Question

Assuming that air contains 78 percent \(\mathrm{N}_{2}, 21\) percent \(\mathrm{O}_{2},\) and 1 percent \(\mathrm{Ar},\) all by volume, how many \(\mathrm{mol}-\) ecules of each type of gas are present in \(1.0 \mathrm{~L}\) of air at STP?

Short answer

The number of molecules of each type of gas present in 1.0 L of air at STP are \(2.09 \times 10^{22}\) molecules of Nitrogen, \(5.64 \times 10^{21}\) molecules of Oxygen and \(2.68 \times 10^{20}\) molecules of Argon.

Step-by-step solution

Determine the Amount of Each Gas

First, find out the volume of each of the three gases in air, this is achieved by multiplying each percentage given by the total volume of the air, in this case, 1.0 L. Therefore, For Nitrogen (\(N_2\)), the volume is \(0.78 \times 1.0 L = 0.78 L\),For Oxygen (\(O_2\)), the volume is \(0.21 \times 1.0 L = 0.21 L\), andFor Argon (\(Ar\)), the volume is \(0.01 \times 1.0 L = 0.01 L\).

Calculate the Moles of Each Gas

To find out the moles of gas, the formula derived from Avogadro’s law is utilized, which is volumegas / 22.4 (the volume per mole at STP). Hence,For \(N_2\), the moles are \(0.78 L / 22.4 L/mol = 0.0348 moles\),For \(O_2\), the moles are \(0.21 L / 22.4 L/mol = 0.00938 moles\), andFor \(Ar\), the moles are \(0.01 L / 22.4 L/mol = 0.000446 moles\).

Calculate the Number of Molecules

To convert moles to molecules, you multiply by Avogadro’s number (\(6.022 \times 10^{23}\) molecules/mol). Therefore,For \(N_2\), the number of molecules will be \(0.0348 moles \times 6.022 \times 10^{23} molecules/mol = 2.09 \times 10^{22}\) molecules,For \(O_2\), the number of molecules is \(0.00938 moles \times 6.022 \times 10^{23} molecules/mol = 5.64 \times 10^{21}\) molecules, andFor \(Ar\), the number of molecules is \(0.000446 moles \times 6.022 \times 10^{23} molecules/mol = 2.68 \times 10^{20}\) molecules.

Key concepts

Avogadro's Law

Avogadro's Law is an essential principle in chemistry that helps us relate the volume of a gas to the amount of substance, or moles, it contains, under specific conditions. According to this law, equal volumes of gases, at the same temperature and pressure, contain an equal number of moles. This relationship is described by the equation \(V \propto n\), where \(V\) is the volume and \(n\) is the number of moles.

At Standard Temperature and Pressure (STP), which is typically defined as 0 degrees Celsius and 1 atmosphere of pressure, one mole of any ideal gas occupies a volume of 22.4 liters. Thus, using Avogadro's Law, if you know the volume of gas, you can calculate how many moles of gas you have by dividing the volume by 22.4 L/mol. This concept is crucial in step 2 of our original solution.

STP (Standard Temperature and Pressure)

Standard Temperature and Pressure, abbreviated as STP, is a set reference often utilized when working with gases to ensure consistency in calculations and results. It denotes a temperature of 0 degrees Celsius (273.15 Kelvin) and pressure of 1 atmosphere (atm).

STP is critical because gases behave predictably under these conditions, thus allowing chemists to standardize measurements and make reliable comparative calculations. For instance, under STP conditions, all gases occupy the same volume of 22.4 liters per mole. This principle is applied in the calculations in the original problem to find the number of moles of each type of gas in 1 liter of air. Essential understanding of STP helps you comprehend why this specific volume is used in the formula to convert gas volume to moles.

Number of Molecules

The number of molecules in a given amount of gas can be determined by using Avogadro's number, which is \(6.022 \times 10^{23}\) molecules per mole. This constant allows for the conversion of moles, which is a measure of quantity of substance, directly into the number of individual molecules.

Once you've calculated the moles using the volume and the molar volume (22.4 L/mol at STP), translating this to molecules involves multiplying by Avogadro's number. This final step in the original solution gives a more specific view of the actual number of individual gas particles present in a sample, which is a common requirement in many practical and theoretical chemistry problems.

Understanding this conversion is vital in chemistry, as it bridges the macroscopic quantities we measure and the microscopic world of molecules.