Question

Assuming that 1 mole \(\left(6.02 \cdot 10^{23}\right.\) molecules) of an ideal gas has a volume of \(22.4 \mathrm{~L}\) at standard temperature and pressure (STP) and that nitrogen, which makes up \(78 \%\) of the air we breathe, is an ideal gas, how many nitrogen molecules are there in an average \(0.5-\mathrm{L}\) breath at STP?

Short answer

Answer: There are approximately \(1.05 \cdot 10^{22}\) nitrogen molecules in an average 0.5 L breath at STP.

Step-by-step solution

Calculate the number of moles of air in a 0.5 L breath at STP

We are given that 1 mole of an ideal gas has a volume of 22.4 L at STP. We have a 0.5 L breath, so we can calculate the number of moles: \(\text{Number of moles} = \frac{\text{Volume of gas at STP}}{\text{Volume occupied by 1 mole at STP}}\) \(\text{Number of moles} = \frac{0.5 \,\mathrm{L}}{22.4 \, \mathrm{L/mol}}\) \(\text{Number of moles} = 0.0223 \, \mathrm{mol}\)

Calculate the number of moles of nitrogen in the 0.5 L breath

Nitrogen makes up 78% of the air we breathe, so we need to multiply the total number of moles of air by 0.78 to find the number of moles of nitrogen: \(\text{Number of moles of nitrogen} = 0.0223 \, \mathrm{mol} \times 0.78\) \(\text{Number of moles of nitrogen} = 0.0174 \, \mathrm{mol}\)

Convert moles of nitrogen to number of molecules

To do this, we will use Avogadro's number, which states that there are \(6.02 \cdot 10^{23}\) molecules in 1 mole. \(\text{Number of molecules of nitrogen} = 0.0174 \, \mathrm{mol} \times (6.02 \cdot 10^{23}\, \mathrm{molecules/mol})\) \(\text{Number of molecules of nitrogen} = 1.05 \cdot 10^{22} \, \mathrm{molecules}\) Thus, there are approximately \(1.05 \cdot 10^{22}\) nitrogen molecules in an average 0.5 L breath at STP.

Key concepts

Avogadro's Number

Avogadro's number, a fundamental constant in chemistry, represents the number of constituent particles, usually atoms or molecules, present in one mole of a substance. Its value is defined as exactly
\(6.02214076 \times 10^{23}\) particles per mole. This number is astonishingly large and can be difficult to conceptualize. However, using Avogadro's number allows chemists to use the mole concept to bridge the gap between the microscopic scale of atoms and the human scale, making it possible to count out practical quantities of material for use in reactions and industry.

To apply this in the context of our exercise, Avogadro's number is used to convert the amount of nitrogen in moles to the actual number of nitrogen molecules in a 0.5 L breath at standard temperature and pressure.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, commonly abbreviated as STP, is a set of conditions for experimental measurements to be established for comparison. By convention, STP is 0 degrees Celsius (273.15 Kelvin) and 1 atmosphere of pressure. Under these specific conditions, one mole of an ideal gas occupies a volume of 22.4 liters, known as the molar volume.

This concept of STP is crucial for solving problems related to gases because it provides a common reference point. In our example, the STP conditions are used to determine the volume that a given number of moles of gas would occupy or in reverse, to calculate the number of moles from the volume of gas.

Molar volume of a gas

Molar volume is defined as the volume occupied by one mole of any gas at a specified temperature and pressure, and for ideal gases at STP, this value is about 22.4 liters per mole. This value is key in converting between moles and volume when dealing with gases.

In the context of the textbook exercise, to calculate the amount of nitrogen in a 0.5 L breath, understanding molar volume allows us to start with the volume of a breath and determine the number of moles of nitrogen it contains. This step is necessary before we can use Avogadro's number to find the total number of nitrogen molecules.

Mole Concept

The mole concept is a method in chemistry used to measure the amount of substance. One mole is defined as exactly \(6.02214076 \times 10^{23}\) elementary entities (this could include atoms, molecules, ions, or other particles), and this value is standard across all substances. It serves as a bridge between the atomic and macroscopic worlds, making calculations and comparisons between different substances both practical and meaningful.

The mole concept is incredibly important for quantitative chemistry work, and in our exercise, it is the key to understanding the relationship between the volume of gas inhaled in a breath and the number of molecules it contains. By calculating the number of moles of nitrogen we are able to develop an understanding of the vast number of molecules we interact with in just a single breath.