Question

The composition of moist air is given on a molar basis to be 78 percent \(\mathrm{N}_{2}, 20\) percent \(\mathrm{O}_{2}\), and 2 percent water vapor. Determine the mass fractions of the constituents of air.

Short answer

Answer: The mass fractions of the constituents of the moist air are approximately: - 76.36% Nitrogen (N2) - 22.38% Oxygen (O2) - 1.26% Water vapor (H2O)

Step-by-step solution

Identify the molecular weights of the constituents

First, we need to find the molecular weights of nitrogen (N2), oxygen (O2), and water vapor (H2O). The molecular weights are: Molecular weight of N2: 28 g/mol Molecular weight of O2: 32 g/mol Molecular weight of H2O: 18 g/mol

Calculate the moles of each constituent

Next, we need to compute the moles of nitrogen (n_N2), oxygen (n_O2), and water vapor (n_H2O) in a sample of moist air. Let's assume we have 100 moles of air. Then, using the given molar percentages: n_N2 = 0.78 * 100 = 78 moles n_O2 = 0.20 * 100 = 20 moles n_H2O = 0.02 * 100 = 2 moles

Calculate the mass of each constituent

Now, we can compute the mass of nitrogen (m_N2), oxygen (m_O2), and water vapor (m_H2O) in the 100 moles of air sample using the molecular weights: m_N2 = n_N2 * molecular weight of N2 m_N2 = 78 * 28 = 2184 g m_O2 = n_O2 * molecular weight of O2 m_O2 = 20 * 32 = 640 g m_H2O = n_H2O * molecular weight of H2O m_H2O = 2 * 18 = 36 g

Calculate the total mass of the air sample

Before determining the mass fractions, we need to calculate the total mass of the air sample (m_total): m_total = m_N2 + m_O2 + m_H2O = 2184 + 640 + 36 = 2860 g

Determine the mass fractions of the constituents

Finally, we can compute the mass fractions of nitrogen (mass_fraction_N2), oxygen (mass_fraction_O2), and water vapor (mass_fraction_H2O) in the air sample: mass_fraction_N2 = m_N2 / m_total = 2184 / 2860 = 0.76364 mass_fraction_O2 = m_O2 / m_total = 640 / 2860 = 0.22378 mass_fraction_H2O = m_H2O / m_total = 36 / 2860 = 0.012587 The mass fractions of the constituents of the moist air are approximately: - 76.36% Nitrogen (N2) - 22.38% Oxygen (O2) - 1.26% Water vapor (H2O)

Key concepts

Molecular Weight Calculation

Understanding the concept behind the molecular weight calculation is essential when working with chemical compositions of mixtures like air. Molecular weight, also termed molecular mass, is the sum of the atomic weights of all atoms in a molecule. It's typically expressed in atomic mass units (u) or grams per mole (g/mol).

In our exercise, we calculate the molecular weight of each constituent in the air mixture: nitrogen (N₂), oxygen (O₂), and water vapor (H₂O). Seeing that nitrogen and oxygen are diatomic molecules (consisting of two atoms each), we multiply the atomic weight of a single nitrogen or oxygen atom by two. For water vapor, the molecular weight is the sum of the atomic weights of two hydrogen atoms and one oxygen atom.

For example:

• The atomic weight of nitrogen (N) is approximately 14 u, hence for nitrogen (N₂), it’s 14 u * 2 = 28 g/mol.
• The atomic weight of oxygen (O) is roughly 16 u, so for oxygen (O₂), it’s 16 u * 2 = 32 g/mol.
• For water vapor (H₂O), it’s (1 u * 2) + 16 u = 18 g/mol, considering that the atomic weight of hydrogen (H) is about 1 u.

The concept of molecular weight is a basic yet crucial element in this calculation, as it bridges the gap between the number of molecules (moles) and their mass.

Mole to Mass Conversion

The mole to mass conversion process is an important step in chemistry, allowing one to translate the amount of substance from moles to grams. A mole is a unit of measurement that denotes a quantity of 6.022 x 10²³ entities, be they atoms, molecules, ions, or other particles.

One mole of any substance has a mass equal to its molecular weight in grams. The conversion relies on this direct relationship. Consequently, to find the mass of a substance, you multiply its amount in moles by its molecular weight.

In our exercise example, we are given moles of each constituent based on their percentage composition in air. We perform the conversion by:

• First calculating the amount of moles for each constituent.
• Then multiplying this amount by the molecular weight of the specific compound.

For instance, if we have 78 moles of nitrogen (N₂), the conversion to mass would be: \( 78 \text{ moles } \times 28 \text{ g/mol} = 2184 \text{ g}. \) This calculation is repeated for each constituent to determine their individual masses in grams.

Mass Fraction Determination

The mass fraction determination is a quantitative analysis method used to express the concentration of each component in a mixture. The mass fraction, also called weight fraction, is a dimensionless number describing the proportion of one substance's mass to the total mass of the mixture.

To calculate mass fractions, we proceed by dividing the mass of each component by the total mass of the mixture, as shown in our exercise. After converting moles to mass for nitrogen, oxygen, and water vapor in step 3, and then determining the total mass in step 4, mass fractions are found using: \( \text{mass fraction} = \frac{\text{mass of component}}{\text{total mass}} \) For example, if the mass of nitrogen in the air sample is 2184 g and the total mass of the air sample is 2860 g, the mass fraction of nitrogen would be approximately: \( \text{mass fraction of N}_2 = \frac{2184}{2860} = 0.76364 \) or 76.36%.

This step gives a clear perspective on the composition of the mixture in terms of mass, which can be more useful than molar composition in various practical applications such as engineering and atmospheric science.