Question

The average concentration of carbon monoxide in air in a city in 2007 was 3.0 ppm. Calculate the number of CO molecules in \(1.0 \mathrm{~L}\) of this air at a pressure of \(100 \mathrm{kPa}\) and a temperature of \(25^{\circ} \mathrm{C}\).

Short answer

In 1.0 L of this air at a pressure of 100 kPa and a temperature of 25°C, there are approximately \(7.27 \times 10^{19}\) CO molecules.

Step-by-step solution

Convert the given concentration to a fraction

The concentration of CO in the air is given as 3.0 ppm (parts per million). We need to convert this to a fraction to use it in our calculations. Since there are one million parts in "parts per million", we have a ratio of CO molecules to total molecules: \[ \frac{3.0}{1,000,000} \]

Use the ideal gas law equation to find the moles of air

The ideal gas law equation is: \[P V = n R T\] where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount in moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. We are given the pressure \(P = 100 \mathrm{kPa}\), the volume \(V = 1.0 \mathrm{~L}\), and the temperature \(T = 25^{\circ} \mathrm{C}\). First, we need to convert temperature to Kelvin: \[T = 25^{\circ} \mathrm{C} + 273.15 = 298.15\mathrm{K}\] And the given pressure should be converted to atm: \[P = \frac{100 \mathrm{kPa}}{101.325} = 0.9869 \mathrm{atm}\] Now, the value of the gas constant \(R\) in L atm/(mol K) is approximately: \[R = 0.0821 \frac{\mathrm{L}\cdot\mathrm{atm}}{\mathrm{mol}\cdot\mathrm{K}}\] Plugging in the known values to the ideal gas law equation, we can now solve for the moles of air (\(n\)): \[n = \frac{P V}{R T} = \frac{0.9869 \mathrm{atm} \cdot 1.0 \mathrm{~L}}{0.0821 \frac{\mathrm{L}\cdot\mathrm{atm}}{\mathrm{mol}\cdot\mathrm{K}} \cdot 298.15\mathrm{K}} = 0.0402 \mathrm{mol}\]

Calculate moles of CO in 1.0 L of air

Now that we know the total number of moles of air in the given volume, we can use the fraction of CO in the air to determine the number of moles of CO present in 1.0 L of air: \[0.0402\ \mathrm{mol} \cdot \frac{3.0}{1,000,000} = 1.21 \times 10^{-4}\ \mathrm{mol}\]

Calculate the number of CO molecules using Avogadro's number

Finally, we can multiply the number of moles of CO by Avogadro's number to obtain the number of CO molecules in 1.0 L of air: \[1.21 \times 10^{-4}\ \mathrm{mol} \cdot 6.022 \times 10^{23}\ \frac{\mathrm{molecules}}{\mathrm{mol}} = 7.27 \times 10^{19}\ \mathrm{molecules}\] In 1.0 L of this air at a pressure of 100 kPa and a temperature of 25°C, there are approximately \(7.27 \times 10^{19}\) CO molecules.

Key concepts

Ideal Gas Law

The Ideal Gas Law is an essential concept in chemistry that helps us understand the behavior of gases. It combines several individual gas laws into one equation and is represented as: \[PV = nRT\]where:

• \(P\) is the pressure of the gas, measured in units such as atmospheres (atm) or Pascals (Pa).
• \(V\) is the volume of the gas, commonly in liters (L).
• \(n\) is the number of moles, which represents the amount of substance present.
• \(R\) is the ideal gas constant, approximately \(0.0821\ \frac{L \, atm}{mol \, K}\) when volume is measured in liters and pressure in atmospheres.
• \(T\) is the temperature in Kelvin (K), determined from the Celsius scale by adding 273.15.

This formula allows us to calculate any one of the four variables (\(P\), \(V\), \(n\), \(T\)) if the others are known.
It's quite powerful when dealing with problems involving gases, as it can relate the measured physical properties to the amount of gas present.
In our problem, using this law, we calculated the number of moles in the air sample.”Concentration of Gases

Concentration of Gases

Concentration of gases is a measure of how much of a particular gas is present in a mixture. A common way to express concentration is ppm, or parts per million.
This indicates the number of parts of a gas in a million parts of the total gas mixture.
For example, a concentration of 3 ppm of carbon monoxide (\(CO\)) means there are 3 molecules of \(CO\) in every million molecules of air.Subsequently, in practical applications and calculations, it is crucial to convert concentrations from ppm to a more usable fraction. To convert 3 ppm to a fraction, you divide by one million, resulting in \(\frac{3}{1,000,000}\).
With this fraction, it then becomes easier to calculate the number of moles of the specific gas using the total moles from the Ideal Gas Law result.
In essence, understanding and calculating gas concentrations as fractions helps in determining the actual number of molecules or moles of substances like CO in a given air sample.”

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry, defined as \(6.022 \times 10^{23}\) molecules per mole.
This number establishes a bridge between the macroscopic scale (what we can measure) and the molecular scale (individual molecules or atoms). It enables converting moles, which we often use to count substances, to specific numbers of molecules or atoms.In calculations, after determining the number of moles (using the Ideal Gas Law), Avogadro's Number is utilized to find the exact count of molecules in that molar amount.
For example, if we have \(1.21 \times 10^{-4}\) moles of a substance, multiplying by Avogadro's Number gives the total number of molecules.
This conversion is critical for quantifying reactions, understanding concentrations at the molecular level, and predicting reaction outcomes in chemistry.
In the exercise, Avogadro's Number helped ascertain the precise number of carbon monoxide molecules in a given volume of air.”