Question

If average molecular wt. of air is 29, then assuming \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) gases are there which options are correct regarding composition of air: (i) \(75 \%\) by mass of \(\mathrm{N}_{2}\) (ii) \(75 \%\) by moles \(\mathrm{N}_{2}\) (iii) \(72.41 \%\) by mass of \(\mathrm{N}_{2}\) (a) only (i) is correct (b) only (ii) is correct (c) both (ii) and (iii) are correct (d) both (i) and (ii) are correct

Short answer

(iii) is correct with 72.41% by mass of \(N_2\). Neither (i) nor (ii) are correct.

Step-by-step solution

- Calculate the mass percentages of N2 and O2

Assume a 100g sample of air. The molar mass of \(N_2\) is 28 g/mol and for \(O_2\) is 32 g/mol. Use the average molecular weight to calculate the mass composition.

- Derive mass percentages from the molecular weight ratio

If \(N_2\) is x percent by mass, then \(O_2\) is \(100-x\) percent by mass. The average molecular weight equation is: \(0.29x + 0.32(100-x) = 29\). Solve for x which represents the mass percentage of \(N_2\).

- Calculate the mole fraction of N2

After finding the mass percentage, calculate moles of \(N_2\) and \(O_2\) in 100g air. The mole fraction is then the moles of \(N_2\) divided by the total moles of air.

- Determine correct option(s)

Compare the calculated mass percentage and mole fraction of \(N_2\) to the given options to identify the correct statements.

Key concepts

Molecular Weight Calculation

Understanding how to calculate the molecular weight of a compound is fundamental in chemistry, especially for analytical and quantitative studies like those needed for the Joint Entrance Examination (JEE). Molecular weight, also known as molecular mass, is the sum of the atomic masses of all the atoms in a molecule.

For example, to calculate the molecular weight of water (H₂O), we add the atomic mass of hydrogen (approximately 1 u for each of the two hydrogen atoms) to the atomic mass of oxygen (approximately 16 u). Thus, the molecular weight of water is around 18 u (1+1+16). Units used are atomic mass units (u) or grams per mole (g/mol).

Similarly, for compounds like nitrogen gas (N₂) and oxygen gas (O₂), which are diatomic molecules, we would calculate their molecular weights by summing the atomic masses of two nitrogen atoms (14 u each for a total of 28 u) and two oxygen atoms (16 u each for a total of 32 u) respectively. This calculation is pivotal when further analyzing the composition of mixtures like air when expressed in terms of average molecular weight.

Mass Percentage in Chemistry

Mass percentage is another key concept in chemistry that indicates the concentration of a component in a mixture or a solution. It is defined as the mass of a solute divided by the total mass of the solution, multiplied by 100%. When dealing with mixtures of gases like air, the mass percentage can give us an insight into the distribution of various gases in the atmosphere.

For instance, if we say that the mass percentage of nitrogen in air is 75%, this means that for every 100 grams of air, 75 grams are nitrogen. The calculation typically involves assuming a convenient total mass for the sample—often 100 grams to simplify calculations—and then using the known molecular weights of the component gases to derive the mass of each gas within that sample. Understanding the relationship between molecular weight and mass percentage allows chemists to find out the composition of mixtures, and this principle is applied when solving problems related to air composition for JEE Physical Chemistry.

Mole Fraction Concept

The mole fraction is an expression of the concentration of a component in a mixture without regard to the masses of the substances involved. It is the ratio of the number of moles of a given component to the total number of moles in the mixture. Mole fraction is dimensionless, as it represents a ratio.

To calculate the mole fraction, one must first determine the number of moles of each component using the formula: number of moles = mass of the component / molar mass of the component. After finding the moles of the components, the mole fraction can be found by dividing the moles of the component of interest by the total moles in the mixture.

In the context of air composition, after calculating the mass percentages of nitrogen (_2) and oxygen (O_2), we would convert these masses to moles given their molecular weights and then compute the mole fractions. This is highly relevant for the JEE Physical Chemistry section where students need to understand how to relate mole fractions to the actual composition of mixtures like air.