Question

The average concentration of carbon monoxide in air in an Ohio city in 2006 was \(3.5 \mathrm{ppm}\). Calculate the number of CO molecules in \(1.0 \mathrm{~L}\). of this air at a pressure of 759 torr and a temperature of \(22^{\circ} \mathrm{C}\)

Short answer

In 1.0 L of air with an average concentration of 3.5 ppm CO, at a pressure of 759 torr and a temperature of \(22^{\circ} \mathrm{C}\), there are approximately \(8.63 \times 10^{16}\) CO molecules.

Step-by-step solution

1 - Convert the given temperature to Kelvin

To use the Ideal Gas Law formula, we must first convert the given temperature in Celsius to Kelvin. Use the formula: \(K = ^{\circ}\mathrm{C} + 273.15\) Temperature in Kelvin: \(22^{\circ}\mathrm{C} + 273.15 = 295.15\mathrm{K}\)

2 - Convert pressure from torr to atmospheres

The given pressure is in torr; we need to convert it to atmospheres (atm). To do this, use the conversion factor: \(1 \mathrm{atm} = 760 \mathrm{torr}\) Pressure in atm: \[\frac{759 \mathrm{~torr}}{760 \mathrm{~torr/atm}} = 0.9987 \mathrm{~atm}\]

3 - Calculate moles of total air using Ideal Gas Law

Now use the Ideal Gas Law formula to find the moles of air in the given volume: \(PV = nRT\) Where: - P is the pressure in atm - V is the volume in liters - n is the number of moles - R is the ideal gas constant (0.0821 L atm/mol K) - T is the temperature in Kelvin Rearrange the formula to solve for n: \(n = \frac{PV}{RT}\) Plug in the values and find out the moles of total air: \[n = \frac{(0.9987 \mathrm{~atm})(1.0 \mathrm{~L})}{(0.0821 \mathrm{~L~atm/mol~K})(295.15 \mathrm{~K})} = 0.04098 \mathrm{~mol}\]

4 - Calculate moles of carbon monoxide in 1.0 L of air

The given concentration of carbon monoxide is 3.5 ppm (parts per million). To find the moles of carbon monoxide, multiply the moles of total air by the concentration in ppm: Moles of CO: \((0.04098 \mathrm{~mol})\times \frac{3.5 \mathrm{~ppm}}{1,000,000 \mathrm{~ppm}} = 1.434 \times 10^{-7} \mathrm{~mol}\)

5 - Calculate the number of CO molecules using Avogadro's number

Finally, to find the number of CO molecules in 1.0 L of air, multiply the moles of CO by Avogadro's number (\(6.022 \times 10^{23}\) molecules/mol): Number of CO molecules: \((1.434 \times 10^{-7} \mathrm{~mol}) \times (6.022 \times 10^{23} \mathrm{~molecules/mol}) = 8.63 \times 10^{16} \mathrm{~molecules}\) Thus, there are approximately \(8.63 \times 10^{16}\) CO molecules in 1.0 L of air with an average concentration of 3.5 ppm CO, at a pressure of 759 torr and a temperature of \(22^{\circ} \mathrm{C}\).

Key concepts

Moles Calculation

To understand how to calculate moles, one must familiarize themselves with the Ideal Gas Law, a crucial formula in chemistry. The Ideal Gas Law links the pressure, volume, temperature, and amount of a gas in a specific setup.
The formula is given as \(PV = nRT\), where:

• \(P\) represents pressure, typically in atmospheres (atm).
• \(V\) indicates volume, often in liters (L).
• \(n\) is the number of moles of the gas.
• \(R\) is the ideal gas constant, which is \(0.0821 \, \text{L atm/mol K}\).
• \(T\) stands for the temperature in Kelvin (K).

To calculate the number of moles \(n\), the equation is rearranged to \(n = \frac{PV}{RT}\). Simply plug in the values for \(P\), \(V\), and \(T\) after converting them to the correct units. For example, using the provided values, \(n = \frac{(0.9987 \, \text{atm})\times(1.0 \, \text{L})}{(0.0821 \, \text{L atm/mol K})\times(295.15 \, \text{K})} = 0.04098 \, \text{mol}\). This tells us the total moles of air in the given conditions.

Temperature Conversion

Temperature conversion is essential in using the Ideal Gas Law because it requires the temperature in Kelvin, the absolute temperature scale. Converting Celsius to Kelvin is straightforward. Simply add 273.15 to the Celsius temperature.
For example, to convert \(22^{\circ}\text{C}\) to Kelvin, use the equation:\[\text{K} = ^{\circ}\text{C} + 273.15\]So, \(22^{\circ}\text{C} + 273.15 = 295.15\text{K}\).
This conversion aligns with the needs of the Ideal Gas Law for thermodynamic calculations. Remember, Kelvin is the scale used for scientific purposes when calculating gas laws due to its starting point at absolute zero.

Avogadro's Number

Avogadro's number is a vital constant in chemistry, particularly when you need to convert moles of a substance to individual molecules. It tells us the number of particles (atoms, molecules, etc.) in one mole.
The value of Avogadro's number is \(6.022\times10^{23}\) particles/mol.
To find out how many molecules are in a given number of moles, multiply the moles by Avogadro's number.

• For example, consider \(1.434\times10^{-7} \, \text{mol}\) of carbon monoxide (CO) as calculated from the given ppm concentration.
• The number of molecules is \((1.434\times10^{-7} \, \text{mol}) \times (6.022\times10^{23} \, \text{molecules/mol}) = 8.63\times10^{16} \, \text{molecules}\).

This conversion enables us to understand the scale and number of individual molecules in any gaseous system.