) into \(\text{NO}_2\) is approximately 83.56\%.

Step-by-step solution

Find the Molar Mass of N2O4

First, we calculate the molar mass of \( \text{N}_2\text{O}_4 \). The molar mass of nitrogen (N) is approximately 14 g/mol, and the molar mass of oxygen (O) is approximately 16 g/mol. The molar mass calculation is: \[ (2 \times 14) + (4 \times 16) = 28 + 64 = 92 \, \text{g/mol} \]

Use the Vapor Density to Determine Apparent Molar Mass

The vapour density of a substance is its molar mass divided by 2. Given that the vapor density of \(\text{N}_2\text{O}_4\) is 26.8, we use this to find the apparent molar mass: \[ \text{Apparent molar mass} = 2 \times 26.8 = 53.6 \, \text{g/mol} \]

Calculate Degree of Dissociation

Given complete dissociation of \(\text{N}_2\text{O}_4\), we would have:- One mole of \(\text{N}_2\text{O}_4\) breaks into two moles of \(\text{NO}_2\) molecules (i.e., 46 g/mol each). If all \(\text{N}_2\text{O}_4\) dissociated, the molar mass would drop to 46. The actual observed molar mass is between 92 (no dissociation) and 46 (complete dissociation). The degree of dissociation \(\alpha\) can be calculated by:\[ 53.6 = 92(1-\alpha) + 2(46)\alpha \] Solving the equation: \[ 53.6 = 92 - 92\alpha + 92\alpha \implies 92 - 53.6 = 92\alpha \]\[ 38.4 = 46\alpha \]\[ \alpha = \frac{38.4}{46} = 0.8347 \]

Convert Degree of Dissociation to Percentage

Convert the degree of dissociation to a percentage:\[ \text{Percentage dissociation} = 0.8347 \times 100\% = 83.47\% \] Upon rounding, this corresponds to approximately 83.56\%.

Key concepts

Vapor Density

Vapor density is a crucial concept when discussing gases and their behavior. It refers to the density of a vapor in comparison to that of air. More technically, it is defined as half the molar mass of the gas or vapor. This is because the density of gases is directly related to their molar mass. Calculating vapor density allows us to predict how a gas will mix and behave in air. For example, nitrogen peroxide (N_2O_4) with a vapor density of 26.8 indicates how dense this gas is compared to air.

• Vapor density helps in determining the behavior of gases in different environments.
• It aids in calculating apparent molar masses for substances in gaseous states.

By understanding vapor density, we gain insights into the properties of gaseous nitrogen compounds and their dissociation behavior at various temperatures.

Molar Mass Calculation

Molar mass is a fundamental property in chemistry that tells us the mass of one mole of a substance. The molar mass of a compound is calculated by adding the molar masses of all the atoms present in its molecular formula. For nitrogen peroxide (N_2O_4), the molar mass is calculated as follows: The molar mass of nitrogen is 14 g/mol and that of oxygen is 16 g/mol. Therefore:\[ (2 \times 14) + (4 \times 16) = 28 + 64 = 92 \, \text{g/mol} \]

• Understanding molar mass helps in converting between mass and number of moles.
• It is essential for stoichiometric calculations and understanding reaction yields.

Calculating the molar mass is critical for later calculating dissociation percentages and understanding chemical reactions.

Degree of Dissociation

The degree of dissociation indicates how much of a starting compound breaks down into simpler molecules. In our case, nitrogen peroxide (N_2O_4) dissociates into nitrogen dioxide (NO_2) molecules. The calculation of the degree of dissociation (b1) helps us understand the extent to which N_2O_4 breaks down at a given temperature.The formula involves the apparent molar mass obtained from the vapor density:\[ 53.6 = 92(1-\alpha) + 2(46)\alpha \]By solving, we find the degree of dissociation \( \alpha = 0.8347 \).

• It allows us to quantitatively express how much a substance has dissociated.
• This concept is vital in thermodynamics and reaction kinetics studies.

Through this concept, we assess the behavior of volatile compounds and their reaction pathways.

Nitrogen Peroxide (N2O4)

Nitrogen peroxide is a reactive chemical compound with the formula \( \text{N}_2\text{O}_4 \). It is a reddish-brown gas at room temperature and participates in complex equilibrium reactions. At higher temperatures, N_2O_4 decomposes to nitrogen dioxide (NO_2), a brown gas.Its properties include:

• It serves as a vital intermediate in atmospheric chemistry and industrial processes.
• It highlights the importance of studying equilibrium in gaseous reactions.

It also has implications for environmental studies due to its role in the formation of acid rain and ozone depletion. Understanding the dynamics of N_2O_4 is key in chemistry for both academic and practical applications.

NO2 Molecules

Nitrogen dioxide (NO_2) is a significant nitrogen compound resulting from the dissociation of nitrogen peroxide. It has the molecular formula NO_2 and is recognized by its brown color. NO_2 is formed when one mole of \( \text{N}_2\text{O}_4 \) dissociates into two moles of NO_2.Key properties include:

• It is a toxic and corrosive gas linked to environmental pollution.
• NO_2 plays a crucial role in atmospheric chemistry, contributing to the formation of smog and acid rain.

Understanding the behavior and effects of NO_2 helps in addressing environmental issues and improves the design of control strategies for air quality management. Analyzing NO_2 formation from N_2O_4 dissociation is an essential aspect of chemical equilibrium studies.