Question

A sample of air contains \(78.08 \%\) nitrogen, \(20.94 \%\) oxygen, \(0.05 \%\) carbon dioxide, and \(0.93 \%\) argon, by volume. How many molecules of each gas are present in \(1.00 \mathrm{~L}\) of the sample at \(25^{\circ} \mathrm{C}\) and 1.00 atm?

Short answer

Nitrogen: \(1.927 \times 10^{22}\) molecules, Oxygen: \(5.179 \times 10^{21}\) molecules, Carbon dioxide: \(1.204 \times 10^{19}\) molecules, Argon: \(2.288 \times 10^{20}\) molecules.

Step-by-step solution

Calculate the total moles of gas using the Ideal Gas Law

Use the Ideal Gas Law to find the total moles of gas. The equation is \[ PV = nRT \]Given: \( P = 1.00 \) atm, \( V = 1.00 \) L, \( R = 0.0821 \) L·atm/(K·mol),\( T = 25^{\circ}C = 298 K \)\[ n = \frac{PV}{RT} = \frac{1.00 \times 1.00}{0.0821 \times 298} = 0.041 mol \]

Calculate the moles of each gas

The total sample volume is split according to the percentages of each gas. Convert the percentages into decimals:- Nitrogen: \( 0.7808 \)- Oxygen: \( 0.2094 \)- Carbon dioxide: \( 0.0005 \)- Argon: \( 0.0093 \)Multiply the total moles \( n = 0.041 \) by each percentage:- Nitrogen: \( 0.041 \times 0.7808 = 0.0320 \) moles- Oxygen: \( 0.041 \times 0.2094 = 0.0086 \) moles- Carbon dioxide: \( 0.041 \times 0.0005 = 0.00002 \) moles- Argon: \( 0.041 \times 0.0093 = 0.00038 \) moles

Convert the moles of each gas to molecules

Use Avogadro's number \(6.022 \times 10^{23}\) to convert moles to molecules:- Nitrogen: \(0.0320 \times 6.022 \times 10^{23} = 1.927 \times 10^{22} \) molecules- Oxygen: \(0.0086 \times 6.022 \times 10^{23} = 5.179 \times 10^{21} \) molecules- Carbon dioxide: \(0.00002 \times 6.022 \times 10^{23} = 1.204 \times 10^{19} \) molecules- Argon: \(0.00038 \times 6.022 \times 10^{23} = 2.288 \times 10^{20} \) molecules

Key concepts

Ideal Gas Law calculation

The Ideal Gas Law is a fundamental equation used to relate the pressure, volume, temperature, and amount of gas present in a sample. The equation is:
\[ PV = nRT \]
where:

• P is the pressure in atmospheres (atm)
• V is the volume in liters (L)
• n is the number of moles of gas
• R is the ideal gas constant, 0.0821 L·atm/(K·mol)
• T is the temperature in Kelvin (K)

To find the total moles of gas in a given sample, use the equation rearranged to solve for n:
\[ n = \frac{PV}{RT} \]
For example, given a pressure of 1 atm, a volume of 1 L, and a temperature of 298 K (25°C), you can find the number of moles of gas:
\[ n = \frac{1.00 \times 1.00}{0.0821 \times 298} = 0.041 \text{ mol} \]
This calculation shows that the sample contains 0.041 moles of gas.

mole calculation

Mole calculation is crucial when dealing with gases. After determining the total moles using the Ideal Gas Law, you can calculate the moles of each gas component in a mixture by using their respective volume percentages.
For example, if you have a gas mixture with 78.08% nitrogen, 20.94% oxygen, 0.05% carbon dioxide, and 0.93% argon, you would start by converting these percentages to decimals:

• Nitrogen: 0.7808
• Oxygen: 0.2094
• Carbon dioxide: 0.0005
• Argon: 0.0093

Then, multiply the total moles of gas by each decimal to find the moles of each gas:

• Nitrogen: 0.041 × 0.7808 = 0.0320 moles
• Oxygen: 0.041 × 0.2094 = 0.0086 moles
• Carbon dioxide: 0.041 × 0.0005 = 0.00002 moles
• Argon: 0.041 × 0.0093 = 0.00038 moles

This way, you distribute the total moles among the gases present in the mixture.

Avogadro's number

Avogadro's number is a constant that is used to convert moles of a substance to the number of molecules or atoms. Avogadro's number is:

\[6.022 \times 10^{23} \text{ molecules/mol}\]
This means that 1 mole of any substance contains exactly 6.022 × 10²³ particles.
Once you have the moles of each gas, you can use Avogadro's number to find the number of molecules. For instance:

• Nitrogen: 0.0320 moles × 6.022 × 10²³ molecules/mol = 1.927 × 10²² molecules
• Oxygen: 0.0086 moles × 6.022 × 10²³ molecules/mol = 5.179 × 10²¹ molecules
• Carbon dioxide: 0.00002 moles × 6.022 × 10²³ molecules/mol = 1.204 × 10¹⁹ molecules
• Argon: 0.00038 moles × 6.022 × 10²³ molecules/mol = 2.288 × 10²⁰ molecules

This conversion helps you determine the actual number of gas molecules present in the sample.

gas percentages

Gas percentages describe the ratio of each gas present within a mixture. In the Ideal Gas Law computation, these percentages play a pivotal role in scattering the total moles calculated among the different gas constituents.

In a situation where a sample of air contains 78.08% nitrogen, 20.94% oxygen, 0.05% carbon dioxide, and 0.93% argon, by volume, these values are used in fragmenting the total moles calculated using the Ideal Gas Law.

• Convert the percentages to decimals
• Multiply each decimal by the total moles to get the moles of each gas

The split of volumes gives insight into the gas composition. For example, since nitrogen represents 78.08% of the air sample, it implies that approximately three-quarters of the gas molecules are nitrogen. Similarly, knowing that oxygen is approximately 20.94% of the sample allows for the determination that about one-fifth of the gas molecules are oxygen.