Question

If \(125 \mathrm{~g}\) of \(\mathrm{N}_{2}\) are mixed with \(175 \mathrm{~g}\) of \(\mathrm{O}_{2}\), what is the mole fraction of each component?

Short answer

The mole fraction of \(\mathrm{N}_2\) is approximately 0.449, and for \(\mathrm{O}_2\) it is approximately 0.551.

Step-by-step solution

Calculate Moles of Nitrogen

First, we determine the number of moles of nitrogen gas (\(\mathrm{N}_2\)). The molar mass of \(\mathrm{N}_2\) is \(28.02\, \mathrm{g/mol}\). Thus, the moles of \(\mathrm{N}_2\) are calculated as follows:\[\text{Moles of } \mathrm{N}_2 = \frac{125 \mathrm{~g}}{28.02 \mathrm{~g/mol}} \approx 4.46 \mathrm{~mol}\]

Calculate Moles of Oxygen

Next, we calculate the number of moles of oxygen gas (\(\mathrm{O}_2\)). The molar mass of \(\mathrm{O}_2\) is \(32.00\, \mathrm{g/mol}\). The moles of \(\mathrm{O}_2\) are:\[\text{Moles of } \mathrm{O}_2 = \frac{175 \mathrm{~g}}{32.00 \mathrm{~g/mol}} \approx 5.47 \mathrm{~mol}\]

Calculate Total Moles

Now, we find the total number of moles by adding the moles of \(\mathrm{N}_2\) and \(\mathrm{O}_2\):\[\text{Total moles} = 4.46 \mathrm{~mol} + 5.47 \mathrm{~mol} = 9.93 \mathrm{~mol}\]

Calculate Mole Fraction of Nitrogen

The mole fraction of \(\mathrm{N}_2\) is given by dividing the moles of \(\mathrm{N}_2\) by the total moles:\[\chi_{\mathrm{N}_2} = \frac{4.46 \mathrm{~mol}}{9.93 \mathrm{~mol}} \approx 0.449\]

Calculate Mole Fraction of Oxygen

Similarly, the mole fraction of \(\mathrm{O}_2\) is calculated by dividing the moles of \(\mathrm{O}_2\) by the total moles:\[\chi_{\mathrm{O}_2} = \frac{5.47 \mathrm{~mol}}{9.93 \mathrm{~mol}} \approx 0.551\]

Key concepts

Moles of Gas

The concept of moles is fundamental in chemistry, especially when dealing with gases. The "mole" measures the amount of substance. It allows chemists to count particles like atoms and molecules in a given mass of material. For gases, this concept simplifies reactions and calculations under similar conditions of temperature and pressure. To determine the moles of a gas, you use the formula:

• Divide the mass of the gas by its molar mass.
• The formula is: \[ ext{Moles of gas} = \frac{\text{mass of gas}}{\text{molar mass}} \]

The step-by-step solution illustrates how to convert grams of nitrogen (\(\mathrm{N}_2\)) and oxygen (\(\mathrm{O}_2\)) into moles by using their respective molar masses. Understanding these conversions is crucial because chemical reactions deal with amounts in moles, not grams directly. This makes it easier to track reactants and products.

Molar Mass

Molar mass is the mass of one mole of a substance. It is expressed in g/mol (grams per mole) and serves as a bridge between the microscopic world of atoms and the macroscopic world we can measure. You determine the molar mass of a compound by summing the atomic masses of all the atoms in its chemical formula.For example:

• Nitrogen gas (\(\mathrm{N}_2\)) has a molecular formula of \(\mathrm{N}_2\). Each nitrogen atom has an atomic mass of approximately 14.01 u (unified atomic mass units).
• Therefore, its molar mass is \(2 \times 14.01\, \mathrm{g/mol} = 28.02\, \mathrm{g/mol}\).
• Similarly, for oxygen gas (\(\mathrm{O}_2\)), with an atomic mass of about 16.00 u, the molar mass becomes \(2 \times 16.00\, \mathrm{g/mol} = 32.00\, \mathrm{g/mol}\).

Knowing the molar mass of substances is key for stoichiometric calculations, as it allows us to change between grams and moles, ensuring accurate measurements in laboratory settings.

Stoichiometry

Stoichiometry is an exciting concept in chemistry that involves the calculation of reactants and products in chemical reactions. At its heart, stoichiometry uses the mole ratio derived from balanced chemical equations to relate quantities of different substances. Here's an overview of how it's applied:

• Start with a balanced chemical equation, which tells you the ratio of moles of reactants and products.
• Use the ratio to convert between: grams of a substance ↔ moles of that substance ↔ moles of another substance.

In the context of gases, stoichiometry can also relate to pressure, volume, and temperature, based on Ideal Gas Law. Even in the absence of a reaction equation, as seen in this problem, stoichiometry—it enables us to determine the mole fractions of elements in a gas mixture. Here, calculating the total moles of two gases, nitrogen and oxygen, helps in finding the mole fractions of each by dividing the specific gas moles by the total moles in the mixture. Mole fraction is a way of expressing the concentration of a component in a mixture, which is especially useful when dealing with gas mixtures.