Question

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

Short answer

There are \(2.097 \times 10^{22}\) \(N_2\) molecules, \(5.645 \times 10^{21}\) \(O_2\) molecules, and \(2.686 \times 10^{20}\) Ar molecules in 1 L of air at STP.

Step-by-step solution

Understand the Concept of STP

Standard Temperature and Pressure (STP) is defined as 0°C (273.15 K) and 1 atm pressure. At STP, one mole of any ideal gas occupies a volume of 22.4 liters.

Determine the Total Moles in 1 Liter of Air

Since 1 mole of air (considered an ideal gas) occupies 22.4 L at STP, 1.0 L of air at STP contains \( \frac{1}{22.4} \) moles of gas. Calculate this fraction: \( \frac{1}{22.4} \approx 0.04464 \) moles.

Calculate Moles of Each Gas

Calculate the moles of each gas in 1.0 L of air using their volume percentages: - \( N_2: \ 0.04464 \times 0.78 = 0.0348272 \) moles- \( O_2: \ 0.04464 \times 0.21 = 0.0093744 \) moles- \( Ar: \ 0.04464 \times 0.01 = 0.0004464 \) moles.

Use Avogadro's Number to Find Number of Molecules

One mole of any substance contains \(6.022 \times 10^{23}\) molecules. Use this to find the number of molecules for each gas:- \( N_2: \ 0.0348272 \times 6.022 \times 10^{23} = 2.097 \times 10^{22} \) molecules- \( O_2: \ 0.0093744 \times 6.022 \times 10^{23} = 5.645 \times 10^{21} \) molecules- \( Ar: \ 0.0004464 \times 6.022 \times 10^{23} = 2.686 \times 10^{20} \) molecules.

Key concepts

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, simply put, is a condition defined specifically for measuring gases. STP is set at a temperature of 0°C, or 273.15 Kelvin. The pressure is 1 atm, which is the average atmospheric pressure at sea level. These conditions are crucial for scientific calculations as they provide a standard reference point.

Imagine it like setting a baseline that everyone agrees on when discussing gases - like how people speak the same language. At STP, 1 mole of any ideal gas will occupy exactly 22.4 liters of volume. This consistency makes it easier to compare and predict gas behaviors in different scenarios.

By knowing the volume one mole of gas occupies at STP, you can also calculate how many moles fit into any other volume when the gas is in standard conditions. This standardization is very handy in science and engineering because it reduces complex variables to a straightforward, manageable form.

Mole Calculation

In chemistry, moles help us count particles like atoms and molecules, which are incredibly tiny. The concept of a mole is similar to a dozen or a gross, just on a much, much larger scale. One mole corresponds to exactly 22.4 liters for gases at Standard Temperature and Pressure (STP).

To perform a mole calculation, you need to understand how many moles or fractions of a mole are present in your sample. If we're given a volume that isn't 22.4 liters, like in the case of the exercise where we have 1 liter, we calculate the number of moles by dividing the volume of the gas by 22.4.

In our exercise: 1 liter of air at STP means we have \( \frac{1}{22.4} = 0.04464 \) moles of gases in total. This number helps determine all chemical interactions and quantities involved in gaseous reactions. Essentially, it becomes the foundation for calculating the number of specific types of molecules within that volume.

Avogadro's number

Avogadro's number is central to understanding the sheer scale of atoms and molecules. This number, \( 6.022 \times 10^{23} \), represents how many atoms or molecules there are in one mole of a substance. It's an unimaginably large number, but it's necessary because atoms and molecules are unimaginably small.

In our exercise, knowing Avogadro's number helps convert moles of gas into actual molecules, which is needed to find the specific number of molecules of each gas present in a given volume. For instance, if you have 0.0348272 moles of nitrogen \( N_2 \) from our calculation, multiplying by Avogadro's number gives you the number of molecules:
\[0.0348272 \times 6.022 \times 10^{23} = 2.097 \times 10^{22}\text{ molecules}\].

This conversion allows chemists to understand reactions at the molecular level. So compellingly, Avogadro’s number bridges our understanding from the macroscopic world to the microscopic world, helping us count tiny particles in large quantities.